Recent zbMATH articles in MSC 35Bhttps://zbmath.org/atom/cc/35B2024-07-17T13:47:05.169476ZWerkzeugFlocking behavior of the Cucker-Smale model on infinite graphs with a central vertex grouphttps://zbmath.org/1536.053352024-07-17T13:47:05.169476Z"Wang, Xinyu"https://zbmath.org/authors/?q=ai:wang.xinyu.1"Xue, Xiaoping"https://zbmath.org/authors/?q=ai:xue.xiaopingSummary: This article investigates the flocking behavior of the Cucker-Smale (CS) model on infinite graphs, considering both standard and cut-off interactions. We introduce the concept of connected infinite graphs with a central vertex group and then derive sufficient conditions for the CS model to produce flocking behavior. For standard interaction, we find that the CS model will exhibit flocking behavior exponentially when the connected infinite graph is equipped with a central vertex group. However, for cut-off interaction, we need the time-varying graph induced by interparticle distance to have a fixed central vertex group and the coupling strength to be above a certain threshold to produce the flocking behavior. Our theoretical analysis shows that if a connected infinite graph has a central vertex group, the second eigenvalue of the corresponding Laplacian is positive, which is crucial for the proof of flocking behavior. The consistent convergence towards flocking may well reveal the advantages and necessities of having a central vertex group in an infinite-particle complex system with sufficient intelligence.Interplay between nonlinear potential theory and fully nonlinear elliptic PDEshttps://zbmath.org/1536.310342024-07-17T13:47:05.169476Z"Harvey, F. Reese"https://zbmath.org/authors/?q=ai:harvey.reese"Payne, Kevin R."https://zbmath.org/authors/?q=ai:payne.kevin-rSummary: We discuss one of the many topics that illustrate the interaction of Blaine Lawson's deep geometric and analytic insights. The first author is extremely grateful to have had the pleasure of collaborating with Blaine over many enjoyable years. The topic to be discussed concerns the fruitful interplay between \textit{nonlinear potential theory}; that is, the study of subharmonics with respect to a general constraint set in the \(2\)-jet bundle and the study of subsolutions and supersolutions of a nonlinear (degenerate) elliptic PDE. The main results include (but are not limited to) the validity of the comparison principle and the existence and uniqueness to solutions to the relevant Dirichlet problems on domains which are suitably ``pseudoconvex''. The methods employed are geometric and flexible as well as being very general on the potential theory side, which is interesting in its own right. Moreover, in many important geometric contexts no natural operator may be present. On the other hand, the potential theoretic approach can yield results on the PDE side in terms of non standard structural conditions on a given differential operator.Novel interpolation spaces and maximal-weighted Hölder regularity results for the fractional abstract Cauchy problemhttps://zbmath.org/1536.340562024-07-17T13:47:05.169476Z"Chen, Yuting"https://zbmath.org/authors/?q=ai:chen.yuting"Fan, Zhenbin"https://zbmath.org/authors/?q=ai:fan.zhenbinIn the paper under review, the authors investigate the maximal-weighted Hölder regularity for the fractional abstract Cauchy problem. The authors introduce two new classes of novel interpolation spaces constructed by resolvent families. The maximal-weighted Hölder regularity of the mild solution is established in weighted Hölder space \(C^{\beta,\gamma}_{0}(X)\), where \(0\leq \gamma \leq \beta\) and \(0<\beta<1\).
Reviewer: Marko Kostić (Novi Sad)Asymptotic analysis of fundamental solutions of hypoelliptic operatorshttps://zbmath.org/1536.350042024-07-17T13:47:05.169476Z"Chkadua, George"https://zbmath.org/authors/?q=ai:chkadua.george"Shargorodsky, Eugene"https://zbmath.org/authors/?q=ai:shargorodsky.eugeneSummary: Asymptotic behavior at infinity is investigated for fundamental solutions of a hypoelliptic partial differential operator
\[
\mathbf{P}(i\partial_x)=(P_1 (i\partial_x))^{m_1} \cdots (P_l (i\partial_x))^{m_l}
\]
with the characteristic polynomial that has real multiple zeros. Based on asymptotic expansions of fundamental solutions, asymptotic classes of functions are introduced and existence and uniqueness of solutions in those classes are established for the equation \(\mathbf{P}(i\partial_x)u=f\) in \(\mathbb{R}^n\). The obtained results imply, in particular, a new uniqueness theorem for the classical Helmholtz equation.An improved fractional Halanay inequality with distributed delayshttps://zbmath.org/1536.350182024-07-17T13:47:05.169476Z"Nguyen Thi Thu Huong"https://zbmath.org/authors/?q=ai:nguyen-thi-thu-huong."Nguyen Nhu Thang"https://zbmath.org/authors/?q=ai:nguyen-nhu-tang."Tran Dinh Ke"https://zbmath.org/authors/?q=ai:tran-dinh-ke.Summary: In this paper, we establish global fractional Halanay-type inequalities with distributed delays in differential and integral forms and consequently give a simple sufficient condition for Mittag-Leffler stability of solutions to fractional differential equations with this type of delays. Our essential tool is the submultiplicative property of the Mittag-Leffler functions which imitates the multiplicative property of the exponential functions in the classical theory of differential equations. The estimation with explicit decaying rate and amplitude constant is a faithful extension of the classical Halanay inequality.
{\copyright} 2023 John Wiley \& Sons, Ltd.Invariance analysis and closed-form solutions for the beam equation in Timoshenko modelhttps://zbmath.org/1536.350212024-07-17T13:47:05.169476Z"Al-Omari, S. M."https://zbmath.org/authors/?q=ai:al-omari.s-m"Hussain, A."https://zbmath.org/authors/?q=ai:hussain.akhtar"Usman, M."https://zbmath.org/authors/?q=ai:usman.mustofa|usman.murat|usman.muhammad|usman.mohammad|usman.mahamood|usman.muhammad-rashid|usman.muhammad.1"Zaman, F. D."https://zbmath.org/authors/?q=ai:zaman.fiazud-dinSummary: Our research focuses on a fourth-order partial differential equation (PDE) that arises from the Timoshenko model for beams. This PDE pertains to situations where the elastic moduli remain constant and an external load, represented as F, is applied. We thoroughly analyze Lie symmetries and categorize the various types of applied forces. Initially, the principal Lie algebra is two-dimensional, but in certain noteworthy cases, it extends to three dimensions or even more. For each specific case, we derive the optimal system, which serves as a foundation for symmetry reductions, transforming the original PDE into ordinary differential equations. In certain instances, we successfully identify exact solutions using this reduction process. Additionally, we delve into the conservation laws using a direct method proposed by Anco, with a particular focus on specific classes within the equation. The findings we have presented in our study are indeed original and innovative. This study serves as compelling evidence for the robustness and efficacy of the Lie symmetry method, showcasing its ability to provide valuable insights and solutions in the realm of mathematical analysis.Point- and contact-symmetry pseudogroups of dispersionless Nizhnik equationhttps://zbmath.org/1536.350222024-07-17T13:47:05.169476Z"Boyko, Vyacheslav M."https://zbmath.org/authors/?q=ai:boyko.vyacheslav-m"Popovych, Roman O."https://zbmath.org/authors/?q=ai:popovych.roman-o"Vinnichenko, Oleksandra O."https://zbmath.org/authors/?q=ai:vinnichenko.oleksandra-oSummary: Applying an original megaideal-based version of the algebraic method, we compute the point-symmetry pseudogroup of the dispersionless (potential symmetric) Nizhnik equation. This is the first example of this kind in the literature, where there is no need to use the direct method for completing the computation. The analogous studies are also carried out for the corresponding nonlinear Lax representation and the dispersionless counterpart of the symmetric Nizhnik system. We also first apply the megaideal-based version of the algebraic method to find the contact-symmetry (pseudo)group of a partial differential equation. It is shown that the contact-symmetry pseudogroup of the dispersionless Nizhnik equation coincides with the first prolongation of its point-symmetry pseudogroup. We check whether the subalgebras of the maximal Lie invariance algebra of the dispersionless Nizhnik equation that naturally arise in the course of the above computations define the diffeomorphisms stabilizing this algebra or its first prolongation. In addition, we construct all the third-order partial differential equations in three independent variables that admit the same Lie invariance algebra. We also find a set of geometric properties of the dispersionless Nizhnik equation that exhaustively defines it.An extension to direct method of Clarkson and Kruskal and Painlavé analysis for the system of variable coefficient nonlinear partial differential equationshttps://zbmath.org/1536.350232024-07-17T13:47:05.169476Z"Gupta, Rajesh Kumar"https://zbmath.org/authors/?q=ai:gupta.rajesh-kumar.1|gupta.rajesh-kumar"Sharma, Manjeet"https://zbmath.org/authors/?q=ai:sharma.manjeetSummary: In this work, direct method of Clarkson and Kruskal has been extended for the system of variable coefficient nonlinear partial differential equations. This extension can be applied to various higher order systems with variable coefficients to obtain novel exact solutions. An example of coupled KdV-Burgers system with variable coefficients has been considered to obtain the new exact solutions by utilizing proposed direct method. The coupled KdV-Burgers system with variable coefficients is especially relevant for modeling shallow water waves in channels with variable width or depth. Moreover, it plays a crucial role in studying the interactions between long-wave and short-wave phenomena in fluid flows with varying viscosity or density. The previously known exact solutions of considered system with constant coefficients have been exploited to derive new solutions of considered system. In this manuscript, the direct method is applied in a generalized manner for the first time to a system of partial differential equations with variable coefficients. The novel exact solutions are in the form of arbitrary function from which the different types of solutions of governed equation can be obtained. The obtained exact solutions have been displayed graphically by taking particular values of arbitrary constants and function. The comprehensive graphical analysis of the wave solutions has been conducted by extracting various standard wave configurations, including kink, bright-dark soliton, dark-bright soliton and periodic waves. The Painlavé analysis of governing system has been also performed by utilizing WTC-Kruskal algorithm which describes non-integrability of system.On the optimal system and series solutions of fifth-order Fujimoto-Watanabe equationshttps://zbmath.org/1536.350242024-07-17T13:47:05.169476Z"Gwaxa, B."https://zbmath.org/authors/?q=ai:gwaxa.b"Jamal, S."https://zbmath.org/authors/?q=ai:jamal.sameerah"Johnpillai, A. G."https://zbmath.org/authors/?q=ai:johnpillai.andrew-gratienSummary: This paper investigates the two fifth-order Fujimoto-Watanabe equations from the perspective of the group theoretic approach. We identify the reduced equations that lead to the solutions of these high order equations. Furthermore, the corresponding solutions are found by power series due to their nonlinear characteristics. As a result, the findings of the study demonstrate the convergence of solutions for such models and identifies the travelling wave solutions.Lie reductions and exact solutions of generalized Kawahara equationshttps://zbmath.org/1536.350252024-07-17T13:47:05.169476Z"Vaneeva, Olena"https://zbmath.org/authors/?q=ai:vaneeva.olena-o"Magda, Olena"https://zbmath.org/authors/?q=ai:magda.olena-v"Zhalij, Alexander"https://zbmath.org/authors/?q=ai:zhalij.alexanderSummary: We complete the classical Lie symmetry analysis of a class of generalized Kawahara equations with time dependent coefficients by classification of Lie reductions of equations from this class. Some exact Lie-invariant solutions are also constructed.
For the entire collection see [Zbl 1515.17004].Asymptotics for singular limits via phase functionshttps://zbmath.org/1536.350262024-07-17T13:47:05.169476Z"Nordmann, Samuel"https://zbmath.org/authors/?q=ai:nordmann.samuel"Schochet, Steve"https://zbmath.org/authors/?q=ai:schochet.steven-hSummary: The asymptotic behavior of solutions as a small parameter tends to zero is determined for a variety of singular-limit PDEs. In some cases even existence for a time independent of the small parameter was not known previously. New examples for which uniform existence does not hold are also presented. Our methods include both an adaptation of geometric optics phase analysis to singular limits and an extension of that analysis in which the characteristic variety determinant condition is supplemented with a periodicity condition.Localized nodal solutions for semiclassical Choquard equations with critical growthhttps://zbmath.org/1536.350272024-07-17T13:47:05.169476Z"Zhang, Bo"https://zbmath.org/authors/?q=ai:zhang.bo|zhang.bo.32|zhang.bo.20|zhang.bo.9|zhang.bo.11|zhang.bo.26|zhang.bo.6|zhang.bo.29|zhang.bo.5|zhang.bo.25|zhang.bo.7|zhang.bo.4|zhang.bo.3"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.31Summary: In this article, we study the existence of localized nodal solutions for semiclassical Choquard equation with critical growth
\[
-\varepsilon^2 \Delta v +V(x)v = \varepsilon^{\alpha-N}\Big(\int_{R^N} \frac{|v(y)|^{2_\alpha^*}}{|x-y|^{\alpha}}\,dy\Big) |v|^{2_\alpha^*-2}v +\theta|v|^{q-2}v,\; x \in R^N,
\]
where \(\theta>0\), \(N\geq 3\), \(0< \alpha<\min \{4,N-1\},\max\{2,2^*-1\}< q< 2^*\), \(2_\alpha^*= \frac{2N-\alpha}{N-2}\), \(V\) is a bounded function. By the perturbation method and the method of invariant sets of descending flow, we establish for small \(\varepsilon\) the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function \(V\).Quantitative convergence of the ``bulk'' free boundary in an oscillatory obstacle problemhttps://zbmath.org/1536.350282024-07-17T13:47:05.169476Z"Abedin, Farhan"https://zbmath.org/authors/?q=ai:abedin.farhan"Feldman, William M."https://zbmath.org/authors/?q=ai:feldman.william-mSummary: We consider an oscillatory obstacle problem where the coincidence set and free boundary are also highly oscillatory. We establish a rate of convergence for a regularized notion of free boundary to the free boundary of a corresponding classical obstacle problem, assuming the latter is regular. The convergence rate is linear in the minimal length scale determined by the fine properties of a corrector function.Homogenization of Helmholtz equation in a periodic layer to study Faraday cage-like shielding effectshttps://zbmath.org/1536.350292024-07-17T13:47:05.169476Z"Aiyappan, S."https://zbmath.org/authors/?q=ai:aiyappan.srinivasan"Griso, Georges"https://zbmath.org/authors/?q=ai:griso.georges"Orlik, Julia"https://zbmath.org/authors/?q=ai:orlik.juliaSummary: The work is motivated by the Faraday cage effect. We consider the Helmholtz equation over a 3D domain containing a thin heterogeneous interface of thickness \(\delta \ll 1\). The layer has a \(\delta\)-periodic structure in the in-plane directions and is cylindrical in the third direction. The periodic layer has one connected component and a collection of isolated regions. The isolated region in the thin layer represents air or liquid, and the connected component represents a solid metal grid with a \(\delta\) thickness. The main issue is created by the contrast of the coefficients in the air and in the grid and that the zero-order term has a complex-valued coefficient in the connected faze while a real-valued in the complement. An asymptotic analysis with respect to \(\delta \to 0\) is provided, and the limit Helmholtz problem is obtained with the Dirichlet condition on the interface. The periodic unfolding method is used to find the limit.Numerical upscaling via the wave equation with perfectly matched layershttps://zbmath.org/1536.350302024-07-17T13:47:05.169476Z"Arjmand, Doghonay"https://zbmath.org/authors/?q=ai:arjmand.doghonaySummary: One of the main ingredients of existing multiscale numerical methods for homogenization problems is an accurate description of the coarse scale quantities, e.g., the homogenized coefficient via local microscopic computations. Typical multiscale frameworks use local problems that suffer from the so-called resonance or cell-boundary error, dominating the all other errors in multiscale computations. Previously, the second order wave equation was used as a local problem to eliminate such an error. Although this approach eliminates the resonance error totally, the computational cost of the method is known to increase with increasing wave speed. In this paper, the possibility of integrating perfectly matched layers to the local wave equation is explored. In particular, questions in relation with accuracy and reduced computational costs are addressed. Numerical simulations are provided in a simplified one-dimensional setting to illustrate the ideas.
For the entire collection see [Zbl 1515.60023].Periodic unfolding method for domains with very small inclusionshttps://zbmath.org/1536.350312024-07-17T13:47:05.169476Z"Avila, Jake"https://zbmath.org/authors/?q=ai:avila.jake"Cabarrubias, Bituin"https://zbmath.org/authors/?q=ai:cabarrubias.bituin-cSummary: This work creates a version of the periodic unfolding method suitable for domains with very small inclusions in \(\mathbb{R}^N\) for \(N\geq 3\). In the first part, we explore the properties of the associated operators. The second part involves the application of the method in obtaining the asymptotic behavior of a stationary heat dissipation problem depending on the parameter \(\gamma < 0\). In particular, we consider the cases when \(\gamma \in (-1,0)\), \(\gamma < -1\) and \(\gamma = -1\). We also include here the corresponding corrector results for the solution of the problem, to complete the homogenization process.A spectral ansatz for the long-time homogenization of the wave equationhttps://zbmath.org/1536.350322024-07-17T13:47:05.169476Z"Duerinckx, Mitia"https://zbmath.org/authors/?q=ai:duerinckx.mitia"Gloria, Antoine"https://zbmath.org/authors/?q=ai:gloria.antoine"Ruf, Matthias"https://zbmath.org/authors/?q=ai:ruf.matthiasSummary: Consider the wave equation with heterogeneous coefficients in the homogenization regime. At large times, the wave interacts in a nontrivial way with the heterogeneities, giving rise to effective dispersive effects. The main achievement of the present work is a new ansatz for the long-time two-scale expansion inspired by spectral analysis. Based on this spectral ansatz, we extend and refine all previous results in the field, proving homogenization up to optimal timescales with optimal error estimates, and covering all the standard assumptions on heterogeneities (both periodic and stationary random settings).Optimal control problem governed by wave equation in an oscillating domain and homogenizationhttps://zbmath.org/1536.350332024-07-17T13:47:05.169476Z"Faella, Luisa"https://zbmath.org/authors/?q=ai:faella.luisa"Raj, Ritu"https://zbmath.org/authors/?q=ai:raj.ritu"Sardar, Bidhan Chandra"https://zbmath.org/authors/?q=ai:sardar.bidhan-chandraSummary: In this article, we consider the optimal control problem governed by the wave equation in a 2-dimensional domain \(\Omega_{\epsilon}\) in which the state equation and the cost functional involves highly oscillating periodic coefficients \(A^{\epsilon}\) and \(B^{\epsilon}\), respectively. This paper aims to examine the limiting behavior of optimal control and state and identify the limit optimal control problem, which involves the influences of the oscillating coefficients.Multiscale topology optimization of modulated fluid microchannels based on asymptotic homogenizationhttps://zbmath.org/1536.350342024-07-17T13:47:05.169476Z"Feppon, F."https://zbmath.org/authors/?q=ai:feppon.florianSummary: Dehomogenization techniques are becoming increasingly popular for enhancing lattice designs of compliant mechanical systems with ultra-large resolutions. Their effectiveness hinges on computing a deformed periodic grid that enable to reconstruct fine-scale designs with modulated and oriented patterns. In this paper, we propose an approach for extending dehomogenization methods to laminar fluid systems. We initiate our methodology by asymptotically deriving Darcy's law on a periodically porous medium deformed by a diffeomorphism. Unlike the mechanical context, we reveal that the homogenized permeability matrix depends not solely on local the orientation but also on the local dilation of the deformed periodic medium. This distinction presents one of the several challenges to be tackled when adapting dehomogenization-based topology optimization techniques to porous media. To accommodate existing methodologies, we formulate a simplified ``poor man's'' homogenized model, which streamlines various aspects, yet still leans on periodic cell problems to estimate the spatially varying permeability matrix. Specifically, we overlook boundary layer effects, we presume periodic grid deformations, and we neglect local dilation, solely considering the relationship with local cell orientations. Subsequently, we present a numerical approach for designing a system that redistributes an input flow across numerous regularly spaced outlets at an output interface. Leveraging the homogenized model, we deduce optimized geometric arrangements of local channel spacing parameters and orientations. We then use established methods to reconstruct grid deformations and fine-scale designs. The fidelity of these reconstructions is then validated through fine-scale simulations. Our observations indicate that while the proposed designs yield satisfactory performance when subjected to the full-scale model, discernible deviations from the homogenized model persist, appealing to future improvements.Corrector estimates and numerical simulations of a system of diffusion-reaction-dissolution-precipitation model in a porous mediumhttps://zbmath.org/1536.350352024-07-17T13:47:05.169476Z"Ghosh, N."https://zbmath.org/authors/?q=ai:ghosh.nivedita|ghosh.nirmalya|ghosh.nikhil|chandra-ghosh.narayan|ghosh.niladri|ghosh.naba-kumar|ghosh.nibedita|ghosh.nandan|ghosh.nirupam|ghosh.nilotpal|ghosh.narayan-ch"Mahato, H. S."https://zbmath.org/authors/?q=ai:mahato.hari-shankar|mahato.hari-shankar.2Summary: A system of diffusion-reaction equations coupled with a dissolution-precipitation model is discussed in this paper. We start by introducing a microscale model together with its homogenized version. In the present paper, we first derive the corrector results to justify the obtained theoretical results. Furthermore, we perform the numerical simulations to compare the outcome of the effective (homogenized) model with the original heterogeneous microscale model.Strange non-local operators homogenizing the Poisson equation with dynamical unilateral boundary conditions: asymmetric particles of critical sizehttps://zbmath.org/1536.350362024-07-17T13:47:05.169476Z"Ildefonso Díaz, Jesus"https://zbmath.org/authors/?q=ai:diaz-diaz.jesus-ildefonso"Shaposhnikova, Tatiana A."https://zbmath.org/authors/?q=ai:shaposhnikova.tatiana-ardolionovna"Podolskiy, Alexander V."https://zbmath.org/authors/?q=ai:podolskiy.alexander-vadimovich|podolskii.alexander-vSummary: We study the homogenization of a nonlinear problem given by the Poisson equation, in a domain with arbitrarily shaped perforations (or particles) and with a dynamic unilateral boundary condition (of Signorini type), with a large coefficient, on the boundary of these perforations (or particles). This problem arises in the study of chemical reactions of zero order. The consideration of a possible asymmetry in the perforations (or particles) is fundamental for considering some applications in nanotechnology, where symmetry conditions are too restrictive. It is important also to consider perforations (or particles) constituted by small different parts and then with several connected components. We are specially concerned with the so-called critical case in which the relation between the coefficient in the boundary condition, the period of the basic structure, and the size of the holes (or particles) leads to the appearance of an unexpected new term in the effective homogenized equation. Because of the dynamic nature of the boundary condition this ``strange term'' becomes now a non-local in time and non-linear operator. We prove a convergence theorem and find several properties of the ``strange operator'' showing that there is a kind of regularization through the homogenization process.Stochastic homogenization and geometric singularities: a study on cornershttps://zbmath.org/1536.350372024-07-17T13:47:05.169476Z"Josien, Marc"https://zbmath.org/authors/?q=ai:josien.marc"Raithel, Claudia"https://zbmath.org/authors/?q=ai:raithel.claudia"Schäffner, Mathias"https://zbmath.org/authors/?q=ai:schaffner.mathiasSummary: In this contribution we are interested in the quantitative homogenization properties of linear elliptic equations with homogeneous Dirichlet boundary data in polygonal domains with corners. To begin our study of this situation, we consider the setting of an angular sector in two dimensions: Unlike in the whole-space, on such a sector there exist nonsmooth harmonic functions (these depend on the angle of the sector). Here we construct extended homogenization correctors corresponding to these harmonic functions and prove growth estimates for these which are quasi-optimal, namely optimal up to a logarithmic loss. Our construction of the \textit{corner correctors} relies on a large-scale regularity theory for \(a\)-harmonic functions in the sector, which we also prove and which, as a by-product, yields a Liouville principle. We also propose a nonstandard 2-scale expansion, which is adapted to the sectoral domain and incorporates the \textit{corner correctors.} Our final result is a quasi-optimal error estimate for this adapted 2-scale expansion.A reiterated homogenization problem for the \(p\)-Laplacian equation in corrugated thin domainshttps://zbmath.org/1536.350382024-07-17T13:47:05.169476Z"Nakasato, Jean Carlos"https://zbmath.org/authors/?q=ai:nakasato.jean-carlos"Pereira, Marcone Corrêa"https://zbmath.org/authors/?q=ai:pereira.marcone-correaSummary: In this paper, we study the asymptotic behavior of the solutions of the \(p\)-Laplacian equation with mixed homogeneous Neumann-Dirichlet boundary conditions. It is posed in a two-dimensional rough thin domain with two different composites periodically distributed. Each composite has its own periodicity and roughness order. Here, we obtain distinct homogenized limit equations which will depend on the relationship among the roughness and thickness orders of each one.Homogenization of the scalar boundary value problem in a thin periodically broken cylinderhttps://zbmath.org/1536.350392024-07-17T13:47:05.169476Z"Nazarov, S. A."https://zbmath.org/authors/?q=ai:nazarov.sergei-aleksandrovich"Slutskii, A. S."https://zbmath.org/authors/?q=ai:slutskij.andrey-sSummary: Homogenization of the Neumann problem for a differential equation in a periodically broken multidimensional cylinder leads to a second-order ordinary differential equation. We study asymptotics for the coefficient of the averaged operator in the case of small transverse cross-sections. The main asymptotic term depends on the ``area'' of cross-sections of the links, their lengths, and the coefficient matrix of the original operator. We find the characteristics of kink zones which affect correction terms, while the asymptotic remainder becomes exponentially small. The justification of the asymptotics is based on Friedrichs's inequality with a coefficient independent of both small parameters: the period of fractures and the relative diameter of cross-sections.Derivation of effective models from heterogenous Cosserat media via periodic unfoldinghttps://zbmath.org/1536.350402024-07-17T13:47:05.169476Z"Nika, Grigor"https://zbmath.org/authors/?q=ai:nika.grigorSummary: We derive two different effective models from a heterogeneous Cosserat continuum taking into account the Cosserat intrinsic length of the constituents. We pass to the limit using homogenization via periodic unfolding and in doing so we provide rigorous proof to the results introduced by \textit{S. Forest} et al. [Int. J. Solids Struct. 38, No. 26--27, 4585--4608 (2001; Zbl 1033.74038)]. Depending on how different characteristic lengths of the domain scale with respect to the Cosserat intrinsic length, we obtain either an effective classical Cauchy continuum or an effective Cosserat continuum. Moreover, we provide some corrector type results for each case.Analysis of the Hopf bifurcation in a diffusive Gierer-Meinhardt modelhttps://zbmath.org/1536.350412024-07-17T13:47:05.169476Z"Asheghi, Rasoul"https://zbmath.org/authors/?q=ai:asheghi.rasoulSummary: In this work, we consider an activator-inhibitor system, known as the Gierer-Meinhardt model. Using the linear stability analysis at the unique positive equilibrium, we derive the conditions of the Hopf bifurcation. We compute the normal form of this bifurcation up to the third degree and obtain the direction of the Hopf bifurcation. Finally, we provide numerical simulations to illustrate the theoretical results of this paper. In this study, we will use the technique of normal form and center manifold theorem.On the bifurcation of solutions to Marguerre-von Kármán equationshttps://zbmath.org/1536.350422024-07-17T13:47:05.169476Z"Bendob, Tourkia"https://zbmath.org/authors/?q=ai:bendob.tourkia"Ghezal, Abderrezak"https://zbmath.org/authors/?q=ai:ghezal.abderrezakSummary: The objective of this work is to study the bifurcation of solutions of the Marguerre-von Kármán equations, which constitute a mathematical model for the buckling of Marguerre-von Kármán shallow shells. More precisely, we reduce the Marguerre-von Kármán equations to a single equation with a cubic operator; its second member depends on the function that defines the middle surface of the shallow shell and the applied forces. Next, we prove a general existence theorem for the reduced equation, by using the main theorem on pseudomonotone operators. Then we study the bifurcation of solutions in the reduced equation, with a second member, is small, at neighborhood of the simple characteristic value of the linearized problem.A qualitative analysis of positive steady-state solutions for a mussel-algae model with diffusionhttps://zbmath.org/1536.350432024-07-17T13:47:05.169476Z"Guo, Gaihui"https://zbmath.org/authors/?q=ai:guo.gaihui"Yang, Xiaoyi"https://zbmath.org/authors/?q=ai:yang.xiaoyi"Yuan, Hailong"https://zbmath.org/authors/?q=ai:yuan.hailongSummary: In this paper, a diffusive mussel-algae model subject to Neumann boundary conditions is considered. The main criteria for the stability and instability of the constant steady-state solutions are presented. Then, by the maximum principle, Hölder inequality and Poincaré inequality, a priori estimates and some characters of positive solutions are given and the nonexistence of the non-constant steady-state solutions for the corresponding elliptic system is investigated. Moreover, the steady-state bifurcations at both simple and double eigenvalues are intensively investigated. In particular, the implicit function theorem and the techniques of space decomposition are used to get the local structure of steady-state bifurcations from double eigenvalues. Next, our analysis focuses on providing specific conditions that can determine the local bifurcation direction and extend the local bifurcation to the global one. Finally, the numerical results are presented to provide support and complement the theoretical analysis findings. More specifically, under various parameters, the evolution processes in spatial patterns are illustrated.Spatial movement with distributed memory and maturation delayhttps://zbmath.org/1536.350442024-07-17T13:47:05.169476Z"Wu, Shuhao"https://zbmath.org/authors/?q=ai:wu.shuhao"Song, Yongli"https://zbmath.org/authors/?q=ai:song.yongliSummary: Distributed memory reflects the phenomenon that memory can decay over time. In this paper, we are concerned with the joint effect of distributed memory and maturation delay on spatiotemporal dynamics of the model that illustrates spatial movement of animals. Without maturation delay, distributed memory with weak temporal kernel gives rise to Turing and double Turing bifurcations and there is no Hopf bifurcation. When maturation delay exists, it has been shown that the joint effect of distributed memory and maturation delay can lead to rich dynamics due to the occurrence of double Hopf, Turing-Hopf, codimension 3 and codimension 4 bifurcations. The Wright-Hutchinson equation and its modification are employed to illustrate the theoretical results and maturation delay-induced stability switches, spatially inhomogeneous steady states/periodic solutions and quasi-periodic solution are found.Stability of periodic waves for the defocusing fractional cubic nonlinear Schrödinger equationhttps://zbmath.org/1536.350452024-07-17T13:47:05.169476Z"Borluk, Handan"https://zbmath.org/authors/?q=ai:borluk.handan"Muslu, Gulcin M."https://zbmath.org/authors/?q=ai:muslu.gulcin-m"Natali, Fábio"https://zbmath.org/authors/?q=ai:natali.fabio-m-amorinSummary: In this paper, we determine the spectral instability of periodic odd waves for the defocusing fractional cubic nonlinear Schrödinger equation. Our approach is based on periodic perturbations that have the same period as the standing wave solution, and we construct real periodic waves by minimizing a suitable constrained problem. The odd solution generates three negative simple eigenvalues for the associated linearized operator, and we obtain all this spectral information by using tools related to the oscillation theorem for fractional Hill operators. Newton's iteration method is presented to generate the odd periodic standing wave solutions and numerical results have been used to apply the spectral stability theory via Krein signature as established in [\textit{T. Kapitula} et al., Physica D 195, No. 3--4, 263--282 (2004; Zbl 1056.37080); ibid. 201, No. 1--2, 199--201 (2005; Zbl 1080.37070)].Spectral analysis and stability of the Moore-Gibson-Thompson-Fourier modelhttps://zbmath.org/1536.350462024-07-17T13:47:05.169476Z"Conti, Monica"https://zbmath.org/authors/?q=ai:conti.monica-c"Dell'Oro, Filippo"https://zbmath.org/authors/?q=ai:delloro.filippo"Liverani, Lorenzo"https://zbmath.org/authors/?q=ai:liverani.lorenzo"Pata, Vittorino"https://zbmath.org/authors/?q=ai:pata.vittorinoSummary: We consider the linear evolution system
\[\begin{cases}
u_{ttt}+\alpha u_{tt} + \beta \Delta^2 u_t + \gamma \Delta^2 u =- \eta \Delta \theta \\
\theta_t - \kappa \Delta \theta = \eta \Delta u_{tt} + \alpha \eta \Delta u_t
\end{cases}\]
describing the dynamics of a thermoviscoelastic plate of MGT type with Fourier heat conduction. The focus is the analysis of the energy transfer between the two equations, particularly when the first one stands in the supercritical regime, and exhibits an antidissipative character. The principal actor becomes then the coupling constant \(\eta\), ruling the competition between the Fourier damping and the MGT antidamping. Indeed, we will show that a sufficiently large \(\eta\) is always able to stabilize the system exponentially fast. One of the features of this model is the presence of the bilaplacian in the first equation. With respect to the analogous model with the Laplacian, this introduces some differences in the mathematical approach. From the one side, the energy estimate method does not seem to apply in a direct way, from the other side, there is a gain of regularity allowing to rely on analytic semigroup techniques.Multi-spike patterns for the Gierer-Meinhardt model with heterogeneity on \(Y\)-shaped metric graphhttps://zbmath.org/1536.350472024-07-17T13:47:05.169476Z"Ishii, Yuta"https://zbmath.org/authors/?q=ai:ishii.yutaSummary: In this paper, we consider the existence and the linear stability of spiky stationary solutions for the Gierer-Meinhardt model with heterogeneity on the \(Y\)-shaped compact metric graph. The existence is shown by the Liapunov-Schmidt reduction method, and the stability is shown by investigating the associated linearized eigenvalue problem. In particular, it is revealed that the location, amplitude, and stability of spikes are decided by the interaction of the heterogeneity functions with the geometry of the graph, represented by the associated Green's function. Moreover, by applying our abstract theorem to several concrete examples, we study the detailed effects of the geometry of the graph and the heterogeneity function on spiky solutions.On orbital stability of solitons for 2D Maxwell-Lorentz equationshttps://zbmath.org/1536.350482024-07-17T13:47:05.169476Z"Komech, Alexander"https://zbmath.org/authors/?q=ai:komech.alexander-ilich"Kopylova, Elena"https://zbmath.org/authors/?q=ai:kopylova.elena-aSummary: We prove the orbital stability of soliton solutions for 2D Maxwell-Lorentz system with extended charged particle. The solitons corresponds to the uniform motion and rotation of the particle. We reduce the corresponding Hamilton system by the canonical transformation via transition to a comoving frame. The solitons are the critical points of the reduced Hamiltonian. The key point of the proof is a lower bound for the Hamiltonian.Diffusion instability domains for systems of parabolic equationshttps://zbmath.org/1536.350492024-07-17T13:47:05.169476Z"Revina, S. V."https://zbmath.org/authors/?q=ai:revina.svetlana-vasilevna|revina.svetlana-vasilevna.1Summary: We consider a system of two reaction-diffusion equations in a bounded domain of the \(m \)-dimensional space with Neumann boundary conditions on the boundary for which the reaction terms \(f(u,v)\) and \(g(u,v)\) depend on two parameters \(a\) and \(b \). Assume that the system has a spatially homogeneous solution \((u_0,v_0) \), with \(f_u(u_0,v_0)>0\) and \(-g_v(u_0,v_0)=F(\operatorname{Det}(\operatorname{J})) \), where \(\operatorname{J}\) is the Jacobian of the corresponding linearized system in the diffusionless approximation and \(F\) is a smooth monotonically increasing function. We propose some method for the analytical description of the domain of necessary and sufficient conditions of Turing instability on the plane of system parameters for a fixed diffusion coefficient \(d \). Also, we show that the domain of necessary conditions of Turing instability on the plane \((\operatorname{Det}(\operatorname{J}),f_u)\) is bounded by the zero-trace curve, the discriminant curve, and the locus of points \(\operatorname{Det(\operatorname{J})}=0 \). Explicit expressions are found for the curves of sufficient conditions and we prove that the discriminant curve is the envelope of the family of these curves. It is shown that one of the boundaries of the Turing instability domain consists of the fragments of the curves of sufficient conditions and is expressed in terms of the function \(F\) and the eigenvalues of the Laplace operator in the domain under consideration. We find the points of intersection of the curves of sufficient conditions and show that their abscissas do not depend on the form of \(F\) and are expressed in terms of the diffusion coefficient and the eigenvalues of the Laplace operator. In the special case \(F(\operatorname{Det}(\operatorname{J}))=\operatorname{Det}(\operatorname{J}) \). For this case, the range of wave numbers at which Turing instability occurs is indicated. We obtain some partition of the semiaxis \(d>1\) into half-intervals each of which corresponds to its own minimum critical wave number. The points of intersection of the curves of sufficient conditions lie on straight lines independent of the diffusion coefficient \(d \). By way of applications of the statements proven, we consider the Schnakenberg system and the Brusselator equations.Hyers-Ulam-Rassias stability of \(n^{\mathrm{th}}\) order linear partial differential equationhttps://zbmath.org/1536.350502024-07-17T13:47:05.169476Z"Sonalkar, V. P."https://zbmath.org/authors/?q=ai:sonalkar.v-p"Mohapatra, A. N."https://zbmath.org/authors/?q=ai:mohapatra.anugraha-nidhi"Valaulikar, Y. S."https://zbmath.org/authors/?q=ai:valaulikar.yeshwant-shivraiSummary: This paper deals with the Hyers-Ulam-Rassias stability of \(n^{\mathrm{th}}\) order linear partial differential equation. Result is obtained by using Laplace transform.Global solutions near homogeneous steady states in a fully cross-diffusive predator-prey system with density-dependent motionhttps://zbmath.org/1536.350512024-07-17T13:47:05.169476Z"Xie, Zhoumeng"https://zbmath.org/authors/?q=ai:xie.zhoumeng"Li, Yuxiang"https://zbmath.org/authors/?q=ai:li.yuxiang.2|li.yuxiang.1|li.yuxiangThe authors study the pursuit-evasion system
\[
\begin{cases} u_t = \nabla \cdot (\phi_1(v) \nabla u - \chi_1(v) u \nabla v) + \mu_1 u(\lambda_1 - u + a_1 v), \\
v_t = \nabla \cdot (\phi_2(u) \nabla v + \chi_2(u) v \nabla u) + \mu_2 v(\lambda_2 - v - a_2 u) \end{cases}
\]
complemented with initial and homogeneous Neumann boundary conditions, in bounded domains \(\Omega \subset \mathbb{R}^n\), \(n \le 3\).
The main result is that certain homogeneous steady states \((u_{\star}, v_{\star})\) (namely those with are stable steady states for the corresponding ODE system) are asymptotically stable with respect to the \(W^{2, 2}\) norm. The main part of the proof consists of showing that a certain linear combination of the \(L^2\) norms of \(u - u_\star\), \(v - v_\star\) and their first and second derivatives forms an energy functional as long as \(\|u - u_\star\|_{L^\infty}\) and \(\|v - v_\star\|_{L^\infty}\) are small. As in [\textit{M. Fuest}, SIAM J. Math. Anal. 52, No. 6, 5865--5891 (2020; Zbl 1458.35222)], which treats constant \(\phi_i, \chi_i > 0\) and which the article is apparently based on, a key point is that one can choose the parameters of the functional in such a way that the most worrisome terms cancel out each other.
However, since \(\chi_1\) and \(\chi_2\) are only assumed to be nonnegative, the authors have to also deal with cases such as \(\chi_1(v_{\star}) = 0\), when such a cancellation is no longer possible. Instead, they can then make use of dissipative terms.
Reviewer: Mario Fuest (Hannover)Nonlocal competition and spatial multi-peak periodic pattern formation in diffusive Holling-Tanner predator-prey modelhttps://zbmath.org/1536.350522024-07-17T13:47:05.169476Z"Geng, Dongxu"https://zbmath.org/authors/?q=ai:geng.dongxu"Wang, Hongbin"https://zbmath.org/authors/?q=ai:wang.hongbin.1|wang.hongbin"Jiang, Weihua"https://zbmath.org/authors/?q=ai:jiang.weihua|jiang.weihua.1Summary: In this paper, we investigate the periodic pattern formations with spatial multi-peaks in a classic diffusive Holling-Tanner predator-prey model with nonlocal intraspecific prey competition. The main innovation is that a spatial dependently kernel is considered in the nonlocal effect, which mathematically complicates the linear stability analysis. We first generate the existences of Hopf, Turing, Turing-Hopf and double-Hopf bifurcations, and determine the stability of the positive equilibrium. It turns out that the stable parameter region for the positive equilibrium decreases with \(\alpha\) increasing, which implies that the parameter region of pattern formation for such kernel is smaller than the spatial average case. For double-Hopf bifurcation, we calculate the normal form up to the third-order term restricted on the center manifold, which is expressed by the original parameters of the system. Via analyzing the equivalent amplitude equations, the system exhibits stable spatially nonhomogeneous periodic patterns, the bistability of such periodic solutions, as well as unstable spatially nonhomogeneous quasi-periodic solutions, all of them possess multiple spatial peaks. Interestingly, some possible strange attractors are found numerically near the double-Hopf singularity. Biologically, the emerging spatio-temporal patterns imply that such nonlocal intraspecific competition can promote the coexistence of the prey and predator species in the form of more complex periodic states.Critical metrics for log-determinant functionals in conformal geometryhttps://zbmath.org/1536.350532024-07-17T13:47:05.169476Z"Esposito, Pierpaolo"https://zbmath.org/authors/?q=ai:esposito.pierpaolo"Malchiodi, Andrea"https://zbmath.org/authors/?q=ai:malchiodi.andreaSummary: We consider critical points of a class of functionals on compact four-dimensional manifolds arising from \textit{Regularized Determinants} for conformally covariant operators, whose explicit form was derived in [\textit{T. P. Branson} and \textit{B. Ørsted}, Proc. Am. Math. Soc. 113, No. 3, 669--682 (1991; Zbl 0762.47019)], extending Polyakov's formula. These correspond to solutions of elliptic equations of Liouville type that are quasilinear, of mixed orders and of critical type. After studying existence, asymptotic behaviour and uniqueness of \textit{fundamental solutions}, we prove a quantization property under blow-up, and then derive existence results via critical point theory.Laminated Timoshenko beam without complementary dissipationhttps://zbmath.org/1536.350542024-07-17T13:47:05.169476Z"Alves, M. S."https://zbmath.org/authors/?q=ai:alves.margareth-silva"Monteiro, R. N."https://zbmath.org/authors/?q=ai:monteiro.rodrigo-nunesSummary: In this study, the stability problem of a laminated beam with only structural damping is analyzed. The results obtained in this study improve the analysis of the problem by investigating stability without introducing additional dissipation. This is accomplished by considering only the usual assumption of equal wave velocities as the stability criterion.Analyticity and stability results for a plate-membrane type transmission problemhttps://zbmath.org/1536.350552024-07-17T13:47:05.169476Z"Barraza Martínez, Bienvenido"https://zbmath.org/authors/?q=ai:barraza-martinez.bienvenido"González Ospino, Jonathan"https://zbmath.org/authors/?q=ai:ospino.jonathan-gonzalez"Hernández Monzón, Jairo"https://zbmath.org/authors/?q=ai:hernandez-monzon.jairoSummary: In this paper, we consider a transmission problem for a system of a thermoelastic plate with (or without) a rotational inertia coupled with a membrane. On the plate a structural damping may or may not act, and on the membrane a damping of Kelvin-Voigt type may or may not be present; free boundary operators for the plate are considered in the coupling with the membrane. We prove well-posedness of the problem and higher regularity of the solution. Depending on the damping and on the presence of the rotational term, we establish strong stability, exponential stability, lack of exponential stability, polynomial stability, and analyticity of the semigroup associated to the transmission problem.
{\copyright} 2023 Wiley-VCH GmbH.Decay rates of strongly damped Infinite laminated beamshttps://zbmath.org/1536.350562024-07-17T13:47:05.169476Z"Bautista, G. J."https://zbmath.org/authors/?q=ai:bautista.george-j"Cabanillas, V. R."https://zbmath.org/authors/?q=ai:cabanillas.victor-r"Potenciano-Machado, L."https://zbmath.org/authors/?q=ai:potenciano-machado.leyter"Méndez, T. Quispe"https://zbmath.org/authors/?q=ai:mendez.teofanes-quispeSummary: In this paper, we study the stability of a Timoshenko laminated beam model with Kelvin-Voigt dampings. We consider both the case of the fully damped and partially damped system in which two dampings are effective on the system. Using the energy method, Fourier analysis and the construction of functionals with suitable weights, we obtain exponential and polynomial decay estimates for the solution of the system and its higher-order derivatives. The polynomial decay rates obtained depend on the regularity of the initial data and vary according to the position of the damping terms.Wave equations with a damping term degenerating near low and high frequency regionshttps://zbmath.org/1536.350572024-07-17T13:47:05.169476Z"Charão, Ruy Coimbra"https://zbmath.org/authors/?q=ai:charao.ruy-coimbra"Ikehata, Ryo"https://zbmath.org/authors/?q=ai:ikehata.ryoSummary: We consider wave equations with a nonlocal polynomial type of damping depending on a small parameter \(\theta \in (0,1)\). This research is a trial to consider a new type of dissipation mechanisms produced by a bounded linear operator for wave equations. These researches were initiated in a series of our previous works with various dissipations modeled by a logarithmic function published in
[\textit{R. C. Charão} et al., Math. Methods Appl. Sci. 44, No. 18, 14003--14024 (2021; Zbl 1479.35089);
\textit{R. C. Charão} and \textit{R. Ikehata}, Z. Angew. Math. Phys. 71, No. 5, Paper No. 148, 26 p. (2020; Zbl 1447.35051);
\textit{A. Piske} et al., J. Differ. Equations 311, 188--228 (2022; Zbl 1481.35067)].
The model of dissipation considered in this work is probably the first defined by more than one sentence and it opens field to consider other more general. We obtain an asymptotic profile and optimal estimates in time of solutions as \(t \rightarrow \infty\) in \(L^2\)-sense, particularly, to the case \(0<\theta <1/ 2\).Asymptotic stability of boundary layer to the multi-dimensional isentropic Euler-Poisson equations arising in plasma physicshttps://zbmath.org/1536.350582024-07-17T13:47:05.169476Z"Chen, Yufeng"https://zbmath.org/authors/?q=ai:chen.yufeng"Ding, Wenjuan"https://zbmath.org/authors/?q=ai:ding.wenjuan"Gao, Junpei"https://zbmath.org/authors/?q=ai:gao.junpei"Lin, Mengyuan"https://zbmath.org/authors/?q=ai:lin.mengyuan"Ruan, Lizhi"https://zbmath.org/authors/?q=ai:ruan.lizhiSummary: This paper is concerned with the initial-boundary value problem on the isentropic Euler-Poisson equations arising in plasma physics in the half space for the spatial dimension \(n =1, 2, 3\). By assuming that the velocity of the positive ion satisfies the Bohm criterion at the far field, we establish the global unique existence and the large time asymptotic stability of boundary layer (i.e., stationary solution) in some weighted Sobolev spaces by weighted energy method. Moreover, the time-decay rates are also obtained.A dynamic approach to heterogeneous elastic wireshttps://zbmath.org/1536.350592024-07-17T13:47:05.169476Z"Dall'Acqua, Anna"https://zbmath.org/authors/?q=ai:dallacqua.anna"Langer, Leonie"https://zbmath.org/authors/?q=ai:langer.leonie"Rupp, Fabian"https://zbmath.org/authors/?q=ai:rupp.fabianSummary: We consider closed planar curves with fixed length and arbitrary winding number whose elastic energy depends on an additional density variable and a spontaneous curvature. Working with the inclination angle, the associated \(L^2\)-gradient flow is a nonlocal quasilinear coupled parabolic system of second order. We show local well-posedness, global existence of solutions, and full convergence of the flow for initial data in a weak regularity class.Convergence to equilibrium for linear parabolic systems coupled by matrix-valued potentialshttps://zbmath.org/1536.350602024-07-17T13:47:05.169476Z"Dobrick, Alexander"https://zbmath.org/authors/?q=ai:dobrick.alexander"Glück, Jochen"https://zbmath.org/authors/?q=ai:gluck.jochenSummary: We consider systems of parabolic linear equations, subject to Neumann boundary conditions on bounded domains in \(\mathbb{R}^d\), that are coupled by a matrix-valued potential \(V\), and investigate under which conditions each solution to such a system converges to an equilibrium as \(t \to \infty\). While this is clearly a fundamental question about systems of parabolic equations, it has been studied, up to now, only under certain positivity assumptions on the potential \(V\). Without positivity, Perron-Frobenius theory cannot be applied and the problem is seemingly wide open. In this paper, we address this problem for all potentials that are \(\ell^p\)-dissipative for some \(p \in [1, \infty]\). While the case \(p = 2\) can be treated by classical Hilbert space methods, the matter becomes more delicate for \(p \neq 2\). We solve this problem by employing recent spectral theoretic results that are closely tied to the geometric structure of \(L^p\)-spaces.
{\copyright} 2023 The Authors. Mathematische Nachrichten published by Wiley-VCH GmbHLamé's system in large size domains: existence, asymptotic behaviour and explicit solutionhttps://zbmath.org/1536.350612024-07-17T13:47:05.169476Z"Guesmia, Senoussi"https://zbmath.org/authors/?q=ai:guesmia.senoussi"Harkat, Soumia"https://zbmath.org/authors/?q=ai:harkat.soumiaSummary: This work is devoted to study the parabolic Lamé system defined in noncylindrical domains. We focus on the asymptotic behaviour of the solution as \(t\) and the state variable domain become very large. Different rates of convergence are established according to the growth of the domain. The ``parabolic-elliptic regularization'' method is used to treat the existence of the solution of such problems in time growing domains with multi-initial conditions. The steady state problem is a Lamé system defined on unbounded domains. In the cases of polynomial data and cylindrical symmetries, the limit solution will be given explicitly and some examples are provided.Asymptotic stabilization for Bresse transmission systems with fractional dampinghttps://zbmath.org/1536.350622024-07-17T13:47:05.169476Z"Hao, Jianghao"https://zbmath.org/authors/?q=ai:hao.jianghao"Wang, Dingkun"https://zbmath.org/authors/?q=ai:wang.dingkunSummary: In this article, we study the asymptotic stability of Bresse transmission systems with two fractional dampings. The dissipation mechanism of control is given by the fractional damping term and acts on two equations. The relationship between the stability of the system, the fractional damping index \(\theta\in [0,1]\) and the different wave velocities is obtained. By using the semigroup method, we obtain the well-posedness of the system. We also prove that when the wave velocities are unequal or equal with \(\theta\neq 0\), the system is not exponential stable, and it is polynomial stable. In addition, the precise decay rate is obtained by the multiplier method and the frequency domain method. When the wave velocities are equal with \(\theta=0\), the system is exponential stable.Global solution of the chemotaxis attraction-repulsion Cauchy problem with the nonlinear signal production in \(\mathbb{R}^N\)https://zbmath.org/1536.350632024-07-17T13:47:05.169476Z"Ha, Tae Gab"https://zbmath.org/authors/?q=ai:ha.tae-gab"Kim, Seyun"https://zbmath.org/authors/?q=ai:kim.seyunSummary: In this paper, we consider the following attraction-repulsion chemotaxis model with a nonlinear signal term:
\[
\begin{alignedat}{2}
u_t &= \nabla \cdot(\nabla u - \xi_1 u \nabla v + \xi_2 u \nabla w), \quad &x \in \mathbb{R}^N,\, t > 0,\\
0 &= \Delta v - \lambda_1 v + f_1(u), &x \in \mathbb{R}^N,\, t > 0,\\
0 &= \Delta w - \lambda_2 w + f_2(u), &x \in \mathbb{R}^N,\, t > 0,
\end{alignedat}
\]
where \(\xi_1, \xi_2, \lambda_1, \lambda_2\) are some positive constants, and
\[
f_1 \in C^1([0, \infty)) \text{ satisfying } 0 \leqslant f_1(s) \leqslant c_1 s^l, \,\forall s \geqslant 0 \text{ and } l > 0,
\]
\[
f_2 \in C^1([0, \infty)) \text{ satisfying } 0 \leqslant f_2(s) \leqslant c_2 s^m,\, \forall s \geqslant 0 \text{ and } m > 0.
\]
We prove that the problem has a unique global solution.Modified scattering for the higher-order KdV-BBM equationshttps://zbmath.org/1536.350642024-07-17T13:47:05.169476Z"Hayashi, Nakao"https://zbmath.org/authors/?q=ai:hayashi.nakao"Naumkin, Pavel I."https://zbmath.org/authors/?q=ai:naumkin.pavel-iSummary: We study the Cauchy problem for the higher-order KdV-BBM type equation
\[
\begin{cases}
\partial_t u+i\boldsymbol{\Lambda} u=\boldsymbol{\Theta}\partial_x u^3, t>0, x\in \mathbb{R}, \\
u(0,x) =u_0 (x), x\in \mathbb{R},
\end{cases}
\]
where \(\boldsymbol{\Lambda} =\mathcal{F}^{-1} \Lambda \mathcal{F}\) and \(\Theta =\mathcal{F}^{-1}\Theta \mathcal{F}\) are the pseudodifferential operators, defined by their symbols \(\Lambda (\xi)\) and \(\Theta (\xi)\), respectively. The aim of the present paper is to develop a general approach through the Factorization Techniques of evolution operators which can be applied for finding the large time asymptotics of small solutions to a wide class of nonlinear dispersive KdV-type equations including the KdV or the improved version of the KdV with higher order dispersion terms.On a repulsion-diffusion equation with immigrationhttps://zbmath.org/1536.350652024-07-17T13:47:05.169476Z"Koepernik, Peter"https://zbmath.org/authors/?q=ai:koepernik.peterSummary: We study a repulsion-diffusion equation with immigration and linear diffusion, whose asymptotic behaviour is related to stability of long-term dynamics in spatial population models and other branching particle systems. We prove well-posedness and find sharp conditions on the repulsion under which a form of the maximum principle and a strong notion of global boundedness of solutions hold. The critical asymptotic strength of the repulsion is \(|x|^{1 - d}\), that of the Newtonian potential.Decay rates for the 4D energy-critical nonlinear heat equationhttps://zbmath.org/1536.350662024-07-17T13:47:05.169476Z"Kosloff, Leonardo"https://zbmath.org/authors/?q=ai:kosloff.leonardo"Niche, César J."https://zbmath.org/authors/?q=ai:niche.cesar-j"Planas, Gabriela"https://zbmath.org/authors/?q=ai:planas.gabrielaSummary: In this paper, we address the decay of solutions to the four-dimensional energy-critical nonlinear heat equation in the critical space \(\dot{H}^1\). Recently, it was proven that the \(\dot{H}^1\) norm of solutions goes to zero when time goes to infinity, but no decay rates were established. By means of the Fourier Splitting Method and using properties arising from the scale invariance, we obtain an algebraic upper bound for the decay rate of solutions.
{\copyright} 2024 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.Stabilization in two-species chemotaxis systems with singular sensitivity and Lotka-Volterra competitive kineticshttps://zbmath.org/1536.350672024-07-17T13:47:05.169476Z"Kurt, Halil Ibrahim"https://zbmath.org/authors/?q=ai:kurt.halil-ibrahim"Shen, Wenxian"https://zbmath.org/authors/?q=ai:shen.wenxianSummary: The current paper is concerned with the stabilization in the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics,
\[
\begin{cases}
u_{t} = \Delta u - \chi_{1} \nabla \cdot (\frac{u}{w} \nabla w) + u(a_{1} - b_{1}u - c_{1}v), \quad &x \in \Omega \\
v_{t} = \Delta v - \chi_{2} \nabla \cdot (\frac{v}{w} \nabla w) + v(a_{2} - b_{2}v - c_{2}u), \quad &x \in \Omega \\
0 = \Delta w - \mu w + \nu u + \lambda v, \quad &x \in \Omega \\
\frac{\partial u}{\partial n} = \frac{\partial v}{\partial n} = \frac{\partial w}{\partial n} = 0, \quad &x \in \partial \Omega,
\end{cases}
\tag{1}
\]
where \(\Omega \subset \mathbb{R}^N\) is a bounded smooth domain, and \(\chi_{i}, a_{i}, b_{i}, c_{i}\) (\(i = 1, 2\)) and \(\mu\), \(\nu\), \(\lambda\) are positive constants. In [J. Differ. Equations 355, 248--295 (2023; Zbl 1517.35045)], among others, we proved that for any given nonnegative initial data \(u_{0}, v_{0} \in C^{0} (\bar\Omega)\) with \(u_{0} + v_{0} \not \equiv 0\), (1) has a unique globally defined classical solution (\(u(t, x; u_{0}, v_{0})\), \(v(t, x; u_{0}, v_{0})\), \(w(t, x; u_{0}, v_{0})\)) with \(u(0, x; u_{0}, v_{0}) = u_{0}(x)\) and \(v(0, x; u_{0}, v_{0}) = v_{0}(x)\) in any space dimensional setting with any positive constants \(\chi_{i}, a_{i}, b_{i}, c_{i}\) (\(i = 1, 2\)) and \(\mu\), \(\nu\), \(\lambda\). In this paper, we assume that the competition in (1) is weak in the sense that
\[
\frac{c_1}{b_2} < \frac{a_1}{a_2}, \quad \frac{c_2}{b_1} < \frac{a_2}{a_1}.
\]
Then (1) has a unique positive constant solution (\(u^{*}, v^{*}, w^{*}\)), where
\[
u^{*} = \frac{a_1b_2 - c_1a_2}{b_1b_2 - c_1c_2}, \quad v^{*} = \frac{b_1a_2 - a_1c_2}{b_1b_2 - c_1c_2}, \quad w^{*} = \frac{\nu}{\mu}u^{*} + \frac{\lambda}{\mu} v^{*}.
\]
We obtain some explicit conditions on \(\chi_{1}, \chi_{2}\) which ensure that the positive constant solution (\(u^{*}, v^{*}, w^{*}\)) is globally stable, that is, for any given nonnegative initial data \(u_{0}, v_{0} \in C^{0} (\bar\Omega)\) with \(u_{0} \not \equiv 0\) and \(v_{0} \not \equiv 0\),
\[
\lim_{t \to\infty} \Big( \| u(t, \cdot; u_{0}, v_{0}) - u^{*} \|_{\infty} + \| v(t, \cdot; u_{0}, v_{0}) - v^{*} \|_{\infty} + \| w(t, \cdot; u_{0}, v_{0}) - w^{*} \| \Big) = 0.
\]Existence and general decay of solution for nonlinear viscoelastic two-dimensional beam with a nonlinear delayhttps://zbmath.org/1536.350682024-07-17T13:47:05.169476Z"Lekdim, Billal"https://zbmath.org/authors/?q=ai:lekdim.billal"Khemmoudj, Ammar"https://zbmath.org/authors/?q=ai:khemmoudj.ammarSummary: We investigate the longitudinal and transversal vibrations of the viscoelastic beam with nonlinear tension and nonlinear delay term under the general decay rate for relaxation function. The existence theorem is proved by the Faedo-Galerkin method and using suitable Lyapunov functional to establish the general decay result.Asymptotic monotonicity of positive solutions for fractional parabolic equation on the right half spacehttps://zbmath.org/1536.350692024-07-17T13:47:05.169476Z"Li, Dongyan"https://zbmath.org/authors/?q=ai:li.dongyan"Dong, Yan"https://zbmath.org/authors/?q=ai:dong.yanSummary: In this paper, we mainly study the asymptotic monotonicity of positive solutions for fractional parabolic equation on the right half space. First, a narrow region principle for antisymmetric functions in unbounded domains is obtained, in which we remarkably weaken the decay condition \(u\rightarrow 0\) at infinity and only assume its growth rate does not exceed \(|x|^{\gamma}\) (\(0< \gamma <2s\)) compared with [\textit{W. Chen} et al., Adv. Math. 377, Article ID 107463, 48 p. (2021; Zbl 1455.35281)]. Then we obtain asymptotic monotonicity of positive solutions of fractional parabolic equation on \(\mathbb{R}^N_+ \times (0,\infty)\).Free boundary problems of a mutualist model with nonlocal diffusionhttps://zbmath.org/1536.350702024-07-17T13:47:05.169476Z"Li, Lei"https://zbmath.org/authors/?q=ai:li.lei.9"Wang, Mingxin"https://zbmath.org/authors/?q=ai:wang.mingxinSummary: A mutualist model with nonlocal diffusions and a free boundary is first considered. We prove that this problem has a unique solution defined for \(t\geq 0\), and its dynamics are governed by a spreading-vanishing dichotomy. Some criteria for spreading and vanishing are also given. Of particular importance is that we find that the solution of this problem has quite rich longtime behaviors, which vary with the conditions satisfied by kernel functions and are much different from those of the counterpart with local diffusion and free boundary. At last, we extend these results to the model with nonlocal diffusions and double free boundaries.Stabilization of the viscoelastic wave equation with variable coefficients and a delay term in nonlocal boundary feedbackhttps://zbmath.org/1536.350712024-07-17T13:47:05.169476Z"Li, Sheng-Jie"https://zbmath.org/authors/?q=ai:li.shengjie"Chai, Shugen"https://zbmath.org/authors/?q=ai:chai.shugenUsing the Faedo-Galerkin approximation, denseness argument, an energy functional and the Riemannian geometry method, the authors present sufficient conditions for existence and uniqueness of strong and weak solutions as well as exponential decay of energy to a class of viscoelastic wave equation with variable coefficients and a delay in nonlinear and nonlocal boundary dissipation.
Reviewer: Jin Liang (Shanghai)Energy decay for wave equations with a potential and a localized dampinghttps://zbmath.org/1536.350722024-07-17T13:47:05.169476Z"Li, Xiaoyan"https://zbmath.org/authors/?q=ai:li.xiaoyan"Ikehata, Ryo"https://zbmath.org/authors/?q=ai:ikehata.ryoSummary: We consider the total energy decay together with the \(L^2\)-bound of the solution itself of the Cauchy problem for wave equations with a short-range potential and a localized damping, where we treat it in the one-dimensional Euclidean space \(\mathbb{R}\). To study these, we adopt a simple multiplier method. In this case, it is essential that compactness of the support of the initial data not be assumed. Since this problem is treated in the whole space, the Poincaré and Hardy inequalities are not available as have been developed for the exterior domain case with \(n \geq 1\). However, the potential is effective for compensating for this lack of useful tools. As an application, the global existence of a small data solution for a semilinear problem is demonstrated.Global well-posedness and stability results for an abstract viscoelastic equation with a non-constant delay term and nonlinear weighthttps://zbmath.org/1536.350732024-07-17T13:47:05.169476Z"Makheloufi, Hocine"https://zbmath.org/authors/?q=ai:makheloufi.hocine"Bahlil, Mounir"https://zbmath.org/authors/?q=ai:bahlil.mounirSummary: In this research work, we consider the second-order viscoelastic equation with a weak internal damping, a time-varying delay term and nonlinear weights
\[
u_{tt}(t)+\mathcal{A}u(t)-\int_0^t g (t-s)\mathcal{A} u(s) ds+\mu_1 (t) u_t (t)+ \mu_2 (t) u_t (t-\tau (t)) =0\; \forall t>0,
\]
together with suitable initial conditions. We first prove the existence of a unique global weak solution by means of the classical Faedo-Galerkin method. Then, by assuming the general condition:
\[
g'(t) \le - \xi (t) H(g(t)), \quad\forall t\geq 0,
\]
where \(H\) is a positive increasing and convex function and \(\xi\) is a positive function which is not necessarily monotone, we establish optimal explicit and general stability estimates which rely on the well-known multipliers method and some properties of convex functions. This study generalizes and improves many earlier ones in the existing literature.Exponential stability of a laminated beam system with thermoelasticity of type III and distributed delayhttps://zbmath.org/1536.350742024-07-17T13:47:05.169476Z"Mpungu, Kassimu"https://zbmath.org/authors/?q=ai:mpungu.kassimuSummary: In this work, we consider a one-dimensional laminated Timoshenko beam system with thermoelasticity of type III and distributed delay. Our concern is to investigate the exponential stability of the vibrations in the system without structural damping. By exploiting the perturbed energy method with appropriate assumptions on the delay feedback and speeds of wave propagation, we establish that the unique dissipation through the thermal effect is sufficient for exponential decay of the solution even in the presence of distributed delay.A nonautonomous model for the interaction between a size-structured consumer and an unstructured resourcehttps://zbmath.org/1536.350752024-07-17T13:47:05.169476Z"Ni, Zhuxin"https://zbmath.org/authors/?q=ai:ni.zhuxin"Huang, Qihua"https://zbmath.org/authors/?q=ai:huang.qihuaSummary: In this paper, we propose and analyze a nonautonomous model that describes the dynamics of a size-structured consumer interacting with an unstructured resource. We prove the existence and uniqueness of the solution of the model using the monotone method based on a comparison principle. We derive conditions on the model parameters that result in persistence and extinction of the population via the upper-lower solution technique. We verify and complement the theoretical results through numerical simulations.Global boundedness and asymptotic behavior in a chemotaxis system with signal-dependent motility and indirect signal absorptionhttps://zbmath.org/1536.350762024-07-17T13:47:05.169476Z"Ren, Guoqiang"https://zbmath.org/authors/?q=ai:ren.guoqiang"Shi, Yu"https://zbmath.org/authors/?q=ai:shi.yuSummary: In this paper we investigate the following chemotaxis system with signal-dependent motility and indirect signal absorption
\[
\begin{cases}
u_t=\Delta (\gamma (v)u)+\mu u(1-u),\ & x\in \Omega,t>0, \\
v_t=\Delta v-vw, & x\in \Omega, t>0, \\
w_t=-\delta w+u, & x\in \Omega, t>0
\end{cases}\tag{*}
\]
in a bounded domain with smooth boundary. We present the global existence of classical solutions to the model (*) in two space dimensions with logistic source and in any dimension for small initial data without logistic source. In addition, the asymptotic behavior of the solutions is studied.
{\copyright} 2022 Wiley-VCH GmbH.Viral infection dynamics with immune chemokines and CTL mobility modulated by the infected cell densityhttps://zbmath.org/1536.350772024-07-17T13:47:05.169476Z"Shu, Hongying"https://zbmath.org/authors/?q=ai:shu.hongying"Jin, Hai-Yang"https://zbmath.org/authors/?q=ai:jin.haiyang"Wang, Xiang-Sheng"https://zbmath.org/authors/?q=ai:wang.xiangsheng"Wu, Jianhong"https://zbmath.org/authors/?q=ai:wu.jianhongSummary: We study a viral infection model incorporating both cell-to-cell infection and immune chemokines. Based on experimental results in the literature, we make a standing assumption that the cytotoxic T lymphocytes (CTL) will move toward the location with more infected cells, while the diffusion rate of CTL is a decreasing function of the density of infected cells. We first establish the global existence and ultimate boundedness of the solution via a priori energy estimates. We then define the basic reproduction number of viral infection \(R_0\) and prove (by the uniform persistence theory, Lyapunov function technique and LaSalle invariance principle) that the infection-free steady state \(E_0\) is globally asymptotically stable if \(R_0 < 1\). When \(R_0 > 1\), then \(E_0\) becomes unstable, and another basic reproduction number of CTL response \(R_1\) becomes the dynamic threshold in the sense that if \(R_1 < 1\), then the CTL-inactivated steady state \(E_1\) is globally asymptotically stable; and if \(R_1 > 1\), then the immune response is uniform persistent and, under an additional technical condition the CTL-activated steady state \(E_2\) is globally asymptotically stable. To establish the global stability results, we need to prove point dissipativity, obtain uniform persistence, construct suitable Lyapunov functions, and apply the LaSalle invariance principle.Uniform decay rate estimates for the 2D wave equation posed in an inhomogeneous medium with exponential growth source term, locally distributed nonlinear dissipation, and dynamic Cauchy-Ventcel-type boundary conditionshttps://zbmath.org/1536.350782024-07-17T13:47:05.169476Z"Simion Antunes, José G."https://zbmath.org/authors/?q=ai:simion-antunes.jose-g"Cavalcanti, Marcelo M."https://zbmath.org/authors/?q=ai:cavalcanti.marcelo-moreira"Cavalcanti, Valéria N. Domingos"https://zbmath.org/authors/?q=ai:domingos-cavalcanti.valeria-nevesSummary: We study the wellposedness and stabilization for a Cauchy-Ventcel problem in an inhomogeneous medium \(\Omega \subset \mathbb{R}^2\) with dynamic boundary conditions subject to a exponential growth source term and a nonlinear damping distributed around a neighborhood \(\omega\) of the boundary according to the geometric control condition. We, in particular, generalize substantially the work due to \textit{A. F. Almeida} et al. [Commun. Contemp. Math. 23, No. 3, Article ID 1950072, 38 p. (2021; Zbl 1458.35268)], in what concerns an exponential growth for source term instead of a polynomial one. We give a proof based on the truncation of a equivalent problem and passage to the limit in order to obtain in one shot, the energy identity as well as the observability inequality, which are the essential ingredients to obtain uniform decay rates of the energy. We show that the energy of the equivalent problem goes uniformly to zero, for all initial data of finite energy taken in bounded sets of finite energy phase space. One advantage of our proof is that the decay rate is independent of the nonlinearity.
{\copyright} 2023 Wiley-VCH GmbH.Sharp polynomial decay for waves damped from the boundary in cylindrical waveguideshttps://zbmath.org/1536.350792024-07-17T13:47:05.169476Z"Wang, Ruoyu P. T."https://zbmath.org/authors/?q=ai:wang.ruoyu-p-tSummary: We study the decay of global energy for the wave equation with Hölder continuous damping placed on the \(C^{1,1}\)-boundary of compact and non-compact waveguides with star-shaped cross-sections. We show there is sharp \(t^{-1/2}\)-decay when the damping is uniformly bounded from below on the cylindrical wall of product cylinders where the Geometric Control Condition is violated. On non-product cylinders, we also show that there is \(t^{-1/3}\)-decay when the damping is uniformly bounded from below on the cylindrical wall.Approaching critical decay in a strongly degenerate parabolic equationhttps://zbmath.org/1536.350802024-07-17T13:47:05.169476Z"Winkler, Michael"https://zbmath.org/authors/?q=ai:winkler.michaelSummary: The Cauchy problem in \(\mathbb{R}^n\), \(n\geq 1\), for the parabolic equation
\[
u_t=u^p \Delta u \qquad \qquad (\star)
\]
is considered in the strongly degenerate regime \(p\geq 1\). The focus is firstly on the case of positive continuous and bounded initial data, in which it is known that a minimal positive classical solution exists, and that this solution satisfies
\[
t^{\frac{1}{p}}\Vert u(\cdot,t)\Vert_{L^\infty (\mathbb{R}^n)} \rightarrow \infty \quad \text{as } t\rightarrow \infty. \tag{0.1}
\]
The first result of this study complements this by asserting that given any positive \(f\in C^0([0,\infty))\) fulfilling \(f(t)\rightarrow +\infty\) as \(t\rightarrow \infty\) one can find a positive nondecreasing function \(\phi \in C^0([0,\infty))\) such that whenever \(u_0\in C^0(\mathbb{R}^n)\) is radially symmetric with \(0< u_0 < \phi (|\cdot |)\), the corresponding minimal solution \(u\) satisfies
\[
\frac{t^{\frac{1}{p}}\Vert u(\cdot,t)\Vert_{L^\infty (\mathbb{R}^n)}}{f(t)} \rightarrow 0 \quad \text{as } t\rightarrow \infty.
\]
Secondly, \((\star)\) is considered along with initial conditions involving nonnegative but not necessarily strictly positive bounded and continuous initial data \(u_0\). It is shown that if the connected components of \(\{u_0>0\}\) comply with a condition reflecting some uniform boundedness property, then a corresponding uniquely determined continuous weak solution to \((\star)\) satisfies
\[
0< \liminf_{t\rightarrow \infty} \Big \{t^{\frac{1}{p}} \Vert u(\cdot,t)\Vert_{L^\infty (\mathbb{R}^n)} \Big\} \leq \limsup_{t\rightarrow \infty} \Big \{t^{\frac{1}{p}} \Vert u(\cdot,t)\Vert_{L^\infty (\mathbb{R}^n)} \Big\} <\infty.
\]
Under a somewhat complementary hypothesis, particularly fulfilled if \(\{u_0>0\}\) contains components with arbitrarily small principal eigenvalues of the associated Dirichlet Laplacian, it is finally seen that (0.1) continues to hold also for such not everywhere positive weak solutions.Well-posedness and asymptotic behavior for a \(p\)-biharmonic pseudo-parabolic equation with logarithmic nonlinearity of the gradient typehttps://zbmath.org/1536.350812024-07-17T13:47:05.169476Z"Zhang, Mengyuan"https://zbmath.org/authors/?q=ai:zhang.mengyuan"Liu, Zhiqing"https://zbmath.org/authors/?q=ai:liu.zhiqing"Zhang, Xinli"https://zbmath.org/authors/?q=ai:zhang.xinliSummary: This paper is concerned with the well-posedness and asymptotic behavior for an initial boundary value problem of a pseudo-parabolic equation with \(p\)-biharmonic operator and logarithmic nonlinearity of the gradient type. The existence of the global weak solution is established by combining the technique of potential-well and the method of Faedo-Galerkin approximation. Meantime, by virtue of the improved logarithmic Sobolev inequality and modified differential inequality, we obtain the results on infinite and finite time blow-up and derive the lifespan of blow-up solutions in various energy levels. Furthermore, the extinction phenomenon with extinction time is presented.
{\copyright} 2023 Wiley-VCH GmbH.Global boundedness and asymptotic stabilization in a chemotaxis system with density-suppressed motility and nonlinear signal productionhttps://zbmath.org/1536.350822024-07-17T13:47:05.169476Z"Zhao, Quanyong"https://zbmath.org/authors/?q=ai:zhao.quanyong"Li, Zhongping"https://zbmath.org/authors/?q=ai:li.zhongpingSummary: In this paper, we study the following chemotaxis model with density-suppressed motility and nonlinear production
\[
\begin{cases}
u_t = \Delta ( \varphi ( v ) u ) + r u - \mu u^\alpha, \quad & x \in \Omega,\, t > 0, \\
v_t = \Delta v - v + w^\beta, & x \in \Omega,\, t > 0, \\
w_t = \Delta w - w + u^\gamma, & x \in \Omega,\, t > 0
\end{cases}
\]
under homogeneous Neumann boundary conditions in a bounded domain \(\Omega \subset \mathbb{R}^n\) (\(n \geq 2\)) with smooth boundary, where \(r \in \mathbb{R}\), \(\mu, \beta, \gamma > 0\) and \(\alpha > 1\). The positive motility function satisfies \(\varphi(s) \in C^3([0, \infty))\) and \(\varphi^\prime(s) \leq 0\) for all \(s \geq 0\). It is showed that the system admits a globally bounded and classical solution under some conditions on \(\alpha, \beta\) and \(\gamma \). Then, under stricter constraints on \(\phi \), we obtained that the parameter \(\beta\) has a wider range than before, which is enough to ensure the global boundedness of the solution. Furthermore, if \(\mu\) is sufficiently large, we proved that the solution converges to \( \left( \left( \frac{r_+}{\mu} \right)^{\frac{1}{\alpha - 1}}, \left( \frac{r_+}{\mu} \right)^{\frac{\beta \gamma}{\alpha - 1}}, \left( \frac{r_+}{\mu} \right)^{\frac{\gamma}{\alpha - 1}} \right)\) in \(L^\infty(\Omega)\) as \(t \to \infty\) for all \(r \in \mathbb{R}\), \(\mu, \beta, \gamma > 0\) and \(\alpha > 1\), where \(r_+ : = \max \{r, 0 \} \). Finally, we showed that the convergence is exponential in cases \(r > 0\) and \(r < 0\).Energy decay analysis for porous elastic system with thermoelasticity of type III: a second spectrum approachhttps://zbmath.org/1536.350832024-07-17T13:47:05.169476Z"Zougheib, Hamza"https://zbmath.org/authors/?q=ai:zougheib.hamza"El Arwadi, Toufic"https://zbmath.org/authors/?q=ai:el-arwadi.touficSummary: Numerous studies have been conducted to investigate porous systems under different damping effects. Recent research has consistently achieved the expected exponential decay of energy solutions when employing stabilization techniques that involve non-physical assumptions of equal wave velocities. In this study, we examine a one-dimensional thermoelastic porous system within the framework of the second frequency spectrum. Remarkably, we demonstrate that exponential decay can be achieved without relying on the assumption of equal wave speeds. We consider the porous system, and we incorporated thermoelastic damping based on the Green-Naghdi law of heat conduction into our study. To begin with, we use the Faedo-Galerkin approximation method to validate the global well-posedness of the system. By utilizing a Lyapunov functional, we establish exponential stability without relying on the assumption of equal wave speed. We then introduce and analyze a numerical scheme. Finally, by assuming additional regularity of the solution, we derive a priori error estimates.Hausdorff and fractal dimensions of attractors for functional differential equations in Banach spaceshttps://zbmath.org/1536.350842024-07-17T13:47:05.169476Z"Hu, Wenjie"https://zbmath.org/authors/?q=ai:hu.wenjie"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomasThe main aim of the present paper is to obtain estimations of Hausdorff dimension and fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations in Banach spaces. The study of attractors and their Hausdorff dimension and fractal dimension estimation is very important, if the attractors have finite Hausdorff dimension or fractal dimension, then the attractors can be described by a finite number of parameters and hence the dynamics of the infinite dynamical systems are likely to be studied by the concepts and methods of finite dimensional dynamical systems.
The authors adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs in order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces. Notice that the current dimensions estimation methods are mainly obtained in Hilbert spaces, but there are many evolution equations arising from real world modelings defined in Banach spaces. The authors generalize the method established in Hilbert spaces to general Banach spaces, and first establish some new criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces by combining the squeezing property and the covering of finite subspace of Banach spaces.
Reviewer: Pengyu Chen (Lanzhou)On global attractor for parabolic partial differential inclusion and its time semidiscretizationhttps://zbmath.org/1536.350852024-07-17T13:47:05.169476Z"Kalita, Piotr"https://zbmath.org/authors/?q=ai:kalita.piotrSummary: In this article we study the operator version of a first order in time partial differential inclusion as well as its time discretization obtained by an implicit Euler scheme. This technique, known as the Rothe method, yields the semidiscrete trajectories that are proved to converge to the solution of the original problem. While both the time continuous problem and its semidiscretization can have nonunique solutions we prove that, as times goes to infinity, all trajectories are attracted towards certain compact and invariant sets, so-called global attractors. We prove that the semidiscrete attractors converge upper-semicontinuously to the global attractor of time continuous problem.
For the entire collection see [Zbl 1346.90007].Limiting dynamics for stochastic delay \(p\)-Laplacian equation on unbounded thin domainshttps://zbmath.org/1536.350862024-07-17T13:47:05.169476Z"Li, Fuzhi"https://zbmath.org/authors/?q=ai:li.fuzhi"Li, Dingshi"https://zbmath.org/authors/?q=ai:li.dingshi"Freitas, Mirelson M."https://zbmath.org/authors/?q=ai:freitas.mirelson-mSummary: We study the long-term behavior of solutions for stochastic delay \(p\)-Laplacian equation with multiplicative noise on unbounded thin domains. We first prove the existence and uniqueness of tempered random attractors for these equations defined on \((n+1)\)-dimensional unbounded thin domains. Then, the upper semicontinuity of these attractors when a family of \((n+1)\)-dimensional thin domains degenerates onto an \(n\)-dimensional domain as the thinness measure approaches zero is established.Pullback attractors with finite fractal dimension for a semilinear transfer equation with delay in some non-cylindrical domainhttps://zbmath.org/1536.350872024-07-17T13:47:05.169476Z"López-Lázaro, Heraclio"https://zbmath.org/authors/?q=ai:lopez-lazaro.heraclio-ledgar"Nascimento, Marcelo J. D."https://zbmath.org/authors/?q=ai:nascimento.marcelo-jose-dias"Takaessu Junior, Carlos R."https://zbmath.org/authors/?q=ai:takaessu.carlos-r-jun"Azevedo, Vinicius T."https://zbmath.org/authors/?q=ai:azevedo.vinicius-tSummary: This paper aims to study a semilinear transfer equation with a delay term defined over a non-cylindrical domain. We prove the existence and regularity of weak solutions as well as the existence, regularity and finiteness of the fractal dimension of pullback attractors on tempered universes that depend on a non-increasing function. We address the problem of estimating the fractal dimension of pullback attractors, under current techniques, which consists of readjusting the proof presented in [\textit{J. Málek} and \textit{D. Pražák}, J. Differ. Equations 181, No. 2, 243--279 (2002; Zbl 1187.37113), Lemma 1.3] and extending it to families of normed spaces parameterized in time, see Theorem 4.11.Global attractors for the three-dimensional tropical climate model with damping termshttps://zbmath.org/1536.350882024-07-17T13:47:05.169476Z"Mao, Rongyan"https://zbmath.org/authors/?q=ai:mao.rongyan"Liu, Hui"https://zbmath.org/authors/?q=ai:liu.hui.5"Miao, Fahe"https://zbmath.org/authors/?q=ai:miao.fahe"Xin, Jie"https://zbmath.org/authors/?q=ai:xin.jieSummary: In this paper, we consider the 3D tropical climate model with damping terms in the equation of \(u, v\) and \(\theta\), respectively. Firstly, we get some uniform estimates of strong solution. Secondly, we derive the result of the continuity of the semigroup \(\{S(t)\}_{t\geq 0}\) in case of \(4\leq \alpha, \beta <5\) and \(\frac{13}{5}<\gamma <5\) via some usual inequalities. Finally, the system (1.1) is shown to possess an \((\mathbb{V},\mathbb{V})\)-global attractor and an \((\mathbb{V},\mathbf{H}^2)\)-global attractor.On the logarithmic Cahn-Hilliard equation with general proliferation termhttps://zbmath.org/1536.350892024-07-17T13:47:05.169476Z"Mheich, Rim"https://zbmath.org/authors/?q=ai:mheich.rim"Petcu, Madalina"https://zbmath.org/authors/?q=ai:petcu.madalina"Talhouk, Raafat"https://zbmath.org/authors/?q=ai:talhouk.raafatSummary: Our aim in this article is to study the well-posedness of the generalized logarithmic nonlinear Cahn-Hilliard equation with regularization and proliferation terms. We are interested in studying the asymptotic behavior, in terms of finite-dimensional attractors, of the dynamical system associated with the problem and majorate the rate of convergence between the solutions of the Cahn-Hilliard equation and the regularized one. Additionally, we present some further regularity results and subsequently prove a strict separation property of the solution. Finally, we provide some numerical simulations to compare the solution with and without the regularization term, and more.Rate of convergence for reaction-diffusion equations with nonlinear Neumann boundary conditions and \(\mathcal{C}^1\) variation of the domainhttps://zbmath.org/1536.350902024-07-17T13:47:05.169476Z"Pereira, Marcone C."https://zbmath.org/authors/?q=ai:pereira.marcone-correa"Pires, Leonardo"https://zbmath.org/authors/?q=ai:pires.leonardoSummary: In this paper, we propose the compact convergence approach to deal with the continuity of attractors of some reaction-diffusion equations under smooth perturbations of the domain subject to nonlinear Neumann boundary conditions. We define a family of invertible linear operators to compare the dynamics of perturbed and unperturbed problems in the same phase space. All continuity arising from small smooth perturbations will be estimated by a rate of convergence given by the domain variation in a \(\mathcal{C}^1\) topology.Meanders, zero numbers and the cell structure of Sturm global attractorshttps://zbmath.org/1536.350912024-07-17T13:47:05.169476Z"Rocha, Carlos"https://zbmath.org/authors/?q=ai:rocha.carlos"Fiedler, Bernold"https://zbmath.org/authors/?q=ai:fiedler.bernoldSummary: We study global attractors \(\mathcal{A}=\mathcal{A}_f\) of semiflows generated by semilinear partial parabolic differential equations of the form \(u_t = u_{xx} + f(x,u,u_x)\), \(0<x<1\), satisfying Neumann boundary conditions. The equilibria \(v\in \mathcal{E}\subset \mathcal{A}\) of the semiflow are the stationary solutions of the PDE, hence they are solutions of the corresponding second order ODE boundary value problem. Assuming hyperbolicity of all equilibria, the dynamic decomposition of \(\mathcal{A}\) into unstable manifolds of equilibria provides a geometric and topological characterization of Sturm global attractors \(\mathcal{A}\) as finite regular signed CW-complexes, the Sturm complexes, with cells given by the unstable manifolds of equilibria. Concurrently, the permutation \(\sigma =\sigma_f\) derived from the ODE boundary value problem by ordering the equilibria according to their values at the boundaries \(x=0,1\), respectively, completely determines the Sturm global attractor \(\mathcal{A}\). Equivalently, we use a planar curve, the meander \(\mathcal{M}=\mathcal{M}_f\), associated to the the ODE boundary value problem by shooting. In the previous paper [J. Dyn. Differ. Equations 34, No. 4, 2787--2818 (2022; Zbl 1503.35040)], we set up to determine the boundary neighbors of any specific unstable equilibrium \(\mathcal{O}\), based exclusively on the information on the corresponding signed hemisphere complex. In addition, a certain minimax property of the boundary neighbors was established. In the signed hemisphere decomposition of the cell boundary of \(\mathcal{O}\), this property identifies the equilibria which are closest to, or most distant from, \(\mathcal{O}\) at the boundaries \(x=0,1\), in each hemisphere. The main objective of the present paper is to derive this minimax property directly from the Sturm permutation \(\sigma\), or equivalently from the Sturm meander \(\mathcal{M}\), based on the Sturm nodal properties of the solutions of the ODE boundary value problem. This minimax result simplifies the task of identifying the equilibria on the cell boundary of each unstable equilibrium, directly from the Sturm meander \(\mathcal{M}\). We emphasize the local aspect of this result by an example for which the identification of the equilibria is obtained from the knowledge of only a segment of the Sturm meander \(\mathcal{M}\).Uniform attractors of non-autonomous suspension bridge equations with memoryhttps://zbmath.org/1536.350922024-07-17T13:47:05.169476Z"Wang, Lulu"https://zbmath.org/authors/?q=ai:wang.lulu"Ma, Qiaozhen"https://zbmath.org/authors/?q=ai:ma.qiaozhen|ma.qiaozhen.1Summary: In this article, we investigate the long-time dynamical behavior of non-autonomous suspension bridge equations with memory and free boundary conditions. We first establish the well-posedness of the system by means of the maximal monotone operator theory. Secondly, the existence of uniformly bounded absorbing set is obtained. Finally, asymptotic compactness of the process is verified, and then the existence of uniform attractors is proved for non-autonomous suspension bridge equations with memory term.Asymptotic behavior of solutions to nonclassical diffusion equations with degenerate memory and a time-dependent perturbed parameterhttps://zbmath.org/1536.350932024-07-17T13:47:05.169476Z"Zhang, Jiangwei"https://zbmath.org/authors/?q=ai:zhang.jiangwei"Xie, Zhe"https://zbmath.org/authors/?q=ai:xie.zhe"Xie, Yongqin"https://zbmath.org/authors/?q=ai:xie.yongqinSummary: This article concerns the asymptotic behavior of solutions for a class of nonclassical diffusion equation with time-dependent perturbation coefficient and degenerate memory. We prove the existence and uniqueness of time-dependent global attractors in the family of time-dependent product spaces, by applying the operator decomposition technique and the contractive function method. Then we study the asymptotic structure of time-dependent global attractors as \(t\to \infty\). It is worth noting that the memory kernel function satisfies general assumption, and the nonlinearity \(f\) satisfies a polynomial growth of arbitrary order.Smooth extensions for inertial manifolds of semilinear parabolic equationshttps://zbmath.org/1536.350942024-07-17T13:47:05.169476Z"Kostianko, Anna"https://zbmath.org/authors/?q=ai:kostianko.anna"Zelik, Sergey"https://zbmath.org/authors/?q=ai:zelik.sergey-vSummary: The paper is devoted to a comprehensive study of smoothness of inertial manifolds (IMs) for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than \(C^{1,\varepsilon}\)-regularity for such manifolds (for some positive, but small \(\varepsilon)\). Nevertheless, as shown in the paper, under natural assumptions, the obstacles to the existence of a \(C^n\)-smooth inertial manifold (where \(n\in\mathbb{N}\) is any given number) can be removed by increasing the dimension and by modifying properly the nonlinearity outside of the global attractor (or even outside the \(C^{1,\varepsilon}\)-smooth IM of a minimal dimension). The proof is strongly based on the Whitney extension theorem.McKean-Vlasov equations involving hitting times: blow-ups and global solvabilityhttps://zbmath.org/1536.350952024-07-17T13:47:05.169476Z"Bayraktar, Erhan"https://zbmath.org/authors/?q=ai:bayraktar.erhan"Guo, Gaoyue"https://zbmath.org/authors/?q=ai:guo.gaoyue"Tang, Wenpin"https://zbmath.org/authors/?q=ai:tang.wenpin"Zhang, Yuming Paul"https://zbmath.org/authors/?q=ai:zhang.yuming-paulSummary: This paper is concerned with the analysis of blow-ups for two McKean-Vlasov equations involving hitting times. Let \((B(t); t\geq 0)\) be standard Brownian motion, and \(\tau := \inf \{t\geq 0 : X(t) \leq 0\}\) be the hitting time to zero of a given process \(X\). The first equation is \(X(t) = X(0-) + B(t) -\alpha \mathbb{P}(\tau\leq t)\). We provide a simple condition on \(\alpha\) and the distribution of \(X(0-)\) such that the corresponding Fokker-Planck equation has no blow-up, and thus the McKean-Vlasov dynamics is well defined for all time \(t\geq 0\). Our approach relies on a connection between the McKean-Vlasov equation and the supercooled Stefan problem, as well as several comparison principles. The second equation is \(X(t) = X(0-) + \beta t+B(t)+\alpha \ln \mathbb{P}(\tau > t), t\geq 0\), whose Fokker-Planck equation is nonlocal. We prove that for \(\beta>0\) sufficiently large and \(\alpha\) no greater than a sufficiently small positive constant, there is no blow-up and the McKean-Vlasov dynamics is well defined for all time \(t\geq 0\). The argument is based on a new transform, which removes the nonlocal term, followed by a relative entropy analysis.Blow up and lifespan of solutions for elastic membrane equation with delayhttps://zbmath.org/1536.350962024-07-17T13:47:05.169476Z"Benzahi, Mourad"https://zbmath.org/authors/?q=ai:benzahi.mourad"Zaraï, Abderrahmane"https://zbmath.org/authors/?q=ai:zarai.abderrahmane"Boulaaras, Salah"https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud"Jan, Rashid"https://zbmath.org/authors/?q=ai:jan.rashid"Iqbal, Mujahid"https://zbmath.org/authors/?q=ai:iqbal.mujahidSummary: The primary objective of this research is to examine a nonlinear elastic membrane equation incorporating delay and source terms within a bounded domain. We obtain sufficient conditions on the initial data and the involved functionals for which the energy of solutions with non positive initial energy as well as positive initial energy blow up in a finite-time. In addition, this research work provides estimates for the lifespan of these solutions.Decay estimate for Yang-Mills fields on ALE spaces and applicationshttps://zbmath.org/1536.350972024-07-17T13:47:05.169476Z"Chen, Youmin"https://zbmath.org/authors/?q=ai:chen.youmin"Zhu, Miaomiao"https://zbmath.org/authors/?q=ai:zhu.miaomiaoSummary: We study the decay estimates at the infinity for Yang-Mills connections on Ricci flat ALE manifolds and explore some applications.On blowup for the supercritical quadratic wave equationhttps://zbmath.org/1536.350982024-07-17T13:47:05.169476Z"Csobo, Elek"https://zbmath.org/authors/?q=ai:csobo.elek"Glogić, Irfan"https://zbmath.org/authors/?q=ai:glogic.irfan"Schörkhuber, Birgit"https://zbmath.org/authors/?q=ai:schorkhuber.birgitSummary: We study singularity formation for the quadratic wave equation in the energy supercritical case, i.e., for \(d \geq 7\). We find in closed form a new, nontrivial, radial, self-similar blow-up solution \(u^*\) which exists for all \(d \geq 7\). For \(d=9\), we study the stability of \(u^*\) without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via \(u^*\). In similarity coordinates, this family represents a codimension-1 Lipschitz manifold modulo translation symmetries. The stability analysis relies on delicate spectral analysis for a non-self-adjoint operator. In addition, in \(d=7\) and \(d=9\), we prove nonradial stability of the well-known ODE blow-up solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates.Gradient blowup profile for the semilinear heat equationhttps://zbmath.org/1536.350992024-07-17T13:47:05.169476Z"Duong, Giao Ky"https://zbmath.org/authors/?q=ai:duong.giao-ky"Ghoul, Tej-Eddine"https://zbmath.org/authors/?q=ai:ghoul.tej-eddine"Zaag, Hatem"https://zbmath.org/authors/?q=ai:zaag.hatemSummary: In this paper, we consider the standard semilinear heat equation
\[
\partial_{t} u = \Delta u + |u|^{p - 1} u, \quad p > 1.
\]
The determination of the (believed to be) generic blowup profile is well established in the literature, with the solution blowing up only at one point. Though the blow-up of the gradient of the solution is a direct consequence of the single-point blow-up property and the mean value theorem, there is no determination of the final blowup profile for the gradient in the literature, up to our knowledge. In this paper, we refine the construction technique of \textit{J. Bricmont} and \textit{A. Kupiainen} [Nonlinearity 7, No. 2, 539--575 (1994; Zbl 0857.35018)] and \textit{F. Merle} and \textit{H. Zaag} [Duke Math. J. 86, No. 1, 143--195 (1997; Zbl 0872.35049)], and derive the following profile for the gradient:
\[
\nabla u (x, T) \sim - \frac{\sqrt{2b}}{p - 1} \frac{x}{|x| \sqrt{|\ln |x||}} \left[ \frac{b|x|^{2}}{2 |\ln|x||} \right]^{-\frac{p + 1}{2(p - 1)}} \text{as } x \to 0,
\]
where \(b = \frac{(p - 1)^2}{4p}\), which is \textbf{as expected} the gradient of the well-known blowup profile of the solution.Blowup for a damped wave equation with mass and general nonlinear memoryhttps://zbmath.org/1536.351002024-07-17T13:47:05.169476Z"Feng, Zhendong"https://zbmath.org/authors/?q=ai:feng.zhendong"Guo, Fei"https://zbmath.org/authors/?q=ai:guo.fei"Li, Yuequn"https://zbmath.org/authors/?q=ai:li.yuequnSummary: We investigate the blowup conditions to the Cauchy problem for a semilinear wave equation with scale-invariant damping, mass and general nonlinear memory term (see Eq. (1.1) in the Introduction). We first establish a local (in time) existence result for this problem by Banach's fixed point theorem, where Palmieri's decay estimates on the solution to the corresponding linear homogeneous equation play an essential role in the proof. We then formulate a blowup result for energy solutions by applying the iteration argument together with the test function method.Blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimensionhttps://zbmath.org/1536.351012024-07-17T13:47:05.169476Z"Iagar, Razvan Gabriel"https://zbmath.org/authors/?q=ai:iagar.razvan-gabriel"Latorre, Marta"https://zbmath.org/authors/?q=ai:latorre.marta"Sánchez, Ariel"https://zbmath.org/authors/?q=ai:sanchez.arielSummary: We classify all the blow-up solutions in self-similar form to the following reaction-diffusion equation
\[
\partial_t u=\Delta u^m +|x|^{\sigma}u^p,
\]
posed for \((x,t)\in \mathbb{R}^N \times (0,T)\), with \(m>1, 1 \leq p<m\) and \(-2(p-1)/(m-1)<\sigma <\infty\). We prove that there are several types of self-similar solutions with respect to the local behavior near the origin, and their existence depends on the magnitude of \(\sigma\). In particular, these solutions have different blow-up sets and rates: some of them have \(x=0\) as a blow-up point, some other only blow up at (space) infinity. We thus emphasize on the effect of the weight on the specific form of the blow-up patterns of the equation. The present study generalizes previous works by the authors limited to dimension \(N=1\) and \(\sigma >0\).On the blow-up of the solution of a nonlinear system of equations of a thermal-electrical modelhttps://zbmath.org/1536.351022024-07-17T13:47:05.169476Z"Korpusov, M. O."https://zbmath.org/authors/?q=ai:korpusov.maksim-olegovich"Perlov, A. Yu."https://zbmath.org/authors/?q=ai:perlov.a-yu"Timoshenko, A. V."https://zbmath.org/authors/?q=ai:timoshenko.a-v"Shafir, R. S."https://zbmath.org/authors/?q=ai:shafir.r-sSummary: In this paper, we propose a system of nonlinear equations for the electric field potential and temperature, which describes the process of heating the semiconductor elements of an electrical board followed by thermal breakdown. For this system of equations, we prove the existence of a classical solution that is not extendable in time and also obtain sufficient conditions for the solution to blow up in finite time.Global existence and asymptotic profile for a damped wave equation with variable-coefficient diffusionhttps://zbmath.org/1536.351032024-07-17T13:47:05.169476Z"Li, Yuequn"https://zbmath.org/authors/?q=ai:li.yuequn"Liu, Hui"https://zbmath.org/authors/?q=ai:liu.hui.5|liu.hui.2|liu.hui.1|liu.hui.4|liu.hui.6|liu.hui.9"Guo, Fei"https://zbmath.org/authors/?q=ai:guo.feiSummary: We considered a Cauchy problem of a one-dimensional semilinear wave equation with variable-coefficient diffusion, time-dependent damping, and perturbations. The global well-posedness and the asymptotic profile are given by employing scaling variables and the energy method. The lower bound estimate of the lifespan to the solution is obtained as a byproduct.Blow-up and general decay of solutions for a Kirchhoff-type equation with distributed delay and variable-exponentshttps://zbmath.org/1536.351042024-07-17T13:47:05.169476Z"Ouchenane, Djamel"https://zbmath.org/authors/?q=ai:ouchenane.djamel"Boulaaras, Salah"https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud"Choucha, Abdelbaki"https://zbmath.org/authors/?q=ai:choucha.abdelbaki"Alnegga, Mohammad"https://zbmath.org/authors/?q=ai:alnegga.mohammadSummary: A nonlinear Kirchhoff-type equation with distributed delay and variableexponents is studied. Under suitable hypothesis the blow-up of solutions is proved, and by using an integral inequality due to Komornik the general decay result is obtained in the case \(b = 0\).The blow-up curve for a weakly coupled system of semilinear wave equations with nonlinearities of derivative-typehttps://zbmath.org/1536.351052024-07-17T13:47:05.169476Z"Sasaki, Takiko"https://zbmath.org/authors/?q=ai:sasaki.takikoSummary: In this paper, we study a blow-up curve for a weakly coupled system of semilinear wave equations with nonlinearities of derivative type in one space dimension. Employing the idea of \textit{L. A. Caffarelli} and \textit{A. Friedman} [Trans. Am. Math. Soc. 297, 223--241 (1986; Zbl 0638.35053)], we prove the blow-up curve becomes Lipschitz continuous under suitable initial conditions. Moreover, we show the blow-up rates of the solution of the wave equations.Blow-up vs. global existence for a Fujita-type heat exchanger systemhttps://zbmath.org/1536.351062024-07-17T13:47:05.169476Z"Tréton, Samuel"https://zbmath.org/authors/?q=ai:treton.samuelSummary: We analyze a reaction-diffusion system on \(\mathbb{R}^N\) which models the dispersal of individuals between two exchanging environments for its diffusive component and incorporates a Fujita-type growth for its reactive component. The originality of this model lies in the coupling of the equations through diffusion, which, to the best of our knowledge, has not been studied in Fujita-type problems. We first consider the underlying diffusive problem, demonstrating that the solutions converge exponentially fast towards those of two uncoupled equations, featuring a dispersive operator that is somehow a combination of Laplacians. By subsequently adding Fujita-type reaction terms to recover the entire system, we identify the critical exponent that separates systematic blow-up from possible global existence.Stability and existence results for a class of nonlinear parabolic equations with three lower order terms and measure data using Lorentz spaceshttps://zbmath.org/1536.351072024-07-17T13:47:05.169476Z"Abdellaoui, Mohammed"https://zbmath.org/authors/?q=ai:abdellaoui.mohammed-aminSummary: We study both existence and stability of \textit{renormalized} solutions for nonlinear parabolic problems with three lower order terms that have, respectively, growth with respect to \(u\) and to the gradient, whose model
\[
(\mathcal{P})
\begin{cases}
u_t -\varDelta_p u-\operatorname{div}[c(t,x)|u|^{\gamma -1}u]+b(t,x)|\nabla u|^{\lambda}+d(t,x)|u|^{\iota}=\mu -\operatorname{div}(E)\quad \text{in }Q, \\
u(0,x)=u_0 (x) \quad \text{in }\varOmega, u(t,x)=0\text{on }(0,T)\times\partial \varOmega,
\end{cases}
\]
where \(Q:=(0,T)\times \varOmega\) (with \(\varOmega\) is an open bounded subset of \(\mathbb{R}^N\) (\(N\geq 2\)) and \(T>0\)), \(1<p<N\), \(\varDelta_p\) is the usual \(p\)-Laplace operator, and \(\mu \in\mathbf{M}(Q)\) is a (general) measure with bounded total variation on \(Q\). As a consequence of our main results, we prove that the conditions \(\gamma =\frac{(N+2)(p-1)}{N+p}\), \(\lambda =\frac{N(p-1)+p}{N+2}\), \(0\leq \iota \leq p-\frac{N-p}{N}\), \(c\in L^{\tau =\frac{N+p}{p-1}}(Q)^N\), \(b\in L^{N+2,1}(Q)\) and \(d\in L^{z',1}(Q)\) (with \(z=\frac{pN-N-p}{\iota N})\) are necessary and sufficient for the existence and the stability of solutions for every sufficiently regular \(u_0 \in L^2 (\varOmega)\), \(E\in L^{p'}(Q)^N\) and irregular \(\mu \in\mathbf{M}(Q)\).Pointwise and weighted Hessian estimates for Kolmogorov-Fokker-Planck-type operatorshttps://zbmath.org/1536.351082024-07-17T13:47:05.169476Z"Ghosh, Abhishek"https://zbmath.org/authors/?q=ai:ghosh.abhishek"Tewary, Vivek"https://zbmath.org/authors/?q=ai:tewary.vivekSummary: In this article, we obtain Hessian estimates for Kolmogorov-Fokker-Planck operators in non-divergence form in several Banach function spaces. Our approach relies on a representation formula and newly developed sparse domination techniques in harmonic analysis. Our result when restricted to weighted Lebesgue spaces yields sharp quantitative Hessian estimates for the Kolmogorov-Fokker-Planck operators.Interior a priori estimates for supersolutions of fully nonlinear subelliptic equations under geometric conditionshttps://zbmath.org/1536.351092024-07-17T13:47:05.169476Z"Goffi, Alessandro"https://zbmath.org/authors/?q=ai:goffi.alessandroSummary: In this note, we prove interior a priori first- and second-order estimates for solutions of fully nonlinear degenerate elliptic inequalities structured over the vector fields of Carnot groups, under the main assumption that \(u\) is semiconvex along the fields. These estimates for supersolutions are new even for linear subelliptic inequalities in nondivergence form, whereas in the nonlinear setting they do not require neither convexity nor concavity on the second derivatives. We complement the analysis exhibiting an explicit example showing that horizontal \(W^{2,q}\) regularity of Calderón-Zygmund type for fully nonlinear subelliptic equations posed on the Heisenberg group cannot be in general expected in the range \(q< Q\), \(Q\) being the homogeneous dimension of the group.
{\copyright} 2024 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.Local-in-time strong solutions of the homogeneous Landau-Coulomb equation with \(L^p\) initial datumhttps://zbmath.org/1536.351102024-07-17T13:47:05.169476Z"Golding, William"https://zbmath.org/authors/?q=ai:golding.william-m"Loher, Amélie"https://zbmath.org/authors/?q=ai:loher.amelieSummary: We consider the homogeneous Landau equation with Coulomb potential and general initial data \(f_{in} \in L^p\), where \(p\) is arbitrarily close to 3/2. We show the local-in-time existence and uniqueness of smooth solutions for such initial data. The constraint \(p > 3/2\) has appeared in several related works and appears to be the minimal integrability assumption achievable with current techniques. We adapt recent ODE methods and conditional regularity results appearing in [\textit{W. Golding} et al., ``Nonlinear regularization estimates and global well-posedness for the Landau-Coulomb equation near equilibrium'', Preprint, \url{arXiv:2303.02281}] to deduce new short time \(L^p \to L^\infty\) smoothing estimates. These estimates enable us to construct local-in-time smooth solutions for large \(L^p\) initial data, and allow us to show directly conditional regularity results for solutions verifying \textit{unweighted} Prodi-Serrin type conditions. As a consequence, we obtain additional stability and uniqueness results for the solutions we construct.Dispersion for the wave equation outside a cylinder in \(\mathbb{R}^3\)https://zbmath.org/1536.351112024-07-17T13:47:05.169476Z"Iandoli, Felice"https://zbmath.org/authors/?q=ai:iandoli.felice"Ivanovici, Oana"https://zbmath.org/authors/?q=ai:ivanovici.oanaSummary: We consider the wave equation with Dirichlet boundary conditions in the exterior of a cylinder in \(\mathbb{R}^3\) and we construct a global in time parametrix to derive sharp dispersion estimates for all frequencies (low and high) and, as a corollary, Strichartz estimates, all matching the \(\mathbb{R}^3\) case.A sharpened Strichartz inequality for the wave equationhttps://zbmath.org/1536.351122024-07-17T13:47:05.169476Z"Negro, Giuseppe"https://zbmath.org/authors/?q=ai:negro.giuseppeSummary: We disprove a conjecture of \textit{D. Foschi} [J. Eur. Math. Soc. (JEMS) 9, No. 4, 739--774 (2007; Zbl 1231.35028)], regarding extremizers for the Strichartz inequality with data in the Sobolev space \(\dot{H}^{1/2} \times \dot{H}^{-1/2} (\mathbb{R}^{d})\), for even \(d \ge 2\). On the other hand, we provide evidence to support the conjecture in odd dimensions and refine his sharp inequality in \(\mathbb{R}^{1 + 3}\), adding a term proportional to the distance of the initial data from the set of extremizers. The proofs use the conformal compactification of the Minkowski space-time given by the Penrose transform.On the weak Harnack estimate for nonlocal equationshttps://zbmath.org/1536.351132024-07-17T13:47:05.169476Z"Prasad, Harsh"https://zbmath.org/authors/?q=ai:prasad.harshSummary: We prove a weak Harnack estimate for a class of doubly nonlinear nonlocal equations modelled on the nonlocal Trudinger equation
\[
\partial_t(|u|^{p-2}u) + (-\Delta_p)^s u = 0
\]
for \(p\in (1, \infty)\) and \(s\in(0, 1)\). Our proof relies on expansion of positivity arguments developed by DiBenedetto, Gianazza and Vespri adapted to the nonlocal setup. Even in the linear case of the nonlocal heat equation and in the time-independent case of fractional \(p\)-Laplace equation, our approach provides an alternate route to Harnack estimates without using Moser iteration, log estimates or Krylov-Safanov covering arguments.On phase-field equations of Penrose-Fife type with non-conserved order parameter under flux boundary conditionhttps://zbmath.org/1536.351142024-07-17T13:47:05.169476Z"Tani, Atusi"https://zbmath.org/authors/?q=ai:tani.atusiSummary: In [Mat. Zamet. SVFU 29, No. 1, 103--121 (2022; Zbl 1536.35138)] we showed the global-in-time solvability of the initial-boundary value problem for the non-conserved phase-field model proposed by \textit{O. Penrose} and \textit{P. C. Fife} [Physica D 43, No. 1, 44--62 (1990; Zbl 0709.76001); Physica D 69, No. 1--2, 107--113 (1993; Zbl 0799.76084)] under the correct form of flux boundary condition for the temperature field in higher space dimensions. In this paper we discuss the uniform boundedness up to the infinite time of its solution in Sobolev-Slobodetski spacesPolyconvex functionals and maximum principlehttps://zbmath.org/1536.351152024-07-17T13:47:05.169476Z"Carozza, Menita"https://zbmath.org/authors/?q=ai:carozza.menita"Esposito, Luca"https://zbmath.org/authors/?q=ai:esposito.luca"Giova, Raffaella"https://zbmath.org/authors/?q=ai:giova.raffaella"Leonetti, Francesco"https://zbmath.org/authors/?q=ai:leonetti.francescoSummary: Let us consider continuous minimizers \(u : \bar \Omega \subset \mathbb{R}^n \to \mathbb{R}^n\) of
\[
\mathcal{F}(v) = \int_{\Omega} [|Dv|^p + |\det Dv|^r] dx,
\]
with \(p > 1\) and \(r > 0 \); then it is known that every component \(u^\alpha\) of \(u = (u^1,\dots, u^n)\) enjoys maximum principle: the set of interior points \(x \), for which the value \(u^\alpha(x)\) is greater than the supremum on the boundary, has null measure, that is, \( \mathcal{L}^n(\{ x \in \Omega: u^\alpha (x) > \sup_{\partial \Omega} u^\alpha \}) = 0 \). If we change the structure of the functional, it might happen that the maximum principle fails, as in the case
\[
\mathcal{F}(v) = \int_{\Omega}[\max\{(|Dv|^p - 1); 0 \} + |\det Dv|^r] dx,
\]
with \(p > 1\) and \(r > 0 \). Indeed, for a suitable boundary value, the set of the interior points \(x \), for which the value \(u^\alpha(x)\) is greater than the supremum on the boundary, has a positive measure, that is \(\mathcal{L}^n(\{ x \in \Omega: u^\alpha (x) > \sup_{\partial \Omega} u^\alpha \}) > 0 \). In this paper we show that the measure of the image of these bad points is zero, that is \(\mathcal{L}^n(u(\{ x \in \Omega: u^\alpha (x) > \sup_{\partial \Omega} u^\alpha \})) = 0 \), provided \(p > n \). This is a particular case of a more general theorem.Maximum principles and consequences for \(\gamma \)-translators in \({\mathbb{R}}^{n+1} \)https://zbmath.org/1536.351162024-07-17T13:47:05.169476Z"Santaella, Jose Torres"https://zbmath.org/authors/?q=ai:santaella.jose-torresSummary: In this paper, we obtain several properties of translating solitons for a general class of extrinsic geometric curvature flow, where the deformation speed is given by a homogeneous smooth symmetric positive function \(\gamma\) defined in a symmetric open cone. The main result of this paper is about the uniqueness of \(\gamma \)-translators in the class of complete graphs defined on a ball.
{\copyright} 2024 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.A comparison principle for doubly nonlinear parabolic partial differential equationshttps://zbmath.org/1536.351172024-07-17T13:47:05.169476Z"Bögelein, Verena"https://zbmath.org/authors/?q=ai:bogelein.verena"Strunk, Michael"https://zbmath.org/authors/?q=ai:strunk.michaelSummary: In this paper, we derive a comparison principle for non-negative weak sub- and super-solutions to doubly nonlinear parabolic partial differential equations whose prototype is
\[
\partial_t u^q - \operatorname{div} \left (|\nabla u|^{p-2}\nabla u \right)=0 \qquad \text{in } \Omega_T,
\]
with \(q> 0\) and \(p> 1\) and \(\Omega_T:=\Omega \times (0,T)\subset \mathbb{R}^{n+1} \). Instead of requiring a lower bound for the sub- or super-solutions in the whole domain \(\Omega_T\), we only assume the lateral boundary data to be strictly positive. The main results yield some applications. Firstly, we obtain uniqueness of non-negative weak solutions to the associated Cauchy-Dirichlet problem. Secondly, we prove that any weak solution is also a viscosity solution.Liouville type theorems for Kirchhoff sub-elliptic equations involving \(\Delta_{\lambda}\)-operatorshttps://zbmath.org/1536.351182024-07-17T13:47:05.169476Z"Nguyen, Thi Thu Huong"https://zbmath.org/authors/?q=ai:nguyen.thi-thu-huong"Quyet, Dao Trong"https://zbmath.org/authors/?q=ai:quyet.dao-trong"Vu, Thi Hien Anh"https://zbmath.org/authors/?q=ai:vu.thi-hien-anhSummary: In this paper, we study the Kirchhoff elliptic equations of the form
\[
-M (\|\nabla_{\lambda} u\|^{2}) \Delta_{\lambda} u = w(x)f(u) \quad \text{in }\mathbb{R}^{N},
\]
where \(M\) is a smooth monotone function, \(w\) is a weight function and \(f(u)\) is of the form \(u^p\), \(e^u\) or \(-u^{-p}\). The operator \(\Delta_{\lambda}\) is strongly degenerate and given by
\[
\Delta_{\lambda} = \sum_{j = 1}^{N} \frac{\partial}{\partial x_{j}}\bigg(\lambda_{j}^{2}(x) \frac{\partial}{\partial x_{j}}\bigg).
\]
We shall prove some classifications of stable solutions to the equation above under general assumptions on \(M\) and \(\lambda_{j}\), \(j = 1, \dots, N\).Liouville theorems and optimal regularity in elliptic equationshttps://zbmath.org/1536.351192024-07-17T13:47:05.169476Z"Tortone, Giorgio"https://zbmath.org/authors/?q=ai:tortone.giorgioSummary: The objective of this paper is to establish a connection between the problem of optimal regularity among solutions to elliptic partial differential equations with measurable coefficients and the Liouville property at infinity. Initially, we address the two-dimensional case by proving an Alt-Caffarelli-Friedman-type monotonicity formula, enabling the proof of optimal regularity and the Liouville property for multiphase problems. In higher dimensions, we delve into the role of monotonicity formulas in characterizing optimal regularity. By employing a hole-filling technique, we present a distinct ``almost-monotonicity'' formula that implies Hölder regularity of solutions. Finally, we explore the interplay between the least growth at infinity and the exponent of regularity by combining blow-up and \(G\)-convergence arguments.
{\copyright} 2024 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.Second order regularity for the \(p (x)\)-Laplace equations with \(L^2\) data on the right-hand sidehttps://zbmath.org/1536.351202024-07-17T13:47:05.169476Z"Cho, Namkyeong"https://zbmath.org/authors/?q=ai:cho.namkyeongThe study of regularity theory for elliptic equations in the form
\[
-\operatorname{div} \mathcal{A}(x,Du)=f\ \text{in}\ \Omega
\]
has been conducted extensively, considering various assumptions on the vector field \(\mathcal{A}(x,\cdot)\). For instance, in the linear Poisson equation, where \(\mathcal{A}(x,Du)=Du\), the classical regularity theory states that if \(\partial \Omega\in C^2\) and \(f\in L^2(\Omega)\) then \(Du\in W^{1,2}(\Omega)\).
The main result of this article states that under Assumption 1.1 on the domain \(\Omega\) and its boundary one can have
\[
(\kappa^2+|Du|^2)^{\frac{p(x)-2}{2}}Du\in W^{1,2}(\Omega).
\]
In order to study the above result, the author considers the entropy solution concept.
Reviewer: Abdolrahman Razani (Qazvīn)Regularity for double phase functionals with two modulating coefficientshttps://zbmath.org/1536.351212024-07-17T13:47:05.169476Z"Kim, Bogi"https://zbmath.org/authors/?q=ai:kim.bogi"Oh, Jehan"https://zbmath.org/authors/?q=ai:oh.jehanSummary: In this paper, we establish regularity results for local minimizers of functionals with non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral with two modulating coefficients
\[
w\mapsto \int [a(x)|Dw|^p +b(x)|Dw|^q] dx, \qquad 1<p<q, \qquad a(\cdot),b(\cdot)\geq 0,
\]
with \(0<\mu \leq a(\cdot)+b(\cdot)\). Here, the coefficient \(b(\cdot)\) is assumed to be Hölder continuous and the coefficient \(a(\cdot)\) is assumed to be uniformly continuous.Local Calderón-Zygmund estimates for parabolic equations in weighted Lebesgue spaceshttps://zbmath.org/1536.351222024-07-17T13:47:05.169476Z"Lee, Mikyoung"https://zbmath.org/authors/?q=ai:lee.mikyoung"Ok, Jihoon"https://zbmath.org/authors/?q=ai:ok.jihoonSummary: We prove local Calderón-Zygmund type estimates for the gradient of weak solutions to degenerate or singular parabolic equations of \(p \)-Laplacian type with \(p > \frac{2n}{n+2}\) in weighted Lebesgue spaces \(L^q_w \). We introduce a new condition on the weight \(w\) which depends on the intrinsic geometry concerned with the parabolic \(p \)-Laplace problems. Our condition is weaker than the one in \(^{[13]} \), where similar estimates were obtained. In particular, in the case \(p = 2 \), it is the same as the condition of the usual parabolic \(A_q\) weight.Local Morrey estimate in Musielak-Orlicz-Sobolev spacehttps://zbmath.org/1536.351232024-07-17T13:47:05.169476Z"Liu, Duchao"https://zbmath.org/authors/?q=ai:liu.duchao"Zhao, Peihao"https://zbmath.org/authors/?q=ai:zhao.peihaoA function $A: \mathbb{R}\rightarrow [0, +\infty)$ is said to be an $N$-modular function, denoted by $A\in N$, if $A(t)>0$ for $t\neq 0$, and if $A$ is even, convex and
\[
\lim_{t\rightarrow0+}\frac{A(t)}{t}=0, \quad \lim_{t\rightarrow +\infty}\frac{A(t)}{t}=+\infty.
\]
Now suppose that $\Omega$ is a smooth domain in $\mathbb{R}^n$. Then a function $A: \Omega \times \mathbb{R}\rightarrow [0, +\infty)$ is called a generalized $N$-modular function, denoted by $N(\Omega)$, if for each $t\in [0, +\infty)$ the function $A(\cdot, t)$ is measurable, and for almost every $x\in\Omega$, we have $A(x, \cdot)\in N$.
We say that a function $A: \Omega\times [0, +\infty)\rightarrow [0, +\infty)$ satisfies the $\Delta_2(\Omega)$ condition, denoted by $A\in \Delta_2(\Omega)$, if there exists a positive constant $k\geq 1$ and a non-negative function $h\in L^1(\Omega)$ such that for $x\in \Omega$ and $t\in [0, +\infty)$
\[
A(x, 2t)\leq kA(x, t)+h(x).
\]
In this paper, the authors obtain a locally uniform Morrey estimate and prove the Hölder continuity of functions in Musielak-Orlicz-Sobolev spaces for a generalized $N$-modular function $A\in N(\Omega)\cap \Delta_2(\Omega)$ under some appropriate assumptions.
Reviewer: Atanu Manna (Bhadohi)Partial regularity for steady double phase fluidshttps://zbmath.org/1536.351242024-07-17T13:47:05.169476Z"Scilla, Giovanni"https://zbmath.org/authors/?q=ai:scilla.giovanni"Stroffolini, Bianca"https://zbmath.org/authors/?q=ai:stroffolini.biancaSummary: We study partial Hölder regularity for nonlinear elliptic systems in divergence form with double-phase growth, modeling double-phase non-Newtonian fluids in the stationary case.Regularity of fundamental solutions for Lévy-type operatorshttps://zbmath.org/1536.351252024-07-17T13:47:05.169476Z"Szczypkowski, Karol"https://zbmath.org/authors/?q=ai:szczypkowski.karolSummary: For a class of non-symmetric non-local Lévy-type operators \(\mathcal{L}^{\kappa}\), which include those of the form
\[
\mathcal{L}^{\kappa}f(x):= \int\limits_{\mathbb{R}^d}(f(x+z)-f(x) - \mathbf{1}_{|z| < 1} \langle z, \nabla f(x)\rangle) \kappa (x,z) J(z)\, dz,
\]
we prove regularity of the fundamental solution \(p^{\kappa}\) to the equation \(\partial_t =\mathcal{L}^{\kappa}\).Pushed fronts in a Fisher-KPP-Burgers system using geometric desingularizationhttps://zbmath.org/1536.351272024-07-17T13:47:05.169476Z"Holzer, Matt"https://zbmath.org/authors/?q=ai:holzer.matt"Kearney, Matthew"https://zbmath.org/authors/?q=ai:kearney.matthew"Molseed, Samuel"https://zbmath.org/authors/?q=ai:molseed.samuel"Tuttle, Katie"https://zbmath.org/authors/?q=ai:tuttle.katie"Wigginton, David"https://zbmath.org/authors/?q=ai:wigginton.davidSummary: We study traveling fronts in a system of reaction-diffusion-advection equations in one spatial dimension motivated by problems in reactive flows. In the limit as a parameter tends to infinity, we construct the approximate front profile and determine the leading order expansion for the selected wavespeed. Such fronts are often constructed as transverse intersections of stable and unstable manifolds of the traveling wave differential equation. However, a re-scaling of the dependent variable leads to a lack of hyperbolicity for one of the end states making the definition of one such manifold unclear. We use geometric blow-up techniques to recover hyperbolicity and following an analysis of the blown up vector field are able to show the existence of a traveling front with a leading order expansion of its speed.Traveling fronts in a reaction-diffusion equation with a memory termhttps://zbmath.org/1536.351282024-07-17T13:47:05.169476Z"Mielke, Alexander"https://zbmath.org/authors/?q=ai:mielke.alexander"Reichelt, Sina"https://zbmath.org/authors/?q=ai:reichelt.sinaSummary: Based on a recent work on traveling waves in spatially nonlocal reaction-diffusion equations, we investigate the existence of traveling fronts in reaction-diffusion equations with a memory term. We will explain how such memory terms can arise from reduction of reaction-diffusion systems if the diffusion constants of the other species can be neglected. In particular, we show that two-scale homogenization of spatially periodic systems can induce spatially homogeneous systems with temporal memory. The existence of fronts is proved using comparison principles as well as a reformulation trick involving an auxiliary speed that allows us to transform memory terms into spatially nonlocal terms. Deriving explicit bounds and monotonicity properties of the wave speed of the arising traveling front, we are able to establish the existence of true traveling fronts for the original problem with memory. Our results are supplemented by numerical simulations.Multiple exp-function solutions, group invariant solutions and conservation laws of a generalized \((2+1)\)-dimensional Hirota-Satsuma-Ito equationhttps://zbmath.org/1536.351292024-07-17T13:47:05.169476Z"Podile, T. J."https://zbmath.org/authors/?q=ai:podile.t-j"Adem, A. R."https://zbmath.org/authors/?q=ai:adem.abdullahi-rashid"Mbusi, S. O."https://zbmath.org/authors/?q=ai:mbusi.sivenathi-oscar"Muatjetjeja, B."https://zbmath.org/authors/?q=ai:muatjetjeja.benSummary: Multiple exp-function technique and group analysis is accomplished for a comprehensive \((2+1)\)-dimensional Hirota-Satsuma-Ito equation that appears in many sectors of nonlinear science such as for example in fluid dynamics. Travelling wave solutions are computed and it is displayed that this underlying equation gives kink solutions. The invariant reductions and further closed-form solutions are processed. Conserved currents are developed and their physical ramifications are illustrated.B-class solitary waves and their persistence under Kuramoto-Sivashinsky perturbationhttps://zbmath.org/1536.351312024-07-17T13:47:05.169476Z"Qian, Zhang"https://zbmath.org/authors/?q=ai:qian.zhangSummary: In this paper, we study the b-class shallow water equation. We take different bifurcation parameters to consider solitary wave solutions as well as their persistence under singular Kuramoto-Sivashinsky perturbation. We apply phase portrait analysis and the method of geometric singular perturbation theory.Solitons and other nonlinear waves for stochastic Schrödinger-Hirota model using improved modified extended tanh-function approachhttps://zbmath.org/1536.351322024-07-17T13:47:05.169476Z"Shehab, Mohammed F."https://zbmath.org/authors/?q=ai:shehab.mohammed-f"El-Sheikh, Mohamed M. A."https://zbmath.org/authors/?q=ai:el-sheikh.mohamed-m-a"Ahmed, Hamdy M."https://zbmath.org/authors/?q=ai:ahmed.hamdy-m"Mabrouk, Amina A. G."https://zbmath.org/authors/?q=ai:mabrouk.amina-a-g"Mirzazadeh, M."https://zbmath.org/authors/?q=ai:mirzazadeh.mohammad"Hashemi, M. S."https://zbmath.org/authors/?q=ai:hashemi.mir-sajjadSummary: The improved modified extended tanh-function approach was used to study optical stochastic soliton solutions and other exact stochastic solutions for the nonlinear Schrödinger-Hirota equation with multiplicative white noise. The derived solutions include stochastic bright solitons, stochastic singular solitons, stochastic periodic solutions, stochastic singular periodic solutions, stochastic exponential solutions, stochastic rational solutions, and stochastic Jacobi elliptic doubly periodic solutions. Constraints on the parameters were taken into account to ensure the existence of the obtained stochastic soliton solutions. Additionally, selected solutions were presented graphically to illustrate the physical characteristics of the stochastic solutions. In this paper, we used Mathematica (11.3) packages to find the coefficients and Matlab (R2015a) packages to plot the graphs.
{\copyright} 2023 John Wiley \& Sons Ltd.Integral representations for the double-diffusivity system on the half-linehttps://zbmath.org/1536.351342024-07-17T13:47:05.169476Z"Chatziafratis, Andreas"https://zbmath.org/authors/?q=ai:chatziafratis.andreas"Aifantis, Elias C."https://zbmath.org/authors/?q=ai:aifantis.elias-c"Carbery, Anthony"https://zbmath.org/authors/?q=ai:carbery.anthony"Fokas, Athanassios S."https://zbmath.org/authors/?q=ai:fokas.athanassios-sSummary: A novel method is presented for explicitly solving inhomogeneous initial-boundary-value problems (IBVPs) on the half-line for a well-known coupled system of evolution partial differential equations. The so-called double-diffusion model, which is based on a simple, yet general, inhomogeneous diffusion configuration, describes accurately several important physical and mechanical processes and thus emerges in miscellaneous applications, ranging from materials science, heat-mass transport and solid-fluid dynamics, to petroleum and chemical engineering. For instance, it appears in nanotechnology and its inhomogeneous version has recently appeared in the area of lithium-ion rechargeable batteries. Our approach is based on the extension of the unified transform (also called the Fokas method), so that it can be applied to systems of coupled equations. First, we derive formally effective solution representations and then justify \textit{a posteriori} their validity rigorously. This includes the reconstruction of the prescribed initial and boundary conditions, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. The novel solution formulae are also utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. In particular, we prove uniform convergence of the solution to the data, its rapid decay at infinity as well as its smoothness up to (and beyond) the boundary axes, provided certain data compatibility conditions at the quarter-plane corner are satisfied. As a sample of important applications of our analysis and investigation of the boundary behavior of the solution and its derivatives, we both prove a novel uniqueness theorem and construct a `non-uniqueness counterexample'. These supplement the preceding `constructive existence' result, within the framework of well-posedness. Moreover, one of the advantages of the unified transform is that it yields representations which are defined on contours in the complex Fourier \(\lambda\)-plane, which exhibit exponential decay for large values of \(\lambda\). This important characteristic of the solutions is expected to allow for an efficient numerical evaluation; this is envisaged in future numerical-analytic investigations. The new formulae and the findings reported herein are also expected to find utility in the study of questions pertaining to well-posedness for nonlinear counterparts too. In addition, our rigorous approach can be extended to IBVPs for other significant models of mathematical physics and potentially also to higher-dimensional and variable-coefficient cases.Existence and uniqueness of the global conservative weak solutions to the cubic Camassa-Holm-type equationhttps://zbmath.org/1536.351372024-07-17T13:47:05.169476Z"Zhu, Min"https://zbmath.org/authors/?q=ai:zhu.min"Guo, Lijuan"https://zbmath.org/authors/?q=ai:guo.lijuan"Wang, Ying"https://zbmath.org/authors/?q=ai:wang.ying.23Summary: This paper is devoted to the cubic Camassa-Holm-Type equation with a nonlocal cubic nonlinearity suggested as an asymptotic model for the 2D full water wave dynamics which is nonlocal with higher-order nonlinearities compared to the classical Camassa-Holm equation. We establish the global existence and uniqueness of the energy conservative weak solution in the energy space.On an elliptic Kirchhoff-Boussinesq type problems with exponential growthhttps://zbmath.org/1536.351532024-07-17T13:47:05.169476Z"Carlos, Romulo D."https://zbmath.org/authors/?q=ai:carlos.romulo-d"Figueiredo, Giovany M."https://zbmath.org/authors/?q=ai:figueiredo.giovany-malcherSummary: In this paper, we prove an existence result of nontrivial solutions for the problem
\[
\Delta^2u\pm\Delta_pu = f(u)\text{ in }\Omega, \text{ and } u = \Delta u = 0\text{ on }\partial\Omega,
\]
where \(\Omega\subset\mathbb{R}^4\) is a smooth bounded domain, \(2 < p < 4\) and \(f:\mathbb{R}\to\mathbb{R}\) is a superlinear continuous function with exponential subcritical or critical growth. We apply the Nehari manifold method in order to prove the main result.
{\copyright} 2023 John Wiley \& Sons Ltd.Energy estimate up to the boundary for stable solutions to semilinear elliptic problemshttps://zbmath.org/1536.351602024-07-17T13:47:05.169476Z"Erneta, Iñigo U."https://zbmath.org/authors/?q=ai:erneta.inigo-uThe author obtains a universal energy estimate up to the boundary for stable solutions of semilinear equations with variable coefficients. More precisely he considers classical solutions \(u\in C^2(\overline{\Omega})\) to \(-L(u)=f(u)\) in a bounded domain \(\Omega\subset{\mathbb R}^n\), where \(L\) denotes a uniformly elliptic operator of the form
\[
L=a_{ij}(x)\partial_{ij}+b_i(x)\partial_i\, , \quad a_{ij}(x)=a_{ji}(x).
\]
A solution \(u\) of the previous equation is called stable if the principal eigenvalue (with respect to Dirichlet conditions) of the linearized equation \(-L-f'(u)\) is nonnegative.
His main result is an estimate for the norm of the gradient of stable solutions vanishing on the flat part of a half-ball, for any nonnegative and nondecreasing \(f\). That is, a bound of the type
\[
\Vert\nabla u\Vert_{L^{2+\gamma}(B^+_{1/2})}\leq C\Vert u\Vert_{L^1 (B^+_1)},
\]
where \(\gamma=\gamma (n)>0\), \(C=C(n,a_{ij},b_i)\) and \(B^+_\rho\) denotes the half-ball of radius \(\rho\).
This bound only requires the elliptic coefficients to be Lipschitz. As a consequence, his estimate continues to hold in general \(C^{1,1}\) domains if we further assume the nonlinearity \(f\) to be convex. This result is new even for the Laplace operator, for which a \(C^3\) regularity assumption on the domain was needed.
A fundamental ingredient in the proof of the previous estimate will be to control the Hessian of a stable solution in half-balls. To this purpose, the author obtains boundary Hessian estimates which can be interpreted as a generalization of a geometric stability condition due to Sternberg and Zumbrun.
Reviewer: Salvador Villegas (Granada)On the first mixed problem for degenerate parabolic equations in stellar domains with Lyapunov boundary in Banach spaceshttps://zbmath.org/1536.351722024-07-17T13:47:05.169476Z"Petrushko, Igor' Meletievich"https://zbmath.org/authors/?q=ai:petrushko.igor-meletievich"Kaputsyna, Tat'yana Vladimirovna"https://zbmath.org/authors/?q=ai:kaputsyna.tatyana-vladimirovna"Petrushko, Maksim Igor'evich"https://zbmath.org/authors/?q=ai:petrushko.maksim-igorevichSummary: The article is devoted to the study of behavior of the solution to a second-order parabolic equation with Tricomi degeneration on the lateral boundary of a cylindrical domain \(Q^T\) , where \(Q\) is a stellar region whose boundary \(\partial Q\) is an \((n -1)\)-dimensional closed surface without boundary of class \(C^{1+ \lambda }\), \(0 < \lambda < 1\). We study the question of unique solvability of the first mixed problem for the equation with the boundary and initial functions belonging to spaces of type \(L_p\), \(p > 1\). This topic goes back to the classical works of Littlewood-Paley and F. Riesz devoted to the boundary values of analytic functions. All directions of taking boundary values for uniformly elliptic equations turn out to be equal, and the solution has a property similar to the continuity with respect to a set of variables. In the case of degeneracy of the equation on the boundary of the domain when the directions are not equal, the situation becomes more complicated. In this case, the statement of the first boundary value problem is determined by the type of degeneracy.Well-posedness and stability for a class of fourth-order nonlinear parabolic equationshttps://zbmath.org/1536.351732024-07-17T13:47:05.169476Z"Li, Xinye"https://zbmath.org/authors/?q=ai:li.xinye"Melcher, Christof"https://zbmath.org/authors/?q=ai:melcher.christofSummary: In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation \(\partial_t u + (-\Delta)^2 u = \nabla \cdot F(\nabla u)\), where \(F\) satisfies a cubic growth conditions. We establish existence and uniqueness of the solution for small initial data in local BMO spaces. In the cubic case \(F(\xi) = \pm | \xi |^2 \xi\), we also examine the large time behaviour and stability of global solutions for arbitrary and small initial data in VMO, respectively.Quantitative steepness, semi-FKPP reactions, and pushmi-pullyu frontshttps://zbmath.org/1536.351762024-07-17T13:47:05.169476Z"An, Jing"https://zbmath.org/authors/?q=ai:an.jing"Henderson, Christopher"https://zbmath.org/authors/?q=ai:henderson.christopher"Ryzhik, Lenya"https://zbmath.org/authors/?q=ai:ryzhik.lenyaThe authors consider a reaction-diffusion equation \[u_t = u_{xx} + f(u), \] where it is assumed that \(f(u) \geq 0\), for \(u\in [0,1]\), \(f(0)=f(1)=0\), \(f^{\prime}(0) >0\), \(f(u)>0\) for \(u\in (0,1)\). The paper is devoted to the analysis of the transition between pushed and pulled fronts that the system is known to have. It is also known that the preference of the equation for a pulled or a pushed front is related to the steepness of a solution to this equation with the initial condition \(u(0,x)=1(x\leq 0)\) compared to the steepness of the slowest monotone front. The authors note that the condition that the solution evolved from this initial condition is steeper than the front moving with the critical speed is equivalent to the fact that any shift of the critical front intersects the profile of that solution just once. The authors define a function which they call the shape defect function that measures the distance between such solutions and the critical front and capture the argument above.
Reviewer: Anna Ghazaryan (Oxford)Global dynamics of a time-delayed nonlocal reaction-diffusion model of within-host viral infectionshttps://zbmath.org/1536.351782024-07-17T13:47:05.169476Z"Li, Zhimin"https://zbmath.org/authors/?q=ai:li.zhimin"Zhao, Xiao-Qiang"https://zbmath.org/authors/?q=ai:zhao.xiaoqiang|zhao.xiao-qiangSummary: In this paper, we study a time-delayed nonlocal reaction-diffusion model of within-host viral infections. We introduce the basic reproduction number \(\mathscr{R}_0\) and show that the infection-free steady state is globally asymptotically stable when \(\mathscr{R}_0\leq 1\), while the disease is uniformly persistent when \(\mathscr{R}_0>1\). In the case where all coefficients and reaction terms are spatially homogeneous, we obtain an explicit formula of \(\mathscr{R}_0\) and the global attractivity of the positive constant steady state. Numerically, we illustrate the analytical results, conduct sensitivity analysis, and investigate the impact of drugs on curtailing the spread of the viruses.An effective analytical method for fractional Brusselator reaction-diffusion systemhttps://zbmath.org/1536.351792024-07-17T13:47:05.169476Z"Nisar, Kottakkaran Sooppy"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaran"Jagatheeshwari, R."https://zbmath.org/authors/?q=ai:jagatheeshwari.r"Ravichandran, C."https://zbmath.org/authors/?q=ai:ravichandran.chokkalingam"Veeresha, P."https://zbmath.org/authors/?q=ai:veeresha.pundikalaSummary: In recent years, reaction-diffusion models have attracted researchers for their wide applications. In this article, we consider Brusselator reaction-diffusion system (BRDS), which is known for its cross diffusion and pattern formations in biology and chemistry. We derive an analytical solution of the fractional Brusselator reaction-diffusion system (FBRDS) with the help of the initial condition by a novel method, residual power series method (RPSM). The system solution has been analyzed by graph.
{\copyright} 2023 John Wiley \& Son Ltd.The dynamics of an eco-epidemiological prey-predator model with infectious diseases in preyhttps://zbmath.org/1536.351812024-07-17T13:47:05.169476Z"Wang, Mingxin"https://zbmath.org/authors/?q=ai:wang.mingxin"Yao, Shaowen"https://zbmath.org/authors/?q=ai:yao.shaowenSummary: In this paper, we first propose an eco-epidemiological prey-predator model with infectious diseases in prey. Then study the ODE model, and diffusive model with the homogeneous Neumann and Dirichlet boundary conditions, respectively. For the ODE model and the diffusive model with the homogeneous Neumann boundary conditions, we give a complete conclusion about the stabilities of nonnegative equilibrium states (nonnegative constant equilibrium solutions). The results show that these two problems has no periodic solutions, and the diffusive model with the homogeneous Neumann boundary conditions has no yet Turing patterns. For the diffusive model with the homogeneous Dirichlet boundary conditions, we first establish the necessary and sufficient conditions for the existence of positive equilibrium solutions, and prove that the positive equilibrium solution is unique when it exists. Then we study the global asymptotic stabilities of trivial and semi-trivial nonnegative equilibrium solutions.Existence of a maximum of time-averaged harvesting in the KPP model on sphere with permanent and impulse harvestinghttps://zbmath.org/1536.351832024-07-17T13:47:05.169476Z"Vinnikov, E. V."https://zbmath.org/authors/?q=ai:vinnikov.e-v"Davydov, A. A."https://zbmath.org/authors/?q=ai:davydov.alexei-aleksandrovich.1"Tunitsky, D. V."https://zbmath.org/authors/?q=ai:tunitskij.d-vSummary: A distributed renewable resource of any nature is considered on a two-dimensional sphere. Its dynamics is described by a model of the Kolmogorov-Petrovsky-Piskunov-Fisher type, and the exploitation of this resource is carried out by constant or periodic impulse harvesting. It is shown that, after choosing an admissible exploitation strategy, the dynamics of the resource tend to limiting dynamics corresponding to this strategy and there is an admissible harvesting strategy that maximizes the time-averaged harvesting of the resource.Invariant manifolds for the thin film equationhttps://zbmath.org/1536.351842024-07-17T13:47:05.169476Z"Seis, Christian"https://zbmath.org/authors/?q=ai:seis.christian"Winkler, Dominik"https://zbmath.org/authors/?q=ai:winkler.dominikSummary: The large-time behavior of solutions to the thin film equation with linear mobility in the complete wetting regime on \(\mathbb{R}^N\) is examined. We investigate the higher order asymptotics of solutions converging towards self-similar Smyth-Hill solutions under certain symmetry assumptions on the initial data. The analysis is based on a construction of finite-dimensional invariant manifolds that solutions approximate to an arbitrarily prescribed order.Solutions with positive components to quasilinear parabolic systemshttps://zbmath.org/1536.351852024-07-17T13:47:05.169476Z"Shamarova, Evelina"https://zbmath.org/authors/?q=ai:shamarova.evelinaSummary: We obtain sufficient conditions for the existence and uniqueness of solutions with non-negative components to general quasilinear parabolic problems
\[
\begin{cases}
\partial_t u^k = \sum_{i, j = 1}^n a_{i j} (t, x, u) \partial_{x_i x_j}^2 u^k + \sum_{i = 1}^n b_i (t, x, u, \partial_x u) \partial_{x_i} u^k + c^k (t, x, u, \partial_x u), \\
u^k (0, x) = \varphi^k (x), \\
u^k (t, \cdot) = 0, \quad \text{on } \partial \mathbb{F}, \\
k = 1, 2, \ldots, m, \quad x \in \mathbb{F}, t > 0.
\end{cases}
\]
Here, \(\mathbb{F}\) is either a bounded domain or \(\mathbb{R}^n\); in the latter case, we disregard the boundary condition. We apply our results to study the existence and asymptotic behavior of componentwise non-negative solutions to the Lotka-Volterra competition model with diffusion. In particular, we show the convergence, as \(t \to + \infty\), of the solution for a 2-species Lotka-Volterra model, whose coefficients vary in space and time, to a solution of the associated elliptic problem.Optimal regularity and fine asymptotics for the porous medium equation in bounded domainshttps://zbmath.org/1536.351872024-07-17T13:47:05.169476Z"Jin, Tianling"https://zbmath.org/authors/?q=ai:jin.tianling"Ros-Oton, Xavier"https://zbmath.org/authors/?q=ai:ros-oton.xavier"Xiong, Jingang"https://zbmath.org/authors/?q=ai:xiong.jingangSummary: We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time \(T^*\). More precisely, we show that solutions are \(C^{2,\alpha}(\overline{\Omega})\) in space, with \(\alpha=\frac{1}{m} \), and \(C^{\infty}\) in time (uniformly in \(x\in\overline{\Omega} )\), for \(t>T^*\). Furthermore, this allows us to refine the asymptotics of solutions for large times, improving the best known results so far in two ways: we establish a faster rate of convergence \(O(t^{-1-\gamma})\), and we prove that the convergence holds in the \(C^{1,\alpha}(\overline{\Omega})\) topology.Well-posedness and longtime dynamics for the finitely degenerate parabolic and pseudo-parabolic equationshttps://zbmath.org/1536.351892024-07-17T13:47:05.169476Z"Liu, Gongwei"https://zbmath.org/authors/?q=ai:liu.gongwei"Tian, Shuying"https://zbmath.org/authors/?q=ai:tian.shuyingSummary: We consider the initial-boundary value problem for degenerate parabolic and pseudo-parabolic equations associated with Hörmander-type operator. Under the subcritical growth restrictions on the nonlinearity \(f(u)\), which are determined by the generalized Métivier index, we establish the global existence of solutions and the corresponding attractors. Finally, we show the upper semicontinuity of the attractors in the topology of \(H_{X,0}^1(\Omega)\).Global existence and finite-time blowup for a mixed pseudo-parabolic \(r(x)\)-Laplacian equationhttps://zbmath.org/1536.351902024-07-17T13:47:05.169476Z"Cheng, Jiazhuo"https://zbmath.org/authors/?q=ai:cheng.jiazhuo"Wang, Qiru"https://zbmath.org/authors/?q=ai:wang.qiruSummary: This article is devoted to the study of the initial boundary value problem for a mixed pseudo-parabolic \(r(x)\)-Laplacian-type equation. First, by employing the imbedding theorems, the theory of potential wells, and the Galerkin method, we establish the existence and uniqueness of global solutions with subcritical initial energy, critical initial energy, and supercritical initial energy, respectively. Then, we obtain the decay estimate of global solutions with sub-sharp-critical initial energy, sharp-critical initial energy, and supercritical initial energy, respectively. For supercritical initial energy, we also need to analyze the properties of \(\omega\)-limits of solutions. Finally, we discuss the finite-time blowup of solutions with sub-sharp-critical initial energy and sharp-critical initial energy, respectively.Initial boundary value problem of pseudo-parabolic Kirchhoff equations with logarithmic nonlinearityhttps://zbmath.org/1536.351912024-07-17T13:47:05.169476Z"Zhao, Qiuting"https://zbmath.org/authors/?q=ai:zhao.qiuting"Cao, Yang"https://zbmath.org/authors/?q=ai:cao.yang.6Summary: In this paper, we consider the initial boundary value problem for a pseudo-parabolic Kirchhoff equation with logarithmic nonlinearity. Using the potential well method, we obtain a threshold result of global existence and finite-time blow-up for the weak solutions with initial energy \(J(u_0) \leq d\). When the initial energy \(J(u_0) > d\), we find another criterion for the vanishing solution and blow-up solution. We also establish the decay rate of the global solution and estimate the life span of the blow-up solution. Meanwhile, we study the existence of the ground state solution to the corresponding stationary problem.
{\copyright} 2023 John Wiley \& Sons Ltd.Regularity and approximation of the solution of a one-sided problem for the Barenblatt-Zheltov-Kochina pseudoparabolic operatorhttps://zbmath.org/1536.351922024-07-17T13:47:05.169476Z"Sazhenkova, Tat'yana Vladimirovna"https://zbmath.org/authors/?q=ai:sazhenkova.tatyana-vladimirovna"Sazhenkov, Sergeĭ Aleksandrovich"https://zbmath.org/authors/?q=ai:sazhenkov.sergei-aleksandrovich"Sazhenkova, Elena Vladimirovna"https://zbmath.org/authors/?q=ai:sazhenkova.elena-vladimirovnaSummary: We consider a one-sided problem for the Barenblatt-Zheltov-Kochina pseudoparabolic operator in the one-dimensional case, supplemented with smooth initial data and homogeneous boundary conditions. This problem is formulated in the form of a variational inequality. From the physical point of view, it models a non-stationary process of filtration of a viscous fluid in a cracky-porous gallery with a restriction on the modulus of the velocity of filtration through the cracks. The existence theorem for a weak solution of this problem is known in the literature in both one-dimensional and multidimensional cases and follows from the results obtained by
\textit{M. Ptashnyk} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 66, No. 12, 2653--2675 (2007; Zbl 1112.35116)]
using the penalty method. In M. Ptashnyk's research, the penalty operator was chosen in a standard form, following the presentation in the monograph by
\textit{J. L. Lions} [Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod; Paris: Gauthier-Villars (1969; Zbl 0189.40603), Theorem 5.1 in Chapter 3].
In this article, we consider an approximate initial-boundary value problem for the pseudoparabolic equation incorporating Kaplan's penalty operator and study the family of its solutions. Due to the specific structure of Kaplan's operator, we obtain higher regularity of the weak solution of the original problem as compared to the previously known regularity properties, and also we find a strengthened property of approximating this solution by a sequence of solutions to the problem with Kaplan's operator. In addition, we establish that the one-sided condition imposed in the original problem is satisfied by the approximate solution on a set of the spatial variable which monotone grows with decrease of the small approximation parameter.Strongly nonlinear parabolic equations with natural growth in general domainshttps://zbmath.org/1536.351942024-07-17T13:47:05.169476Z"Azroul, Elhoussine"https://zbmath.org/authors/?q=ai:azroul.elhoussine"Khouakhi, Moussa"https://zbmath.org/authors/?q=ai:khouakhi.moussa"Masmodi, Mohamed"https://zbmath.org/authors/?q=ai:masmodi.mohamed"Yazough, Chihab"https://zbmath.org/authors/?q=ai:yazough.chihabSummary: In this paper, we deal with the existence and boundedness of solutions for nonlinear parabolic problem whose model is
\[
\begin{cases} \partial u_t -\Delta_pu +\mu \vert u\vert^{p-2}u= {\mathcal{L}}(x,t,u) &\text{in }\Omega \times (0,T), \\
u(x,t)=0 &\text{on } \partial \Omega \times (0,T),\\
u(x,0)= u_0 &\text{in } \Omega, \end{cases}
\]
where \(\Omega\) is unbounded domain, \({\mathcal{L}}(x,t,\nabla u)=d(x,t)\vert \nabla u\vert^p+f(x,t)-div \text{ } g(x,t)\), \(T\) is a positive number, \(1< p <N, d\in L^{\infty}(\Omega \times (0,T)), \Delta_pu\) is the p-Laplace operator and the lower order terms have a power growth of order \(p\) with respect to \(\nabla u\). The assumptions on the source terms lead to the existence results though with exponential integrability.Flow by powers of the Gauss curvature in space formshttps://zbmath.org/1536.351962024-07-17T13:47:05.169476Z"Chen, Min"https://zbmath.org/authors/?q=ai:chen.min.6|chen.min.3|chen.min.5|chen.min.1"Huang, Jiuzhou"https://zbmath.org/authors/?q=ai:huang.jiuzhouSummary: In this paper, we prove that convex hypersurfaces under the flow by powers \(\alpha > 0\) of the Gauss curvature in space forms \(\mathbb{N}^{n + 1}(\kappa)\) of constant sectional curvature \(\kappa\) \((\kappa = \pm 1)\) contract to a point in finite time \(T^\ast\). Moreover, convex hypersurfaces under the flow by power \(\alpha > \frac{1}{n + 2}\) of the Gauss curvature converge (after rescaling) to a limit which is the geodesic sphere in \(\mathbb{N}^{n + 1}(\kappa)\). This extends the known results in Euclidean space to space forms.Blow-up and energy decay for a class of wave equations with nonlocal Kirchhoff-type diffusion and weak dampinghttps://zbmath.org/1536.351972024-07-17T13:47:05.169476Z"Liao, Menglan"https://zbmath.org/authors/?q=ai:liao.menglan"Tan, Zhong"https://zbmath.org/authors/?q=ai:tan.zhong.1|tan.zhongSummary: The purpose of this paper is to study the following equation driven by a nonlocal integro-differential operator \(\mathcal{L}_K\):
\[
u_{tt} + [u]_s^{2(\theta -1)}\mathcal{L}_Ku + a|u_t|^{m-1}u_t = b|u|^{p-1}u,
\]
with homogeneous Dirichlet boundary condition and initial data, where \([u]_s^2\) is the Gagliardo seminorm, \(a \geq 0\), \(b > 0\), \(0 < s < 1\), and \(\theta\in[1, 2_s^\ast/2)\) with \(2_s^\ast = 2N/(N - 2s)\), \(N\) is the space dimension. By virtue of a differential inequality technique, an upper bound of the blow-up time is obtained with a bounded initial energy if \(m < p\) and some additional conditions are satisfied. For \(m\equiv 1\), in particular, the blow-up result with high initial energy also is showed by constructing a new control functional and combining energy inequalities with the concavity argument. Moreover, an estimate for the lower bound of the blow-up time is investigated. Finally, the energy decay estimate is proved as well. These results improve and complement some recent works.
{\copyright} 2023 John Wiley \& Sons, Ltd.Explicit formula of radiation fields of free waves with applications on channel of energyhttps://zbmath.org/1536.351982024-07-17T13:47:05.169476Z"Li, Liang"https://zbmath.org/authors/?q=ai:li.liang|li.liang.4"Shen, Ruipeng"https://zbmath.org/authors/?q=ai:shen.ruipeng"Wei, Lijuan"https://zbmath.org/authors/?q=ai:wei.lijuanSummary: We give a few explicit formulas regarding the radiation fields of linear free waves. We then apply these formulas on the channel-of-energy theory. We characterize all the radial weakly nonradiative solutions in all dimensions and give a few new exterior energy estimates.A mixed problem for a class of second-order nonlinear hyperbolic systems with Dirichlet and Poincaré boundary conditionshttps://zbmath.org/1536.352012024-07-17T13:47:05.169476Z"Dzhokhadze, O. M."https://zbmath.org/authors/?q=ai:dzhokhadze.otar-mikhajlovich"Kharibegashvili, S. S."https://zbmath.org/authors/?q=ai:kharibegashvili.s-s"Shavlakadze, N. N."https://zbmath.org/authors/?q=ai:shavlakadze.n-nSummary: For a certain class of second-order hyperbolic systems, a mixed problem with Dirichlet and Poincaré boundary conditions is studied. In the linear case, an explicit representation of a soultion of this problem is given and questions related to its uniqueness and existence are studied depending on the character of nonlinearities in the system.On the unique solvability of radiative transfer equations with polarizationhttps://zbmath.org/1536.352072024-07-17T13:47:05.169476Z"Bosboom, Vincent"https://zbmath.org/authors/?q=ai:bosboom.vincent"Schlottbom, Matthias"https://zbmath.org/authors/?q=ai:schlottbom.matthias"Schwenninger, Felix L."https://zbmath.org/authors/?q=ai:schwenninger.felix-lSummary: We investigate the well-posedness of the radiative transfer equation with polarization and varying refractive index. The well-posedness analysis includes non-homogeneous boundary value problems on bounded spatial domains, which requires the analysis of suitable trace spaces. Additionally, we discuss positivity, Hermiticity, and norm-preservation of the matrix-valued solution. As auxiliary results, we derive new trace inequalities for products of matrices.Stabilization of a semilinear wave equation with delayhttps://zbmath.org/1536.352152024-07-17T13:47:05.169476Z"Gonzalez Martinez, Victor Hugo"https://zbmath.org/authors/?q=ai:gonzalez-martinez.victor-hugo"Marchiori, Talita Druziani"https://zbmath.org/authors/?q=ai:marchiori.talita-druziani"de Souza Franco, Alisson Younio"https://zbmath.org/authors/?q=ai:de-souza-franco.alisson-younioSummary: We study the wellposedness and the stabilization of solutions of a semilinear wave equation with delay and locally distributed dissipation. The novelty of this paper is that we deal with the semilinear wave equation subject to delay and locally distributed damping without smallness conditions in the initial data or in the delay term. In order to address this, the argumentation requires the use of Strichartz estimates and some microlocal analysis results such as propagation of microlocal defect measures and the Gárard's linearizability property. To obtain the observability estimate in the critical case we prove a Unique Continuation Property for the semilinear wave equation and apply it to our problem. Once we establish essential observability properties for the solutions, it is not difficult to prove that the solutions decay exponentially to 0.Asymptotic property for eigenelements of the Laplace operator in a domain with an oscillating boundaryhttps://zbmath.org/1536.352252024-07-17T13:47:05.169476Z"Jaouabi, Ahlem"https://zbmath.org/authors/?q=ai:jaouabi.ahlem"Khelifi, Abdessatar"https://zbmath.org/authors/?q=ai:khelifi.abdessatarSummary: This work deals with the variations of eigenvalues and eigenfunctions for the Laplace operator in a bounded domain of \(\mathbf{R}^2\) with Dirichlet boundary conditions, a part of which boundary, depending on a small parameter \(\epsilon\), is rapidly oscillating. By using surface potentials we show that the eigenvalues are exactly built with the characteristic values of meromorphic operator-valued functions that are of Fredholm type with index 0. Then, we proceed from the generalized Rouché's theorem to study the splitting problem.
{\copyright} 2023 Wiley-VCH GmbH.Multiple tubular excisions and large Steklov eigenvalueshttps://zbmath.org/1536.352282024-07-17T13:47:05.169476Z"Brisson, Jade"https://zbmath.org/authors/?q=ai:brisson.jadeSummary: Given a closed Riemannian manifold \(M\) and \(b\ge 2\) closed connected submanifolds \(N_j\subset M\) of codimension at least 2, we prove that the first nonzero eigenvalue of the domain \(\Omega_\varepsilon \subset M\) obtained by removing the tubular neighbourhood of size \(\varepsilon\) around each \(N_j\) tends to infinity as \(\varepsilon\) tends to 0. More precisely, we prove a lower bound in terms of \(\varepsilon \), \(b\), the geometry of \(M\) and the codimensions and the volumes of the submanifolds and an upper bound in terms of \(\varepsilon\) and the codimensions of the submanifolds. For eigenvalues of index \(k=b,b+1,\ldots \), we have a stronger result: their order of divergence is \(\varepsilon^{-1}\) and their rate of divergence is only depending on \(m\) and on the codimensions of the submanifolds.Some weighted fourth-order Hardy-Hénon equationshttps://zbmath.org/1536.352322024-07-17T13:47:05.169476Z"Deng, Shengbing"https://zbmath.org/authors/?q=ai:deng.shengbing"Tian, Xingliang"https://zbmath.org/authors/?q=ai:tian.xingliangSummary: By using a suitable transform related to Sobolev inequality, we investigate the sharp constants and optimizers in radial space for the following weighted Caffarelli-Kohn-Nirenberg-type inequalities:
\[
\begin{aligned}
&\int_{\mathbb{R}^N} | x |^\alpha | \Delta u |^2 d x \geq S^{r a d}(N, \alpha) \left( \int_{\mathbb{R}^N} | x |^{- \alpha} | u |^{p_\alpha^\ast}\, d x \right)^{\frac{ 2}{ p_\alpha^\ast}}, \\
&u \in C_c^\infty( \mathbb{R}^N),
\end{aligned}
\] where \(N \geq 3\), \(4 - N < \alpha < 2\), \(p_\alpha^\ast = \frac{ 2 ( N - \alpha )}{ N - 4 + \alpha} \). Then we obtain the explicit form of the unique (up to scaling) radial positive solution \(U_{\lambda , \alpha}\) to the weighted fourth-order Hardy (for \(\alpha > 0\)) or Hénon (for \(\alpha < 0\)) equation:
\[
\Delta(| x |^\alpha \Delta u) = | x |^{- \alpha} u^{p_\alpha^\ast - 1}, \quad u > 0 \text{ in } \mathbb{R}^N.
\]
For \(\alpha \neq 0\), it is known the solutions of above equation are invariant for dilations \(\lambda^{\frac{ N - 4 + \alpha}{ 2}} u(\lambda x)\) but not for translations. However we show that if \(\alpha\) is a negative even integer, there exist new solutions to the linearized problem, which related to above equation at \(U_{1 , \alpha} \), that ``replace'' the ones due to the translations invariance. This interesting phenomenon was first shown by \textit{F. Gladiali} et al. [Adv. Math. 249, 1--36 (2013; Zbl 1335.35077)] for the second-order Hénon problem. Finally, as applications, we investigate the remainder term of above inequality and also the existence of solutions to some related perturbed equations.Construction of Boltzmann and McKean-Vlasov type flows (the sewing lemma approach)https://zbmath.org/1536.352342024-07-17T13:47:05.169476Z"Alfonsi, Aurélien"https://zbmath.org/authors/?q=ai:alfonsi.aurelien"Bally, Vlad"https://zbmath.org/authors/?q=ai:bally.vladSummary: We are concerned with a mixture of Boltzmann and McKean-Vlasov-type equations, this means (in probabilistic terms) equations with coefficients depending on the law of the solution itself, and driven by a Poisson point measure with the intensity depending also on the law of the solution. Both the analytical Boltzmann equation and the probabilistic interpretation initiated by
\textit{H. Tanaka} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 46, 67--105 (1978; Zbl 0389.60079); J. Fac. Sci., Univ. Tokyo, Sect. I A 34, 351--369 (1987; Zbl 0639.60105)]
have intensively been discussed in the literature for specific models related to the behavior of gas molecules. In this paper, we consider general abstract coefficients that may include mean field effects and then we discuss the link with specific models as well. In contrast with the usual approach in which integral equations are used in order to state the problem, we employ here a new formulation of the problem in terms of flows of self-maps on the space of probability measure endowed with the Wasserstein distance. This point of view already appeared in the framework of rough differential equations. Our results concern existence and uniqueness of the solution, in the formulation of flows, but we also prove that the ``flow solution'' is a solution of the classical integral weak equation and admits a probabilistic interpretation. Moreover, we obtain stability results and regularity with respect to the time for such solutions. Finally we prove the convergence of empirical measures based on particle systems to the solution of our problem, and we obtain the rate of convergence. We discuss as examples the homogeneous and the inhomogeneous Boltzmann (Enskog) equation with hard potentials.Global existence for an isotropic modification of the Boltzmann equationhttps://zbmath.org/1536.352352024-07-17T13:47:05.169476Z"Snelson, Stanley"https://zbmath.org/authors/?q=ai:snelson.stanleySummary: Motivated by the open problem of large-data global existence for the non-cutoff Boltzmann equation, we introduce a model equation that in some sense disregards the anisotropy of the Boltzmann collision kernel. We refer to this model equation as \textit{isotropic Boltzmann} by analogy with the isotropic Landau equation introduced by \textit{J. Krieger} and \textit{R. M. Strain} [Commun. Partial Differ. Equations 37, No. 4--6, 647--689 (2012; Zbl 1247.35087)]. The collision operator of our isotropic Boltzmann model converges to the isotropic Landau collision operator under a scaling limit that is analogous to the grazing collisions limit connecting (true) Boltzmann with (true) Landau.
Our main result is global existence for the isotropic Boltzmann equation in the space homogeneous case, for certain parts of the ``very soft potentials'' regime in which global existence is unknown for the space homogeneous Boltzmann equation. The proof strategy is inspired by the work of \textit{M. Gualdani} and \textit{N. Guillen} [J. Funct. Anal. 283, No. 6, Article ID 109559, 25 p. (2022; Zbl 1492.35064)] on isotropic Landau, and makes use of recent progress on weighted fractional Hardy inequalities.Spectrum structure and solution behavior of the Boltzmann equation with soft potentialshttps://zbmath.org/1536.352362024-07-17T13:47:05.169476Z"Yang, Tong"https://zbmath.org/authors/?q=ai:yang.tong"Yu, Hongjun"https://zbmath.org/authors/?q=ai:yu.hongjunSummary: We first analyze the spectrum structure of the Boltzmann equation for the whole range of soft potentials by deducing some new estimates on its related linear operators. With the estimates on the semigroup generated by the linearized Boltzmann operator with spatial frequency bounded away zero, we obtain the existence and large-time behavior of perturbative solutions in a torus. In addition, we deduce the time decay rate of the semigroup near zero spatial frequency and obtain the existence and the optimal time decay rate of perturbative solutions in the whole space. These results give a complete description of the spectrum structure and solution behavior of the perturbative solutions to the Boltzmann equation for soft potentials that generalize the classical results by
\textit{R. E. Caflisch} [Commun. Math. Phys. 74, 71--95 (1980; Zbl 0434.76065); ibid. 74, 97--109 (1980; Zbl 0434.76066)] and
\textit{S. Ukai} and \textit{K. Asano} [Publ. Res. Inst. Math. Sci. 18, 477--519 (1982; Zbl 0538.45011)]
in a torus and whole space, respectively.A geometric trapping approach to global regularity for 2D Navier-Stokes on manifoldshttps://zbmath.org/1536.352392024-07-17T13:47:05.169476Z"Bulut, Aynur"https://zbmath.org/authors/?q=ai:bulut.aynur"Khang, Manh Huynh"https://zbmath.org/authors/?q=ai:khang.manh-huynhSummary: In this paper, we use frequency decomposition techniques to give a direct proof of global existence and regularity for the Navier-Stokes equations on two-dimensional Riemannian manifolds without boundary. Our techniques are inspired by an approach of \textit{J. C. Mattingly} and \textit{Ya. G. Sinai} [Commun. Contemp. Math. 1, No. 4, 497--516 (1999; Zbl 0961.35112)] which was developed in the context of periodic boundary conditions on a flat background, and which is based on a maximum principle for Fourier coefficients.
The extension to general manifolds requires several new ideas, connected to the less favorable spectral localization properties in our setting. Our arguments make use of frequency projection operators, multilinear estimates that originated in the study of the non-linear Schrödinger equation, and ideas from microlocal analysis.A generalized blow up criteria with one component of velocity for 3D incompressible MHD systemhttps://zbmath.org/1536.352412024-07-17T13:47:05.169476Z"Han, Bin"https://zbmath.org/authors/?q=ai:han.bin|han.bin.1"Xiong, Xi"https://zbmath.org/authors/?q=ai:xiong.xiSummary: In this paper, the authors study the global regularity of the 3D magnetohydrodynamics system in terms of one velocity component. In particular, they establish a new Prodi-Serrin type regularity criterion in the framework of weak Lebesgue spaces both in time and space variables.On the Leray problem for steady flows in two-dimensional infinitely long channels with slip boundary conditionshttps://zbmath.org/1536.352442024-07-17T13:47:05.169476Z"Sha, Kaijian"https://zbmath.org/authors/?q=ai:sha.kaijian"Wang, Yun"https://zbmath.org/authors/?q=ai:wang.yun.3"Xie, Chunjing"https://zbmath.org/authors/?q=ai:xie.chunjingSummary: In this paper, we investigate the Leray problem for steady Navier-Stokes system with full slip boundary conditions in a two-dimensional channel with straight outlets. The existence of solutions with arbitrary flux in a general channel supplemented with slip boundary conditions, which tend to the associated shear flows at far fields, is established. Furthermore, if the flux is suitably small, the solution is proved to be unique. One of the crucial ingredients is to construct an appropriate flux carrier and to show a Hardy type inequality for flows with full slip boundary conditions.Global solutions of 3D isentropic compressible Navier-Stokes equations with two slow variableshttps://zbmath.org/1536.352452024-07-17T13:47:05.169476Z"Yang, NanNan"https://zbmath.org/authors/?q=ai:yang.nannanSummary: Motivated by \textit{Y. Lu} and \textit{P. Zhang} [J. Differ. Equations 376, 406--468 (2023; Zbl 1527.35215)], we prove the global existence of solutions to the three-dimensional isentropic compressible Navier-Stokes equations with smooth initial data slowly varying in two directions. In such a setting, the \(L^2\)-norms of the initial data are of order \(O(\varepsilon^{-1})\), which are large.Existence of homogeneous Euler flows of degree \(-\alpha \notin [-2, 0]\)https://zbmath.org/1536.352462024-07-17T13:47:05.169476Z"Abe, Ken"https://zbmath.org/authors/?q=ai:abe.kenSummary: We consider (\(-\alpha\))-homogeneous solutions to the stationary incompressible Euler equations in \(\mathbb{R}^{3} \backslash \{0\}\) for \(\alpha \geqq 0\) and in \(\mathbb{R}^{3}\) for \(\alpha < 0\).
\textit{R. Shvydkoy} [Trans. Am. Math. Soc. 370, No. 4, 2517--2535 (2018; Zbl 1386.35333)]
demonstrated the \textit{nonexistence} of (\(-1\))-homogeneous solutions \((u, p) \in C^{1} (\mathbb{R}^{3} \backslash \{0\})\) and (\(-\alpha\))-homogeneous solutions in the range \(0 \leqq \alpha \leqq 2\) for the Beltrami and axisymmetric flow; namely, that no (\(-\alpha\))-homogeneous solutions \((u, p) \in C^{1} (\mathbb{R}^{3} \backslash \{0\})\) for \(1 \leqq \alpha \leqq 2\) and \((u, p) \in C^{2} (\mathbb{R}^{3} \backslash \{0\})\) for \(0 \leqq \alpha < 1\) exist among these particular classes of flows other than irrotational solutions for integers \(\alpha\). The nonexistence result of the Beltrami (\(-\alpha\))-homogeneous solutions \((u, p) \in C^{2} (\mathbb{R}^{3} \backslash \{0\})\) holds for all \(\alpha < 1\). We show the nonexistence of axisymmetric (\(-\alpha\))-homogeneous solutions without swirls \((u, p) \in C^{2} (\mathbb{R}^{3} \backslash \{0\})\) for \(-2 \leqq \alpha < 0\). The main result of this study is the \textit{existence} of axisymmetric (\(-\alpha\))-homogeneous solutions in the complementary range \(\alpha \in \mathbb{R} \backslash[0, 2]\). More specifically, we show the existence of axisymmetric Beltrami (\(-\alpha\))-homogeneous solutions \((u, p) \in C^{1} (\mathbb{R}^{3} \backslash \{0\})\) for \(\alpha > 2\) and \((u, p) \in C (\mathbb{R}^3)\) for \(\alpha < 0\) and axisymmetric (\(-\alpha\))-homogeneous solutions with a nonconstant Bernoulli function \((u, p) \in C^{1} (\mathbb{R}^{3} \backslash \{0\})\) for \(\alpha > 2\) and \((u, p) \in C (\mathbb{R}^3)\) for \(\alpha < -2\), including axisymmetric (\(-\alpha\))-homogeneous solutions without swirls \((u, p) \in C^{2} (\mathbb{R}^{3} \backslash \{0\})\) for \(\alpha > 2\) and \((u, p) \in C^{1}(\mathbb{R}^{3} \backslash \{0\}) \cap C (\mathbb{R}^3)\) for \(\alpha < -2\). This is the first existence result on (\(-\alpha\))-homogeneous solutions with no explicit forms. The level sets of the axisymmetric stream function of the irrotational (\(-\alpha\))-homogeneous solutions in the cross-section are the Jordan curves for \(\alpha = 3\). For \(2 < \alpha < 3\), we show the existence of axisymmetric (\(-\alpha\))-homogeneous solutions whose stream function level sets are the Jordan curves. They provide new examples of the Beltrami/Euler flows in \(\mathbb{R}^{3} \backslash \{0\}\) whose level sets of the proportionality factor/Bernoulli surfaces are nested surfaces created by the rotation of the sign ``\(\infty\)''.Construction of the free-boundary 3D incompressible Euler flow under limited regularityhttps://zbmath.org/1536.352472024-07-17T13:47:05.169476Z"Aydin, Mustafa Sencer"https://zbmath.org/authors/?q=ai:aydin.mustafa-sencer.1"Kukavica, Igor"https://zbmath.org/authors/?q=ai:kukavica.igor"Ożański, Wojciech S."https://zbmath.org/authors/?q=ai:ozanski.wojciech-s"Tuffaha, Amjad"https://zbmath.org/authors/?q=ai:tuffaha.amjad-mSummary: We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. In the Lagrangian setting, we construct a unique local-in-time solution for \(u_0 \in H^{2.5 + \delta}\) such that the Rayleigh-Taylor condition holds and \(\operatorname{curl} u_0 \in H^{2 + \delta}\) in an arbitrarily small neighborhood of the free boundary. We show that the result is optimal in the sense that \(H^{3 + \delta}\) regularity of the Lagrangian deformation near the free boundary can be ensured if and only if the initial vorticity has \(H^{2 + \delta}\) regularity near the free boundary.\(C^\gamma\) well-posedness of some non-linear transport equationshttps://zbmath.org/1536.352492024-07-17T13:47:05.169476Z"Cantero, J. C."https://zbmath.org/authors/?q=ai:cantero.juan-carlosSummary: Given \(k:\mathbb{R}^n\setminus\{0\}\to\mathbb{R}^n\) a kernel of class \(C^2\) and homogeneous of degree \(1 - n\), we prove global-in-time existence and uniqueness of Hölder regular solutions for some non-linear transport equations with velocity fields given by convolution of the density with \(k\). The 3D quasi geostrophic and the 2D Euler equations can be recovered for particular choices of \(k\).Self-similar finite-time blowups with smooth profiles of the generalized Constantin-Lax-Majda modelhttps://zbmath.org/1536.352502024-07-17T13:47:05.169476Z"Huang, De"https://zbmath.org/authors/?q=ai:huang.de"Qin, Xiang"https://zbmath.org/authors/?q=ai:qin.xiang"Wang, Xiuyuan"https://zbmath.org/authors/?q=ai:wang.xiuyuan.1"Wei, Dongyi"https://zbmath.org/authors/?q=ai:wei.dongyiSummary: We show that the \(a\)-parameterized family of the generalized Constantin-Lax-Majda model, also known as the Okamoto-Sakajo-Wunsch model, admits exact self-similar finite-time blowup solutions with interiorly smooth profiles for all \(a \leq 1\). Depending on the value of \(a\), these self-similar profiles are either smooth on the whole real line or compactly supported and smooth in the interior of their closed supports. The existence of these profiles is proved in a consistent way by considering the fixed-point problem of an \(a\)-dependent nonlinear map, based on which detailed characterizations of their regularity, monotonicity, and far-field decay rates are established. Our work unifies existing results for some discrete values of \(a\) and also explains previous numerical observations for a wide range of \(a\).Degenerate Cauchy-Goursat problem for 2-D steady isentropic Euler system with van der Waals gashttps://zbmath.org/1536.352512024-07-17T13:47:05.169476Z"Srivastava, H."https://zbmath.org/authors/?q=ai:srivastava.h-h|srivastava.h-s-p|srivastava.harish|srivastava.hari-mohan|srivastava.h-k"Zafar, M."https://zbmath.org/authors/?q=ai:zafar.m-khurram|zafar.madiha|zafar.m-asim|zafar.mudasar|zafar.md-nadim|zafar.muhammad-naveed|zafar.muhammad-i|zafar.muhammad-bilal|zafar.m-asim.1Summary: This study concerns with the existence-uniqueness of local classical sonic-supersonic solution to a degenerate Cauchy-Goursat problem that arises in transonic phenomena. The flow is governed by 2-D steady isentropic Euler system with a polytropic van der Waals gas. The idea of characteristic decomposition has been used to convert the Euler system into a new system involving the angle variables \((\varTheta, \mu )\). To overcome the parabolic degeneracy caused at the sonic curve, the partial hodograph transformation and a variety of dependent-independent variables have been introduced to transform the nonlinear system into a linear one with explicit singularity-regularity structure. The uniform convergence of the sequences \((W^{(m)},Z^{(m)})\) has been discussed by employing the mathematical induction. Eventually, the inversion of the solution from partial hodograph plane to the original plane has been established.
{\copyright} 2023 Wiley Periodicals LLC.The convexity for compressible subsonic cavity flowshttps://zbmath.org/1536.352522024-07-17T13:47:05.169476Z"Wang, Xiaohui"https://zbmath.org/authors/?q=ai:wang.xiaohuiSummary: In this paper, we first show that the flow velocity takes its maximum on the free boundary, provided that the nozzle wall and obstacle satisfy some corresponding geometric assumptions. Second, the convexity of the free boundary to the compressible subsonic cavity flows will be established. Finally, the optimal regularity of the free boundary at the detachment point is obtained.
{\copyright} 2023 Wiley Periodicals LLC.Long-time solvability for the 2D inviscid Boussinesq equations with borderline regularity and dispersive effectshttps://zbmath.org/1536.352532024-07-17T13:47:05.169476Z"Angulo-Castillo, V."https://zbmath.org/authors/?q=ai:angulo-castillo.vladimir"Ferreira, L. C. F."https://zbmath.org/authors/?q=ai:ferreira.lucas-catao-de-freitas"Kosloff, L."https://zbmath.org/authors/?q=ai:kosloff.leonardoSummary: We are concerned with the long-time solvability for 2D inviscid Boussinesq equations for a larger class of initial data which covers the case of borderline regularity. First we show the local solvability in Besov spaces uniformly with respect to a parameter \(\kappa\) associated with the stratification of the fluid. Afterwards, employing a blow-up criterion and Strichartz-type estimates, the long-time solvability is obtained for large \(\kappa\) regardless of the size of initial data.Relative energy and weak-strong uniqueness of a two-phase viscoelastic phase separation modelhttps://zbmath.org/1536.352552024-07-17T13:47:05.169476Z"Brunk, Aaron"https://zbmath.org/authors/?q=ai:brunk.aaron"Lukáčová-Medvid'ová, Mária"https://zbmath.org/authors/?q=ai:lukacova-medvidova.mariaSummary: The aim of this paper is to analyze a viscoelastic phase separation model. We derive a suitable notion of the relative energy taking into account the nonconvex nature of the energy law for the viscoelastic phase separation. This allows us to prove the weak-strong uniqueness principle. We will provide the estimates for the full model in two space dimensions. For a reduced model, we present the estimates in three space dimensions and derive conditional relative energy estimates.
{\copyright} 2022 The Authors. \textit{ZAMM - Journal of Applied Mathematics and Mechanics} published by Wiley-VCH GmbH.Some remarks about the stationary micropolar fluid equations: existence, regularity and uniquenesshttps://zbmath.org/1536.352562024-07-17T13:47:05.169476Z"Chamorro, Diego"https://zbmath.org/authors/?q=ai:chamorro.diego"Llerena, David"https://zbmath.org/authors/?q=ai:llerena.david"Vergara-Hermosilla, Gastón"https://zbmath.org/authors/?q=ai:vergara-hermosilla.gastonSummary: We consider here the stationary micropolar fluid equations, which are a special generalization of the usual Navier-Stokes system where the microrotations of the fluid particles must be taken into account. We thus obtain two coupled equations: one mainly based on the velocity field \(\overrightarrow{u}\) and the other on the microrotation field \(\overrightarrow{\omega} \). In this work we will study some problems related to the existence of weak solutions as well as some regularity and uniqueness properties. Our main result establishes the uniqueness of the trivial solution under some suitable infinity decay conditions for the velocity field only.Global finite-energy solutions of the compressible Euler-Poisson equations for general pressure laws with large initial data of spherical symmetryhttps://zbmath.org/1536.352582024-07-17T13:47:05.169476Z"Chen, Gui-Qiang G."https://zbmath.org/authors/?q=ai:chen.gui-qiang-g"Huang, Feimin"https://zbmath.org/authors/?q=ai:huang.feimin"Li, Tianhong"https://zbmath.org/authors/?q=ai:li.tianhong"Wang, Weiqiang"https://zbmath.org/authors/?q=ai:wang.weiqiang.2"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.7Summary: We are concerned with global finite-energy solutions of the three-dimensional compressible Euler-Poisson equations with \textit{gravitational potential} and \textit{general pressure law}, especially including the constitutive equation of \textit{white dwarf stars}. In this paper, we construct global finite-energy solutions of the Cauchy problem for the Euler-Poisson equations with large initial data of spherical symmetry as the inviscid limit of the solutions of the corresponding Cauchy problem for the compressible Navier-Stokes-Poisson equations. The strong convergence of the vanishing viscosity solutions is achieved through entropy analysis, uniform estimates in \(L^p\), and a more general compensated compactness framework via several new ingredients. A key estimate is first established for the integrability of the density over unbounded domains independent of the vanishing viscosity coefficient. Then a special entropy pair is carefully designed via solving a Goursat problem for the entropy equation such that a higher integrability of the velocity is established, which is a crucial step. Moreover, the weak entropy kernel for the general pressure law and its fractional derivatives of the required order near vacuum \((\rho =0)\) and far-field \((\rho =\infty)\) are carefully analyzed. Owing to the generality of the pressure law, only the \(W^{-1,p}_{\text{loc}}\)-compactness of weak entropy dissipation measures with \(p \in [1,2)\) can be obtained; this is rescued by the equi-integrability of weak entropy pairs which can be established by the estimates obtained above, so that the div-curl lemma still applies. Finally, based on the above analysis of weak entropy pairs, the \(L^p\) compensated compactness framework for the compressible Euler equations with general pressure law is established. This new compensated compactness framework and the techniques developed in this paper should be useful for solving further nonlinear problems with similar features.Evolution of initial discontinuities in a particular case of two-step initial problem for the defocusing complex modified KdV equationhttps://zbmath.org/1536.352592024-07-17T13:47:05.169476Z"Chen, Jing"https://zbmath.org/authors/?q=ai:chen.jing.14"Zhou, Ao"https://zbmath.org/authors/?q=ai:zhou.ao"Xue, Yushan"https://zbmath.org/authors/?q=ai:xue.yushanSummary: In this paper, the complete classification of solutions of defocusing complex modified KdV equation with a particular case of two-step initial condition is investigated by the finite-gap integration approach and Whitham modulation theory. The periodic wave solution and corresponding Whitham modulation equations for zero-phase, one-phase, two-phase are found. The self-similar wave structures in each case are composed of different building blocks: the plateau, vacuum, rarefaction wave, dispersive shock wave and two-genus region, which are studied in detail. The results of direct numerical simulations for the defocusing complex modified KdV equation agree well with those of Whitham modulation theory.Space-time decay rate for strong solutions to the viscous liquid-gas two-phase flow with magnetic fieldhttps://zbmath.org/1536.352602024-07-17T13:47:05.169476Z"Chen, Yuang"https://zbmath.org/authors/?q=ai:chen.yuang"Luo, Zhengyan"https://zbmath.org/authors/?q=ai:luo.zhengyan"Zhang, Yinghui"https://zbmath.org/authors/?q=ai:zhang.yinghuiSummary: We investigate the viscous liquid-gas two-phase model with magnetic field in a weighted Sobolev space \(H^2_a (\mathbb{R}^3)\). Based on precise weighted energy estimation, we establish the space-time decay rates of the \(k(\in [0, 2])\)th order spatial derivative of strong solutions. The main difficulty comes from the lack of dissipative structure of \(\|\nabla P\|_{L^2_a}^2 \), and we need to construct an interactive weighted energy functional to solve it.Anomalous dissipation and lack of selection in the Obukhov-Corrsin theory of scalar turbulencehttps://zbmath.org/1536.352612024-07-17T13:47:05.169476Z"Colombo, Maria"https://zbmath.org/authors/?q=ai:colombo.maria"Crippa, Gianluca"https://zbmath.org/authors/?q=ai:crippa.gianluca"Sorella, Massimo"https://zbmath.org/authors/?q=ai:sorella.massimoThis paper studies anomalous dissipation and non-uniqueness phenomena in linear transport theory, for the equation
\[
\partial_t \theta + u \cdot \nabla \theta = 0 \quad \text{ in } \quad ]0,T[ \times \mathbb{T}^2 \, , \tag{1}
\]
posed for a scalar variable \(\theta:]0,T[\times \mathbb{T}^2\rightarrow \mathbb{R}\) and the velocity field \(u:]0,T[\times \mathbb{T}^2\rightarrow \mathbb{R}^2\), where \(\mathbb{T}^2 \) denotes the two-dimensional torus and \((0,T)\) is the time-interval. Periodic spacial extension is assumed and \(\theta\) is additionally subject to the initial condition \(\theta = \theta_{\mathrm{in}}\) at initial time. The velocity field is assumed to be Hölderian in space and, according to the context, either \(p\)-integrable or Hölderian in time, this means \(u \in L^p(]0,T[ ; C^{\alpha}(\mathbb{T}^2))\) or \(u \in C^{\alpha}(]0,T[\times \mathbb{T}^2)\). Moreover, it ought to be divergence-free in the weak sense. An important concern of the paper is to study in connection with (1) the asymptotic behaviour of the vanishing diffusivity solutions \(\theta = \theta_{\kappa}\) to the parabolic equation
\[
\partial_t \theta + u \cdot \nabla \theta = \kappa \, \Delta \theta \quad \text{ in } \quad ]0,T[ \times \mathbb{T}^2 \, , \tag{2}
\]
subject to periodic boundary conditions and the initial condition \(\theta_{\mathrm{in}}\).
In a first Theorem A, the authors show that for \(p \geq 2\) and \(0 \leq \alpha <1\), there exist a velocity field \(u \in L^p(]0,T[ ; C^{\alpha}(\mathbb{T}^2))\) and an initial value \(\theta_{\mathrm{in}} \in C^{\infty}(\mathbb{T}^2)\) such that with \(p^{\circ} = 2p/(p-1)\) and any \(0 \leq \beta < 1/2\) satisfying \(\alpha + 2\beta < 1\), the problem (1) admits a solution \(\theta \in L^{p^{\circ}}(0,T; \, C^{\beta}(\mathbb{T}^2))\) that strictly dissipates the \(L^2\)-norm, in the sense that \(\|\theta(T,\cdot)\|_{L^2(\mathbb{T}^2)} < \|\theta_{\mathrm{in}}\|_{L^2(\mathbb{T}^2)}\). This solution can be gained as the limit of the sequence \(\{\theta_{\kappa}\}\) of solutions to (2) that exhibit the phenomenon anomalous dissipation:
\[
\limsup_{\kappa \rightarrow 0} \kappa \, \int_0^T\int_{\mathbb{T}^2} |\nabla \theta_{\kappa}|^2 \, dxdt > 0 \, .
\]
In a second surprising Theorem B, the authors moreover show that for every \(\alpha \in [0,1[\) there are a velocity field \(u \in C^{\alpha}(]0,T[ \times \mathbb{T}^2)\) and initial data \(\theta_{\mathrm{in}} \in C^{\infty}(\mathbb{T}^2)\) such that the problem (1) possesses at least two solutions, the one of which conserves the \(L^2\)-norm, while the other will not. Both solutions are accumulation points of the sequence \(\{\theta_{\kappa}\}\) of solutions to (2), the first without and the second exhibiting anomalous dissipation. In this sense, the vanishing diffusion limit would not provide in its own a valid selection criterium for (1).
The theorems rely on concrete, though highly technical, construction techniques for the singular velocity field and the initial data, while the convergence analysis is based on the introduction of an appropriate stochastic flow and the associated Feynman-Kac representation of the solution to (2).
Reviewer: Pierre-Étienne Druet (Darmstadt)Asymptotic behavior of solutions to stochastic 3D globally modified Navier-Stokes equations with unbounded delayshttps://zbmath.org/1536.352622024-07-17T13:47:05.169476Z"Cung The Anh"https://zbmath.org/authors/?q=ai:cung-the-anh."Vu Manh Toi"https://zbmath.org/authors/?q=ai:vu-manh-toi."Phan Thi Tuyet"https://zbmath.org/authors/?q=ai:phan-thi-tuyet.Summary: This paper studies the existence of weak solutions and the stability of stationary solutions to stochastic 3D globally modified Navier-Stokes equations with unbounded delays in the phase space \(BCL_{-\infty}(H)\). We first prove the existence and uniqueness of weak solutions by using the classical technique of Galerkin approximations. Then we study stability properties of stationary solutions by using several approach methods. In the case of proportional delays, some sufficient conditions ensuring the polynomial stability in both mean square and almost sure senses will be provided.Calculation of Reynolds equation for the generalized non-Newtonian fluids and its asymptotic behavior in a thin domainhttps://zbmath.org/1536.352632024-07-17T13:47:05.169476Z"Dilmi, Mohamed"https://zbmath.org/authors/?q=ai:dilmi.mohamed"Benseghir, Aissa"https://zbmath.org/authors/?q=ai:benseghir.aissa"Dilmi, Mourad"https://zbmath.org/authors/?q=ai:dilmi.mourad"Benseridi, Hamid"https://zbmath.org/authors/?q=ai:benseridi.hamidSummary: Three-dimensional boundary-value problem describing a generalized non-Newtonian fluid with nonlinear Tresca friction type in a thin domain \(\Omega^{\varepsilon}\) are considered. We study the asymptotic behavior when one dimension of the fluid domain tends to zero. We prove some weak convergence of the velocity and the pressure of the fluid. Then the limit problem in two-dimensional domain and the specific Reynolds equation are obtained.The Clausius-Mossotti formulahttps://zbmath.org/1536.352652024-07-17T13:47:05.169476Z"Duerinckx, Mitia"https://zbmath.org/authors/?q=ai:duerinckx.mitia"Gloria, Antoine"https://zbmath.org/authors/?q=ai:gloria.antoineThis paper provides a proof of the Clausius-Mossotti formula for the effective conductivity in the dilute regime; here inclusions are allowed to overlap and that no upper bound is assumed on the number of points per unit volume. It is assumed that the shape of inclusions is spherical and the underlying point process assumes stationarity and ergodicity. The paper provides an estimate for the error that mostly comes from `interactions' between inclusions; the estimate relies on elliptic regularity theory. The approach is based on a short proof that the authors provided for Einstein's formula in a separate publication. This is a well-written paper with several details and it should be useful for someone working on dilute systems in many other contexts.
Reviewer: Vishnu Dutt Sharma (Ghandinagar)Global solutions for the incompressible Hall-magnetohydrodynamics system around constant equilibrium stateshttps://zbmath.org/1536.352682024-07-17T13:47:05.169476Z"Fujii, Mikihiro"https://zbmath.org/authors/?q=ai:fujii.mikihiro"Nakasato, Ryosuke"https://zbmath.org/authors/?q=ai:nakasato.ryosukeSummary: We consider the initial value problem for the magnetohydrodynamics system with the Hall effect, around a constant equilibrium state \((u, B) = (0, \overline{B}) \in \mathbb{R}^3 \times \mathbb{R}^3\). Then, the perturbed system possesses the dispersive nature due to the constant magnetic field \(\overline{B} \). The aim of this paper is to control the dispersion and prove the well-posedness of the perturbed system. To this end, we consider scaling critical function spaces of Besov type based on the Fourier-Lebesgue spaces \(\widehat{L^p}( \mathbb{R}^3)\), namely critical Fourier-Besov spaces, and show the global well-posedness for all \(1 \leqslant p \leqslant \infty \). Moreover, we focus on the weak dispersive case \(| \overline{B} | \ll 1\) and the strong dispersive case \(| \overline{B} | \gg 1\). In the former case, the dispersion is negligible and we show the well-posedness in the scaling critical Besov spaces based on the usual Lebesgue space \(L^p\). For the latter case, we prove that the global unique solution is established even for arbitrarily large data in the scaling critical \(L^2\)-Besov spaces by virtue of the Strichartz type estimates for the linear solutions.Optimal lower bound for the blow-up rate of the magnetic Zakharov system without the skin effecthttps://zbmath.org/1536.352692024-07-17T13:47:05.169476Z"Gan, Zaihui"https://zbmath.org/authors/?q=ai:gan.zaihui"Wang, Yuchen"https://zbmath.org/authors/?q=ai:wang.yuchen"Wang, Yue"https://zbmath.org/authors/?q=ai:wang.yue.4|wang.yue.8|wang.yue.3|wang.yue.1|wang.yue.15|wang.yue.6|wang.yue.10|wang.yue.11|wang.yue.7|wang.yue.2|wang.yue.9"Yu, Jialing"https://zbmath.org/authors/?q=ai:yu.jialingSummary: We focus on the following Cauchy problem of the magnetic Zakharov system in two-dimensional space:
\[
\begin{cases}
iE_{1t} + \Delta E_{1} - nE_{1} + \eta E_{2} \left( E_{1} \overline{E_{2}} - \overline{E_{1}} E_{2} \right) = 0, \\
iE_{2t} + \Delta E_{2} - nE_{2} + \eta E_{1} \left( \overline{E_{1}} E_{2} - E_{1} \overline{E_{2}} \right) = 0, \\
n_{t} + \nabla \cdot \mathbf{v} = 0, \\
\mathbf{v}_{t} + \nabla n + \nabla \left( |E_{1}|^{2} + |E_{2}|^{2} \right) = 0,
\tag{G-Z}
\end{cases}
\]
\[
(E_{1}, E_{2}, n, \mathbf{v}) (0, x) = (E_{10}, E_{20}, n_{0}, \mathbf{v}_{0}) (x).
\tag{G-Z-I}
\]
System (G-Z) describes the spontaneous generation of a magnetic field without the skin effect in a cold plasma, and \(\eta > 0\) is the magnetic coefficient. The nonlinear cubic coupling terms \(E_{2} \left( E_{1}\overline{E_{2}} - \overline{E_{1}} E_{2} \right)\) and \(E_{1} \left( \overline{E_{1}} E_{2} - E_{1} \overline{E_{2}} \right)\) generated by the cold magnetic field bring additional difficulties compared with the classical Zakharov system. For when the initial mass meets a presettable condition
\[
\frac{||Q||_{L^{2}(\mathbb{R}^{2})}^{2}}{1 + \eta} < ||E_{10}||_{L^{2}(\mathbb{R}^{2})}^{2} + ||E_{20}||_{L^{2}(\mathbb{R}^{2})}^{2} < \frac{||Q||_{L^{2}(\mathbb{R}^{2})}^{2}}{\eta},
\]
where \(Q\) is the unique radially positive solution of the equation\(-\Delta V + V = V^{3}\), we prove that there is a constant \(c > 0\) depending only on the initial data such that for \(t\) near \(T\) (the blow-up time),
\[
\| \left( E_{1}, E_{2}, n, \mathbf{v} \right) \|_{H^{1}(\mathbb{R}^{2}) \times H^{1}(\mathbb{R}^{2}) \times L^{2}(\mathbb{R}^{2}) \times L^{2}(\mathbb{R}^{2})} \geqslant \frac{c}{T - t}.
\]
As the magnetic coefficient \(\eta\) tends to 0, the blow-up rate recovers the result for the classical 2-D Zakharov system due to \textit{F. Merle} [Commun. Pure Appl. Math. 49, No. 8, 765--794 (1996; Zbl 0856.35014)]. On the other hand, for any positive \(\eta\), the result of this paper reveals a rigorous justification that the optimal lower bound of the blow-up rates is not affected by the presence of a magnetic field without the skin effect in a cold plasma.On periodic and compactly supported least energy solutions to semilinear elliptic equations with non-Lipschitz nonlinearityhttps://zbmath.org/1536.352702024-07-17T13:47:05.169476Z"Giacomoni, Jacques"https://zbmath.org/authors/?q=ai:giacomoni.jacques"Il'yasov, Yavdat"https://zbmath.org/authors/?q=ai:ilyasov.yavdat-sh"Kumar, Deepak"https://zbmath.org/authors/?q=ai:kumar.deepakSummary: We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: \(- \Delta u = \lambda u^p - u^q\) in \(\mathbb{R}^{N + 1}\), where \(0 < q < p < 1\) and \(\lambda \in \mathbb{R}\). The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter \(\lambda\), where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of \(p, q\). Important properties of the solutions are derived for suitable ranges of the parameters, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space \(\mathbb{R}^{N + 1}\).On the low Mach number limit for 2D Navier-Stokes-Korteweg systemshttps://zbmath.org/1536.352712024-07-17T13:47:05.169476Z"Hientzsch, Lars Eric"https://zbmath.org/authors/?q=ai:hientzsch.lars-ericSummary: This paper addresses the low Mach number limit for two-dimensional Navier-Stokes-Korteweg systems. The primary purpose is to investigate the relevance of the capillarity tensor for the analysis. For the sake of a concise exposition, our considerations focus on the case of the quantum Navier-Stokes (QNS) equations. An outline for a subsequent generalization to general viscosity and capillarity tensors is provided. Our main result proves the convergence of finite energy weak solutions of QNS to the unique Leray-Hopf weak solutions of the incompressible Navier-Stokes equations, for general initial data without additional smallness or regularity assumptions. We rely on the compactness properties stemming from energy and BD-entropy estimates. Strong convergence of acoustic waves is proven by means of refined Strichartz estimates that take into account the alteration of the dispersion relation due to the capillarity tensor. For both steps, the presence of a suitable capillarity tensor is pivotal.Stability of time-dependent motions for fluid-rigid ball interactionhttps://zbmath.org/1536.352722024-07-17T13:47:05.169476Z"Hishida, Toshiaki"https://zbmath.org/authors/?q=ai:hishida.toshiakiSummary: We aim at the stability of time-dependent motions, such as time-periodic ones, of a rigid body in a viscous fluid filling the exterior to it in 3D. The fluid motion obeys the incompressible Navier-Stokes system, whereas the motion of the body is governed by the balance for linear and angular momentum. Both motions are affected by each other at the boundary. Assuming that the rigid body is a ball, we adopt a monolithic approach to deduce \(L^q\)-\(L^r\) decay estimates of solutions to a non-autonomous linearized system. We then apply those estimates to the full nonlinear initial value problem to find temporal decay properties of the disturbance. Although the shape of the body is not allowed to be arbitrary, the present contribution is the first attempt at analysis of the large time behavior of solutions around nontrivial basic states, that can be time-dependent, for the fluid-structure interaction problem and provides us with a stability theorem which is indeed new even for steady motions under the self-propelling condition or with wake structure.Global well-posedness and stability of the 2D Boussinesq equations with partial dissipation near a hydrostatic equilibriumhttps://zbmath.org/1536.352742024-07-17T13:47:05.169476Z"Kang, Kyungkeun"https://zbmath.org/authors/?q=ai:kang.kyungkeun"Lee, Jihoon"https://zbmath.org/authors/?q=ai:lee.jihoon"Nguyen, Dinh Duong"https://zbmath.org/authors/?q=ai:nguyen.dinh-duongSummary: The paper is devoted to investigating the well-posedness, stability and large-time behavior near the hydrostatic balance for the 2D Boussinesq equations with partial dissipation. More precisely, the global well-posedness is obtained in the case of partial viscosity and without thermal diffusion for the initial data belonging to \(H^\delta( \mathbb{R}^2) \times H^s( \mathbb{R}^2)\) for \(\delta \in [s - 1, s + 1]\) if \(s \in \mathbb{R}\), \(s > 2\), for \(\delta \in(1, s + 1]\) if \(s \in(0, 2]\) and for \(\delta \in [0, 1]\) if \(s = 0\). In addition, if one has either horizontal or vertical thermal diffusion then the stability and large-time behavior are provided in \(H^m( \mathbb{R}^2)\), \(m \in \mathbb{N}\) and in \(\dot{H}^{m - 1}( \mathbb{R}^2)\) with \(m \in \mathbb{N}\), \(m \geq 2\), respectively.The low Mach number limit of non-isentropic magnetohydrodynamic equations with large temperature variations in bounded domainshttps://zbmath.org/1536.352752024-07-17T13:47:05.169476Z"Liang, Min"https://zbmath.org/authors/?q=ai:liang.min"Ou, Yaobin"https://zbmath.org/authors/?q=ai:ou.yaobinSummary: This paper verifies the low Mach number limit of the non-isentropic compressible magnetohydrodynamic (MHD) equations with or without the magnetic diffusion in a three-dimensional bounded domain when the temperature variation is large but finite. The uniform estimates of strong solutions are established in a short time interval independent of the Mach number, provided that the slip boundary condition for the velocity and the Neumann boundary condition for the temperature are imposed and the initial data is well-prepared.Ill-posedness for the Burgers equation in Sobolev spaceshttps://zbmath.org/1536.352762024-07-17T13:47:05.169476Z"Li, Jinlu"https://zbmath.org/authors/?q=ai:li.jinlu.1"Yu, Yanghai"https://zbmath.org/authors/?q=ai:yu.yanghai"Zhu, Weipeng"https://zbmath.org/authors/?q=ai:zhu.weipengSummary: In this paper, we consider the Cauchy problem for the Burgers equation in the line. We shall prove that this problem is ill-posed in the Sobolev space \(H^s({\mathbb{R}})\) with \(1\le s<\frac{3}{2}\) in the sense of ``norm inflation'' by constructing an explicit example of initial data.Nonlinear stability of travelling wave solution of the Navier-Stokes-Poisson system for ions with \(\gamma\)-law pressurehttps://zbmath.org/1536.352772024-07-17T13:47:05.169476Z"Liu, Jinjing"https://zbmath.org/authors/?q=ai:liu.jinjing"Yao, Lei"https://zbmath.org/authors/?q=ai:yao.leiSummary: In this paper, we consider the existence, uniqueness, and nonlinear stability of the travelling wave solution of the Navier-Stokes-Poisson system with density-dependent viscosity for ions. First, we derive a system (Navies-Stokes-Poisson system for ions) from a bipolar Navier-Stokes-Poisson system with \(\gamma\)-law pressure and density-dependent viscosity. Second, we establish the existence and uniqueness of the traveling wave solution by Center Manifold Theorem for the system. Finally, we prove that the travelling wave solution is asymptotically stable for small initial perturbations with integral zero.Well-posedness for the free boundary hard phase model in general relativityhttps://zbmath.org/1536.352792024-07-17T13:47:05.169476Z"Miao, Shuang"https://zbmath.org/authors/?q=ai:miao.shuang"Shahshahani, Sohrab"https://zbmath.org/authors/?q=ai:shahshahani.sohrabSummary: The hard phase model describes a relativistic barotropic and irrotational fluid with sound speed equal to the speed of light. In the framework of general relativity, the fluid, as a matter field, affects the geometry of the background spacetime. Therefore the motion of the fluid must be coupled to the Einstein equations which describe the structure of the underlying spacetime. In this work we prove a priori estimates and well-posedness in Sobolev spaces for this model with free boundary. Estimates for the curvature are derived using the Bianchi equations in a frame that is parallel transported by the fluid velocity. The fluid velocity is also decomposed with respect to this parallel frame, and its components are estimated using a coupled interior-boundary system of wave equations.Existence and stability of \(L^p\) solutions of the steady state magnetohydrodynamic equations with rough external forceshttps://zbmath.org/1536.352812024-07-17T13:47:05.169476Z"Uddhao, S. V."https://zbmath.org/authors/?q=ai:uddhao.swapna-v"Raiter, P. D."https://zbmath.org/authors/?q=ai:raiter.p-d"Saraykar, R. V."https://zbmath.org/authors/?q=ai:saraykar.ravindra-v|saraykar.ravi-vSummary: In this paper, following the work of \textit{C. Bjorland} et al. [Commun. Partial Differ. Equations 36, No. 1--3, 216--246 (2011; Zbl 1284.76088)] we prove the existence, the asymptotic behavior and stability of solutions to the steady state 3D magnetohydrodynamic equations in \(L^p\) and \(L^{p,\infty}\), \(3/2<p\leq \infty\) with rough external forces.Nonrelativistic limit of normalized solutions to a class of nonlinear Dirac equationshttps://zbmath.org/1536.352842024-07-17T13:47:05.169476Z"Chen, Pan"https://zbmath.org/authors/?q=ai:chen.pan"Ding, Yanheng"https://zbmath.org/authors/?q=ai:ding.yanheng"Guo, Qi"https://zbmath.org/authors/?q=ai:guo.qi"Wang, Hua-Yang"https://zbmath.org/authors/?q=ai:wang.huayangSummary: In this paper, we investigate the nonrelativistic limit of normalized solutions to a nonlinear Dirac equation as given below:
\[
\begin{cases}
-ic\sum \limits_{k=1}^3\alpha_k\partial_k u + mc^2 \beta{u} - \Gamma\ast(K|{u}|^\kappa) K|{u}|^{\kappa - 2}u - P|u|^{s - 2}u = \omega u, \\
\int_{\mathbb{R}^3}|u|^2 dx = 1.
\end{cases}
\]
Here, \(c > 0\) represents the speed of light, \(m > 0\) is the mass of the Dirac particle, \(\omega\in\mathbb{R}\) emerges as an indeterminate Lagrange multiplier, \(\Gamma\), \(K\), \(P\) are real-valued function defined on \(\mathbb{R}^3\), also known as potential functions. Our research first confirms the presence of normalized solutions to the Dirac equation under high-speed light conditions. We then illustrate that these solutions converge to normalized ground states of nonlinear Schrödinger equations, and we also show uniform boundedness and exponential decay of these solutions. Our results form the first discussion on nonrelativistic limit of normalized solutions to nonlinear Dirac equations. This not only aids in the study of normalized solutions of the nonlinear Schrödinger equations, but also physically explains that the normalized ground states of high-speed particles and low-speed motion particles are consistent.Stability of a one-dimensional full viscous quantum hydrodynamic systemhttps://zbmath.org/1536.352852024-07-17T13:47:05.169476Z"Han, Xiaoying"https://zbmath.org/authors/?q=ai:han.xiaoying"Qin, Yuming"https://zbmath.org/authors/?q=ai:qin.yuming"Sun, Wenlong"https://zbmath.org/authors/?q=ai:sun.wenlongSummary: A full viscous quantum hydrodynamic system for particle density, current density, energy density and electrostatic potential coupled with a Poisson equation in one dimensional bounded intervals is studied. First, the existence and uniqueness of a steady-state solution to the quantum hydrodynamic system is established. Then, utilizing the fact that the third order perturbation term has an appropriate sign, the local-in-time existence of the solution is investigated by introducing a fourth order viscous regularization and using the entropy dissipation method. In the end, the exponential stability of the steady-state solution is shown by constructing a uniform a-priori estimate.Long-time error bounds of low-regularity integrators for nonlinear Schrödinger equationshttps://zbmath.org/1536.352872024-07-17T13:47:05.169476Z"Feng, Yue"https://zbmath.org/authors/?q=ai:feng.yue"Maierhofer, Georg"https://zbmath.org/authors/?q=ai:maierhofer.georg"Schratz, Katharina"https://zbmath.org/authors/?q=ai:schratz.katharinaSummary: We introduce a new non-resonant low-regularity integrator for the cubic nonlinear Schrödinger equation (NLSE) allowing for long-time error estimates which are optimal in the sense of the underlying partial differential equation. The main idea thereby lies in treating the zeroth mode exactly within the discretization. For long-time error estimates, we rigorously establish the error bounds of different low-regularity integrators for the NLSE with small initial data characterized by a dimensionless parameter \(\varepsilon \in (0, 1]\). We begin with the low-regularity integrator for the quadratic NLSE in which the integral is computed exactly and the improved uniform first-order convergence in \(H^r\) is proven at \(O(\varepsilon \tau)\) for solutions in \(H^r\) with \(r > 1/2\) up to the time \(T_{\varepsilon} = T/\varepsilon\) with fixed \(T > 0\). Then, the improved uniform long-time error bound is extended to a symmetric second-order low-regularity integrator in the long-time regime. For the cubic NLSE, we design new non-resonant first-order and symmetric second-order low-regularity integrators which treat the zeroth mode exactly and rigorously carry out the error analysis up to the time \(T_{\varepsilon} = T/\varepsilon^2\). With the help of the regularity compensation oscillation technique, the improved uniform error bounds are established for the new non-resonant low-regularity schemes, which further reduces the long-time error by a factor of \(\varepsilon^2\) compared with classical low-regularity integrators for the cubic NLSE. Numerical examples are presented to validate the error estimates and compare with the classical time-splitting methods in the long-time simulations.Puiseux asymptotic expansions for convection-dominated transport problems in thin graph-like networks: strong boundary interactionshttps://zbmath.org/1536.352882024-07-17T13:47:05.169476Z"Mel'nyk, Taras"https://zbmath.org/authors/?q=ai:melnyk.taras-a"Rohde, Christian"https://zbmath.org/authors/?q=ai:rohde.christianSummary: This article completes the study of the influence of the intensity parameter \(\alpha\) in the boundary condition \(\varepsilon \partial_{\boldsymbol{\nu}_\varepsilon} u_\varepsilon - u_\varepsilon \overrightarrow{V_\varepsilon} \boldsymbol{\cdot} \boldsymbol{\nu}_\varepsilon = \varepsilon^\alpha \varphi_\varepsilon\) given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order \(\mathcal{O} (\varepsilon)\). Inside of the thin network a time-dependent convection-diffusion equation with high Péclet number of order \(\mathcal{O} (\varepsilon^{-1})\) is considered. The novelty of this article is the case of \(\alpha < 1\), which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity \(\varphi_\varepsilon\) (the cases \(\alpha = 1\) and \(\alpha > 1\) were previously studied by the same authors).
A complete Puiseux asymptotic expansion is constructed for the solution \(u_\varepsilon\) as \(\varepsilon \to 0\), i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter \(\varepsilon\).Conservation laws and exact solutions of a generalized Kadomtsev-Petviashvili (KP)-like equationhttps://zbmath.org/1536.352892024-07-17T13:47:05.169476Z"Iqbal, Anjum"https://zbmath.org/authors/?q=ai:iqbal.anjum"Naeem, Imran"https://zbmath.org/authors/?q=ai:naeem.imran(no abstract)On the Cauchy problem for a four-component Novikov system with peaked solutionshttps://zbmath.org/1536.352932024-07-17T13:47:05.169476Z"Wang, Haiquan"https://zbmath.org/authors/?q=ai:wang.haiquan"Chen, Miaomiao"https://zbmath.org/authors/?q=ai:chen.miaomiaoSummary: Considered herein is the Cauchy problem for a four-component Novikov system with peaked solutions. We first investigate the local Gevrey regularity and analyticity of the solutions by a generalized Ovsyannikov theorem. Then, based on the local well-posedness of this problem, the results with respect to the nonuniformly continuous dependence on initial data of the solutions in Besov spaces \(\left(B_{2,1}^{5/2}(\mathbb{T})\right)^2 \times \left(B_{2,1}^{3/2}(\mathbb{T})\right)^2\) and \(\left(B^s_{p, r}(\mathbb{R})\right)^2\times\left(B_{p, r}^{s-1}(\mathbb{R})\right)^2(s>\max\{5/2,2+1/p\}\),1\(\leq p\), \(r \leq \infty)\) are established by constructing new approximate solutions and initial data.
{\copyright} 2023 John Wiley \& Sons, Ltd.Perturbation at blow-up time of self-similar solutions for the modified Korteweg-de Vries equationhttps://zbmath.org/1536.352962024-07-17T13:47:05.169476Z"Correia, Simão"https://zbmath.org/authors/?q=ai:correia.simao"Côte, Raphaël"https://zbmath.org/authors/?q=ai:cote.raphaelSummary: We prove a first stability result of self-similar blow-up for the modified Korteweg-de Vries equation on the line. More precisely, given a self-similar solution and a sufficiently small regular profile, there is a unique global solution which behaves at \(t = 0\) as the sum of the self-similar solution and the smooth perturbation.The KP limit of a reduced quantum Euler-Poisson equationhttps://zbmath.org/1536.352972024-07-17T13:47:05.169476Z"Liu, Huimin"https://zbmath.org/authors/?q=ai:liu.huimin"Pu, Xueke"https://zbmath.org/authors/?q=ai:pu.xuekeSummary: In this paper, we consider the derivation of the Kadomtsev-Petviashvili (KP) equation for cold ion-acoustic wave in the long wavelength limit of a two-dimensional reduced quantum Euler-Poisson system under different scalings for varying directions in the Gardner-Morikawa transform. It is shown that the types of the KP equation depend on the scaled quantum parameter \(H>0\). The KP-I is derived for \(H>2\), KP-II for \(0<H<2\), and the dispersiveless KP (dKP) equation for the critical case \(H=2\). The rigorous proof for these limits is given in the well-prepared initial data case, and the norm that is chosen to close the proof is anisotropic in the two directions, in accordance with the anisotropic structure of the KP equation as well as the Gardner-Morikawa transform. The results can be generalized in several directions.
{\copyright} 2023 Wiley Periodicals LLC.Finite point blowup for the critical generalized Korteweg-de Vries equationhttps://zbmath.org/1536.352982024-07-17T13:47:05.169476Z"Martel, Yvan"https://zbmath.org/authors/?q=ai:martel.yvan"Pilod, Didier"https://zbmath.org/authors/?q=ai:pilod.didierSummary: In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de Vries equation, including the determination of sufficient conditions for blowup, the stability of blowup in a refined topology and the classification of minimal mass blowup. Exotic blow-up solutions with a continuum of blow-up rates and multi-point blow-up solutions were also constructed. However, all these results, as well as numerical simulations, involve the bubbling of a solitary wave going to infinity at the blow-up time, which means that the blow-up dynamics and the residue are eventually uncoupled. Even at the formal level, the question whether blowup at a finite point could occur for this equation remained open. In this article, we answer this question by constructing solutions that blow up in finite time under the form of a single bubble concentrating the ground state at a finite point with an unforeseen blow-up rate. The fact that we find a blow-up rate intermediate a blow-up rate intermediate between the self-similar rate and other rates previously known also reopens the question of which blow-up rates are actually possible for this equation.A note on the \(H^s\)-critical inhomogeneous nonlinear Schrödinger equationhttps://zbmath.org/1536.353002024-07-17T13:47:05.169476Z"An, JinMyong"https://zbmath.org/authors/?q=ai:an.jinmyong"Kim, JinMyong"https://zbmath.org/authors/?q=ai:kim.jinmyongSummary: In this paper, we consider the Cauchy problem for the \(H^s\)-critical inhomogeneous nonlinear Schrödinger (INLS) equation
\[
i\, u_t + \Delta u=|x|^{-b} f(u), \quad u(0)=u_0 \in H^s (\mathbb{R}^n),
\]
where \(n\in \mathbb{N}, 0\leq s<\frac{n}{2}, 0<b<\min \{ 2,n-s,1+\frac{n-2s}{2}\}\) and \(f(u)\) is a nonlinear function that behaves like \(\lambda |u|^{\sigma}u\) with \(\lambda \in\mathbb{C}\) and \(\sigma =\frac{4-2b}{n-2s}\). First, we establish the local well-posedness as well as the small data global well-posedness in \(H^s (\mathbb{R}^n)\) for the \(H^s\)-critical INLS equation by using the contraction mapping principle based on the Strichartz estimates in Sobolev-Lorentz spaces. Next, we obtain some standard continuous dependence results for the \(H^s\)-critical INLS equation. Our results about the well-posedness and standard continuous dependence for the \(H^s\)-critical INLS equation improve the ones of
\textit{L. Aloui} and \textit{S. Tayachi} [Discrete Contin. Dyn. Syst. 41, No. 11, 5409--5437 (2021; Zbl 1479.35759)]
and
\textit{J. An} and \textit{J. Kim} [Evol. Equ. Control Theory 12, No. 3, 1039--1055 (2023; Zbl 1512.35527)]
by extending the validity of \(s\) and \(b\). Based on the local well-posedness in \(H^1 (\mathbb{R}^n)\), we finally establish the blow-up criteria for \(H^s\)-solutions to the focusing energy-critical INLS equation. In particular, we prove the finite time blow-up for finite-variance, radially symmetric or cylindrically symmetric initial data.Error estimates of the time-splitting methods for the nonlinear Schrödinger equation with semi-smooth nonlinearityhttps://zbmath.org/1536.353012024-07-17T13:47:05.169476Z"Bao, Weizhu"https://zbmath.org/authors/?q=ai:bao.weizhu"Wang, Chushan"https://zbmath.org/authors/?q=ai:wang.chushanSummary: We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schrödinger equation (NLSE) with semi-smooth nonlinearity \(f(\rho) = \rho^\sigma\), where \(\rho =|\psi|^2\) is the density with \(\psi\) the wave function and \(\sigma > 0\) is the exponent of the semi-smooth nonlinearity. Under the assumption of \(H^2\)-solution of the NLSE, we prove error bounds at \(O(\tau^{\frac{1}{2}+\sigma} + h^{1+2\sigma})\) and \(O(\tau + h^2)\) in \(L^2\)-norm for \(0<\sigma \leq \frac{1}{2}\) and \(\sigma \geq \frac{1}{2}\), respectively, and an error bound at \(O(\tau^\frac{1}{2} + h)\) in \(H^1\)-norm for \(\sigma \geq \frac{1}{2}\), where \(h\) and \(\tau\) are the mesh size and time step size, respectively. In addition, when \(\frac{1}{2}<\sigma <1\) and under the assumption of \(H^3\)-solution of the NLSE, we show an error bound at \(O(\tau^{\sigma} + h^{2\sigma})\) in \(H^1\)-norm. Two key ingredients are adopted in our proof: one is to adopt an unconditional \(L^2\)-stability of the numerical flow in order to avoid an a priori estimate of the numerical solution for the case of \(0 < \sigma \leq \frac{1}{2}\), and to establish an \(l^\infty\)-conditional \(H^1\)-stability to obtain the \(l^\infty\)-bound of the numerical solution by using the mathematical induction and the error estimates for the case of \(\sigma \geq \frac{1}{2}\); and the other one is to introduce a regularization technique to avoid the singularity of the semi-smooth nonlinearity in obtaining improved local truncation errors. Finally, numerical results are reported to demonstrate our error bounds.Orthonormal Strichartz estimate for dispersive equations with potentialshttps://zbmath.org/1536.353032024-07-17T13:47:05.169476Z"Hoshiya, Akitoshi"https://zbmath.org/authors/?q=ai:hoshiya.akitoshiSummary: In this paper we prove the orthonormal Strichartz estimates for the higher order and fractional Schrödinger, wave, Klein-Gordon and Dirac equations with potentials. As in the case of the Schrödinger operator, the proofs are based on the smooth perturbation theory by \textit{T. Kato} [Math. Ann. 162, 258--279 (1966; Zbl 0139.31203)]. However, for the Klein-Gordon and Dirac equations, we also use a method of the microlocal analysis in order to prove the estimates for wider range of admissible pairs. As applications we prove the global existence of a solution to the higher order or fractional Hartree equation with potentials which describes the dynamics of infinitely many particles. We also give a local existence result for the semi-relativistic Hartree equation with electromagnetic potentials. As another application, the refined Strichartz estimates are proved for higher order and fractional Schrödinger, wave and Klein-Gordon equations.Long-time asymptotics to the defocusing generalized nonlinear Schrödinger equation with the decaying initial value problemhttps://zbmath.org/1536.353062024-07-17T13:47:05.169476Z"Li, Jian"https://zbmath.org/authors/?q=ai:li.jian.22"Xia, Tiecheng"https://zbmath.org/authors/?q=ai:xia.tie-chengSummary: In this paper, the main work is to study the long-time asymptotics of the defocusing generalized nonlinear Schrödinger equation with the decaying initial value. The Riemann-Hilbert method and the nonlinear steepest descent method by Deift-Zhou have made great contributions to obtain it. Starting from the Lax pair of the defocusing generalized nonlinear Schrödinger equation, the associated oscillatory Riemann-Hilbert problem can be obtained. Then, via the stationary point, the steepest decent contours, and the trigonometric decomposition of jump matrix, we get the solvable Riemann-Hilbert problem from the associated oscillatory Riemann-Hilbert problem. Based on the decaying initial value in Schwartz space, the Weber equation, and the standard parabolic cylinder function, the expression of the solution for the generalized nonlinear Schrödinger equation can be given.
{\copyright} 2023 John Wiley \& Sons, Ltd.Long-time dynamics for the radial focusing fractional INLShttps://zbmath.org/1536.353072024-07-17T13:47:05.169476Z"Majdoub, Mohamed"https://zbmath.org/authors/?q=ai:majdoub.mohamed"Saanouni, Tarek"https://zbmath.org/authors/?q=ai:saanouni.tarekSummary: We consider the following fractional NLS with focusing inhomogeneous power-type nonlinearity:
\[
i \partial_t u-(-\Delta)^s u+|x|^{-b}|u|^{p-1}u=0, \,\, (t, x)\in \mathbb{R}\times \mathbb{R}^N,
\]
where \(N\geq 2\), \(1/2<s<1\), \(0<b<2s\), and \(1+\frac{2(2s-b)}{N}<p<1+\frac{2(2s-b)}{N-2s}\). We prove the ground state threshold of global existence and scattering versus finite time blowup of energy solutions in the inter-critical regime with spherically symmetric initial data. The scattering is proved by the new approach of Dodson-Murphy. This method is based on Tao's scattering criteria and Morawetz estimates. We describe the threshold using some non-conserved quantities in the spirit of the recent paper by Dinh. The radial assumption avoids a loss of regularity in Strichartz estimates. The challenge here is to overcome two main difficulties. The first one is the presence of a non-local fractional Laplacian operator. The second one is the presence of a singular weight in the nonlinearity. The greater part of this paper is devoted to the scattering of global solutions in \(H^s(\mathbb{R}^N)\). The Lorentz spaces and the Strichartz estimates play crucial roles in our approach.
{\copyright} 2023 John Wiley \& Sons Ltd.Global well-posedness and scattering of the defocusing energy-critical inhomogeneous nonlinear Schrödinger equation with radial datahttps://zbmath.org/1536.353092024-07-17T13:47:05.169476Z"Park, Dongjin"https://zbmath.org/authors/?q=ai:park.dongjinSummary: We consider the defocusing energy-critical inhomogeneous nonlinear Schrödinger equation (INLS) \(i u_t + \Delta u = | x |^{- b} | u |^k u\) in \(\mathbb{R} \times \mathbb{R}^n\) where \(n \geq 3\), \(0 < b < \min(2, n / 2)\), and \(k = (4 - 2 b) /(n - 2)\). We show that for every spherically symmetric initial data \(\phi\) in \(H^1( \mathbb{R}^n)\), or preferably in \(\dot{H}^1( \mathbb{R}^n)\), the associated solution is globally well-posed and scatters for every such \(n\) and \(b\) except for \(n = 4\) with \(1 \leq b < 2\) and \(n = 5\) with \(1 / 2 \leq b \leq 5 / 4\). We mainly apply the arguments of \textit{T. Tao} [New York J. Math. 11, 57--80 (2005; Zbl 1119.35092)], but inspired by the work of \textit{L. Aloui} and \textit{S. Tayachi} [Discrete Contin. Dyn. Syst. 41, No. 11, 5409--5437 (2021; Zbl 1479.35759)], we utilize Lorentz spaces to define spacetime norms. This method is distinct from the concentration compactness principle and establishes a quantitative bound for the solution's spacetime norm. The bound has an exponential form \(C \exp(C E [ \phi ]^C)\) in terms of the energy \(E [\phi]\), similar to Tao's work [loc. cit.].Curvature driven complexity in the defocusing parametric nonlinear Schrödinger systemhttps://zbmath.org/1536.353102024-07-17T13:47:05.169476Z"Promislow, Keith"https://zbmath.org/authors/?q=ai:promislow.keith"Ramadan, Abba"https://zbmath.org/authors/?q=ai:ramadan.abba-iSummary: The parametric nonlinear Schrödinger equation models a variety of parametrically forced and damped dispersive waves. For the defocusing regime, we derive a normal velocity for the evolution of curved dark-soliton fronts that represent a \(\pi \)-phase shift across a thin interface. We establish a simple mechanism through which the parametric term transitions the normal velocity evolution from a curvature-driven flow to motion against curvature regularized by surface diffusion of curvature. In the former case interfacial length shrinks, while in the latter case interface length generically grows until self-intersection followed by a transition to complex motion.Sharp decay rates for localized perturbations to the critical front in the Ginzburg-Landau equationhttps://zbmath.org/1536.353122024-07-17T13:47:05.169476Z"Avery, Montie"https://zbmath.org/authors/?q=ai:avery.montie"Scheel, Arnd"https://zbmath.org/authors/?q=ai:scheel.arndSummary: We revisit the nonlinear stability of the critical invasion front in the Ginzburg-Landau equation. Our main result shows that the amplitude of localized perturbations decays with rate \(t^{-3/2}\), while the phase decays diffusively. We thereby refine earlier work of \textit{J. Bricmont} and \textit{A. Kupiainen} [Commun. Math. Phys. 159, No. 2, 287--318 (1994; Zbl 0808.35131)] as well as \textit{J.-P. Eckmann} and \textit{C. E. Wayne} [Commun. Math. Phys. 161, No. 2, 323--334 (1994; Zbl 0817.35036)], who separately established nonlinear stability but with slower decay rates. On a technical level, we rely on sharp linear estimates obtained through analysis of the resolvent near the essential spectrum via a far-field/core decomposition which is well suited to accurately describing the dynamics of separate neutrally stable modes arising from far-field behavior on the left and right.Stochastic Landau-Lifshitz-Bloch equation with transport noise: well-posedness, dissipation enhancementhttps://zbmath.org/1536.353132024-07-17T13:47:05.169476Z"Qiu, Zhaoyang"https://zbmath.org/authors/?q=ai:qiu.zhaoyang"Sun, Chengfeng"https://zbmath.org/authors/?q=ai:sun.chengfengSummary: The Landau-Lifshitz-Bloch equation is the only valid model describing the simulation of heat-assisted magnetic recording around the Curie temperature. In order to explain the noise-induced phenomenon more comprehensively between different equilibrium states, we consider a special type of noise: multiplicative transport noise, to perturb the equation on a torus \(\mathbb{T}^d, d=2,3\). The existence of martingale weak solution is proved for \(d=2,3\). For \(d=2\), we show the uniqueness, then the strong pathwise solution is established. Compared with other type of Wiener noise, we further show that the transport noise provides the regularizing effect, thus, the energy dissipation is enhanced.Relativistic equations with singular potentialshttps://zbmath.org/1536.353142024-07-17T13:47:05.169476Z"Arcoya, David"https://zbmath.org/authors/?q=ai:arcoya.david"Sportelli, Caterina"https://zbmath.org/authors/?q=ai:sportelli.caterinaAuthors' abstract: ``The first part of this paper concern with the study of the Lorentz force equation
\[
\left( \frac{q'}{\sqrt{1-|q'|^2}}\right) '= \overrightarrow{E}(t,q)+q'\times \overrightarrow{B}(t,q)
\]
in the relevant physical configuration where the electric field \(\overrightarrow{E}\) has a singularity in zero. By using Szulkin's critical point theory, we prove the existence of T-periodic solutions provided that T and the electric and magnetic fields interact properly. In the last part, we employ both a variational and a topological argument to prove that the scalar relativistic pendulum-type equation
\[
\left( \frac{q'}{\sqrt{1-(q')^2}}\right) ' +q = G^{\prime }(q) +h(t),
\]
admits at least a periodic solution when \( h\in L^1 (0, T) \) and \(G\) is singular at zero.''
Reviewer's remarks: The paper proves the existence and multiplicity of solutions of relativistic Lorentz equation with singular potentials. From Section 4 onwards, the authors examine scalar relativistic equation for a pendulum.
The outcome of this paper is already used as a critical tool in other works. For example see [\textit{C. Bereanu}, Calc. Var. Partial Differ. Equ. 63, No. 2, Paper No. 29, 25 p. (2024; Zbl 1531.58015)].
Notation: Following notations are implicitly used throughout this work:
\begin{itemize}
\item \(q\) refers to position vector
\item Dash superscript (e.g. \(q'\)) indicates time derivative.
\end{itemize}
Reviewer: Morteza Shahpari (Adelaide)Existence and exponential decay of solutions for magnetic effected piezoelectric beams with second sound and distributed delay termhttps://zbmath.org/1536.353152024-07-17T13:47:05.169476Z"Douib, Madani"https://zbmath.org/authors/?q=ai:douib.madaniSummary: This paper is concerned with a system of magnetic effected piezoelectric beams with distributed delay term, where the heat flux is given by Cattaneo's law (second sound). We prove the existence and the uniqueness of the solution using the semigroup theory. Then, we establish the exponential stability of the solution by introducing a suitable Lyapunov functional.Optimal stability for laminated beams with Kelvin-Voigt damping and Fourier's lawhttps://zbmath.org/1536.353202024-07-17T13:47:05.169476Z"Cabanillas Zannini, Victor R."https://zbmath.org/authors/?q=ai:cabanillas-zannini.victor-r"Méndez, Teófanes Quispe"https://zbmath.org/authors/?q=ai:mendez.teofanes-quispe"Ramos, A. J. A."https://zbmath.org/authors/?q=ai:ramos.anderson-j-aSummary: This article deals with the asymptotic behavior of a mathematical model for laminated beams with Kelvin-Voigt dissipation acting on the equations of transverse displacement and dimensionless slip. We prove that the evolution semigroup is exponentially stable if the damping is effective in the two equations of the model. Otherwise, we prove that the semigroup is polynomially stable and find the optimal decay rate when damping is effective only in the slip equation. Our stability approach is based on the Gearhart-Prüss-Huang Theorem, which characterizes exponential stability, while the polynomial decay rate is obtained using the Borichev and Tomilov Theorem.Global attractors for porous elastic system with memory and nonlinear frictional dampinghttps://zbmath.org/1536.353212024-07-17T13:47:05.169476Z"Duan, Yu-Ying"https://zbmath.org/authors/?q=ai:duan.yu-ying"Xiao, Ti-Jun"https://zbmath.org/authors/?q=ai:xiao.ti-junSummary: This paper is concerned with the long-time behavior of a porous-elastic system with infinite memory and nonlinear frictional damping. We prove that the dynamical system generated by the solutions of the equations is dissipative, only under the basic conditions (for the well-posedness) on the memory kernel \(g\) and the frictional damping \(h\). Further, we come up with a condition on \(g\), being more general than the usual one \(g^\prime (t) \leq - cg(t)\) (with a positive constant \(c\)), under which we prove the asymptotic smoothness and quasi-stability (the latter needs some stronger condition on \(h\)) of the dynamical system. Accordingly, we obtain the existence of a global attractor and show the finite dimensionality of the attractor.
{\copyright} 2023 John Wiley \& Sons, Ltd.Fractal dimension of global attractors for a Kirchhoff wave equation with a strong damping and a memory termhttps://zbmath.org/1536.353222024-07-17T13:47:05.169476Z"Qin, Yuming"https://zbmath.org/authors/?q=ai:qin.yuming"Wang, Hongli"https://zbmath.org/authors/?q=ai:wang.hongli"Yang, Bin"https://zbmath.org/authors/?q=ai:yang.bin.3Summary: This paper is concerned with the dimension of the global attractors for a time-dependent strongly damped subcritical Kirchhoff wave equation with a memory term. A careful analysis is required in the proof of a stabilizability inequality. The main result establishes the finite dimensionality of the global attractor.Homogenization of an eigenvalue problem through rough surfaceshttps://zbmath.org/1536.353242024-07-17T13:47:05.169476Z"Avila, Jake"https://zbmath.org/authors/?q=ai:avila.jake"Monsurrò, Sara"https://zbmath.org/authors/?q=ai:monsurro.sara"Raimondi, Federica"https://zbmath.org/authors/?q=ai:raimondi.federicaSummary: In a bounded cylinder with a rough interface we study the asymptotic behaviour of the spectrum and its associated eigenspaces for a stationary heat propagation problem. The main novelty concerns the proof of the uniform a priori estimates for the eigenvalues. In fact, due to the peculiar geometry of the domain, standard techniques do not apply and a suitable new approach is developed.Global solutions and relaxation limit to the Cauchy problem of a hydrodynamic model for semiconductorshttps://zbmath.org/1536.353252024-07-17T13:47:05.169476Z"Lu, Yun-guang"https://zbmath.org/authors/?q=ai:lu.yunguangSummary: In this paper, we study the Cauchy problem for the one-dimensional Euler-Poisson (or hydrodynamic) model for semiconductors, where the energy equation is replaced by a pressure-density relation. First, the existence of global entropy solutions is proved by using the vanishing artificial viscosity method, where, a special flux approximate is introduced to ensure the uniform boundedness of the electric field \(E\) and the a-priori \(L^\infty\) estimate, \(0 < 2 \delta \leq \rho^{\varepsilon , \delta} \leq M(t)\), \(| u^{\varepsilon , \delta} | \leq M(t)\), where \(M(t)\) could tend to infinity as the time \(t\) tends to infinity, on the viscosity-flux approximate solutions \(( \rho^{\varepsilon , \delta}, u^{\varepsilon , \delta})\); Second, the compensated compactness theory is applied to prove the pointwise convergence of \(( \rho^{\varepsilon , \delta}, u^{\varepsilon , \delta})\) as \(\varepsilon, \delta\) go to zero, and that the limit \((\rho(x, t), u(x, t))\) is a global entropy solution; Third, a technique, to apply the maximum principle to the combination of the Riemann invariants and \(\int_{- \infty}^x \rho^{\varepsilon , \delta}(x, t) - 2 \delta d x\), deduces the uniform \(L^\infty\) estimate, \(0 < 2 \delta \leq \rho^{\varepsilon , \delta} \leq M\), \(| u^{\varepsilon , \delta} | \leq M\), independent of the time \(t\) and \(\varepsilon, \delta \); Finally, as a by-product, the known compactness framework
[\textit{S. Junca} and \textit{M. Rascle}, Q. Appl. Math. 58, No. 3, 511--521 (2000; Zbl 1127.35354); \textit{P. Marcati} and \textit{R. Natalini}, Arch. Ration. Mech. Anal. 129, No. 2, 129--145 (1995; Zbl 0829.35128)]
is applied to show the relaxation limit, as the relation time \(\tau\) and \(\varepsilon, \delta\) go to zero, for general pressure \(P(\rho)\).Small data solutions for the Vlasov-Poisson system with a repulsive potentialhttps://zbmath.org/1536.353272024-07-17T13:47:05.169476Z"Ruiz, Anibal Velozo"https://zbmath.org/authors/?q=ai:ruiz.anibal-velozo.1"Velozo Ruiz, Renato"https://zbmath.org/authors/?q=ai:velozo-ruiz.renatoSummary: In this paper, we study small data solutions for the Vlasov-Poisson system with the simplest external potential, for which unstable trapping holds for the associated Hamiltonian flow. We prove sharp decay estimates in space and time for small data solutions to the Vlasov-Poisson system with the repulsive potential \(\frac{-|x|^2}{2}\) in dimension two or higher. The proofs are obtained through a commuting vector field approach. We exploit the uniform hyperbolicity of the Hamiltonian flow, by making use of the commuting vector fields contained in the stable and unstable invariant distributions of phase space for the linearized system. In dimension two, we make use of modified vector field techniques due to the slow decay estimates in time. Moreover, we show an explicit teleological construction of the trapped set in terms of the non-linear evolution of the force field.On a spatially inhomogeneous nonlinear Fokker-Planck equation: Cauchy problem and diffusion asymptoticshttps://zbmath.org/1536.353282024-07-17T13:47:05.169476Z"Anceschi, Francesca"https://zbmath.org/authors/?q=ai:anceschi.francesca"Zhu, Yuzhe"https://zbmath.org/authors/?q=ai:zhu.yuzheSummary: We investigate the Cauchy problem and the diffusion asymptotics for a spatially inhomogeneous kinetic model associated to a nonlinear Fokker-Planck operator. We derive the global well-posedness result with instantaneous smoothness effect, when the initial data lies below a Maxwellian. The proof relies on the hypoelliptic analog of classical parabolic theory, as well as a positivity-spreading result based on the Harnack inequality and barrier function methods. Moreover, the scaled equation leads to the fast diffusion flow under the low field limit. The relative phi-entropy method enables us to see the connection between the overdamped dynamics of the nonlinearly coupled kinetic model and the correlated fast diffusion. The global-in-time quantitative diffusion asymptotics is then derived by combining entropic hypocoercivity, relative phi-entropy, and barrier function methods.Existence, stability and regularity of periodic solutions for nonlinear Fokker-Planck equationshttps://zbmath.org/1536.353302024-07-17T13:47:05.169476Z"Luçon, Eric"https://zbmath.org/authors/?q=ai:lucon.eric"Poquet, Christophe"https://zbmath.org/authors/?q=ai:poquet.christopheSummary: We consider a class of nonlinear Fokker-Planck equations describing the dynamics of an infinite population of units with mean-field interaction. Relying on a slow-fast viewpoint and on the theory of approximately invariant manifolds we obtain the existence of a stable periodic solution for the PDE, consisting of probability measures. Moreover we establish the existence of a smooth isochron map in the neighborhood of this periodic solution.Turing patterns in a \(p\)-adic FitzHugh-Nagumo system on the unit ballhttps://zbmath.org/1536.353332024-07-17T13:47:05.169476Z"Chacón-Cortés, L. F."https://zbmath.org/authors/?q=ai:chacon-cortes.leonardo-fabio"Garcia-Bibiano, C. A."https://zbmath.org/authors/?q=ai:garcia-bibiano.c-a"Zúñiga-Galindo, W. A."https://zbmath.org/authors/?q=ai:zuniga-galindo.wilson-aSummary: We introduce discrete and \(p\)-adic continuous versions of the FitzHugh-Nagumo system on the one-dimensional \(p\)-adic unit ball. We provide criteria for the existence of Turing patterns. We present extensive simulations of some of these systems. The simulations show that the Turing patterns are traveling waves in the \(p\)-adic unit ball.A chemotaxis-Stokes system with nonhomogeneous boundary condition: global large-datum solution and asymptotic behaviorhttps://zbmath.org/1536.353352024-07-17T13:47:05.169476Z"Jin, Chunhua"https://zbmath.org/authors/?q=ai:jin.chunhua|jin.chunhua.1Summary: In this paper, we study the global solvability and stabilization to a chemotaxis-Stokes model with porous medium diffusion \(\Delta u^m\) and mixed nonhomogeneous boundary value conditions in three-dimensional space. When \(m\) is slightly bigger than 1, we can get a solution with strong regularity, but when \(m\) is close to 1, the regularity of the solution becomes weak. Specifically, our results are divided into two cases: (i) \(m>\frac{11}{4}-\sqrt 3(\approx 1.018)\) and (ii) \(1<m\le \frac{11}{4}-\sqrt 3\). For case (i), we obtain a global bounded weak existence with good regularity for any initial datum, and for decay incoming oxygen, we also prove that the bounded solution will converge to a constant steady state. But for case (ii), it is hard to obtain the boundedness of solutions, and a global ``very'' weak solution is obtained.
{\copyright} 2022 Wiley-VCH GmbH.Decay estimates in time for fractional evolution equationshttps://zbmath.org/1536.353472024-07-17T13:47:05.169476Z"Achache, Mahdi"https://zbmath.org/authors/?q=ai:achache.mahdiSummary: We consider the homogeneous time-fractional diffusion equation
\[
\partial^{\alpha} (u-u_0) (t)+Au(t)=0, \quad t\text{-a.e.}
\]
Here, \(\partial^{\alpha}\) denotes the Riemann-Liouville fractional derivative of order \(\alpha\in (0,1)\) with respect to time and \(A\) is a uniformly elliptic operator on \(\mathbb{R}^d\). We prove decay estimates in time and other regularity properties for the solution of the above equation.Three positive solutions for the indefinite fractional Schrödinger-Poisson systemshttps://zbmath.org/1536.353492024-07-17T13:47:05.169476Z"Che, Guofeng"https://zbmath.org/authors/?q=ai:che.guofeng"Wu, Tsung-Fang"https://zbmath.org/authors/?q=ai:wu.tsungfangSummary: In this paper, we are concerned with the following fractional Schrödinger-Poisson systems with concave-convex nonlinearities:
\[
\begin{cases}
(-\Delta)^{s}u + u + \mu l(x) \phi u = f(x) |u|^{p - 2}u + g(x) |u|^{q - 2}u \ &\text{in } \mathbb{R}^{3}, \\
(-\Delta)^{t} \phi = l(x)u^{2} \ &\text{in } \mathbb{R}^{3},
\end{cases}
\]
where \(1/2 < t \leq s < 1\), \(1 < q < 2 < p < \min \{4, 2_{s}^{\ast}\}\), \(2_{s}^{\ast} = 6/(3 - 2s)\), and \(\mu > 0\) is a parameter, \(f \in C (\mathbb{R}^3)\) is sign-changing in \(\mathbb{R}^3\) and \(g \in L^{p/(p - q)} (\mathbb{R}^3)\). Under some suitable assumptions on \(l(x)\), \(f(x)\) and \(g(x)\), we explore that the energy functional corresponding to the system is coercive and bounded below on \(H^{\alpha} (\mathbb{R}^3)\) which gets a positive solution. Furthermore, we constructed some new estimation techniques, and obtained other two positive solutions. Recent results from the literature are generally improved and extended.A class of fractional parabolic reaction-diffusion systems with control of total mass: theory and numericshttps://zbmath.org/1536.353512024-07-17T13:47:05.169476Z"Daoud, Maha"https://zbmath.org/authors/?q=ai:daoud.maha"Laamri, El-Haj"https://zbmath.org/authors/?q=ai:laamri.el-haj"Baalal, Azeddine"https://zbmath.org/authors/?q=ai:baalal.azeddineSummary: In this paper, we prove global-in-time existence of strong solutions to a class of fractional parabolic reaction-diffusion systems posed in a bounded domain of \(\mathbb{R}^N\). The nonlinear reactive terms are assumed to satisfy natural structure conditions which provide nonnegativity of the solutions and uniform control of the total mass. The diffusion operators are of type \(u_i \mapsto d_i (-\Delta )^s u_i\) where \(0<s<1\). Global existence of strong solutions is proved under the assumption that the nonlinearities are at most of polynomial growth. Our results extend previous results obtained when the diffusion operators are of type \(u_i \mapsto -d_i \Delta u_i\). On the other hand, we use numerical simulations to examine the global existence of solutions to systems with exponentially growing right-hand sides, which remains so far an open theoretical question even in the case \(s=1\).Approximate solutions to hyperbolic partial differential equation with fractional differential and fractional integral forcing functionshttps://zbmath.org/1536.353552024-07-17T13:47:05.169476Z"Gupta, Nishi"https://zbmath.org/authors/?q=ai:gupta.nishi"Maqbul, Md."https://zbmath.org/authors/?q=ai:maqbul.mdSummary: This manuscript deals with a hyperbolic partial differential equation with fractional differential and fractional integral forcing functions. Semidiscretization method is used to establish a unique strong solution and also approximate solutions. Error estimates and continuous dependence of the strong solution on the given conditions have also been discussed. At the end, we illustrated the results with an example.On Barenblatt-Zeldovich intermediate asymptoticshttps://zbmath.org/1536.353572024-07-17T13:47:05.169476Z"Kostin, V. A."https://zbmath.org/authors/?q=ai:kostin.v-a"Kostin, D. V."https://zbmath.org/authors/?q=ai:kostin.dmitrii-vladimirovich"Kostin, A. V."https://zbmath.org/authors/?q=ai:kostin.andrey-viktorovichSummary: The concept of intermediate asymptotics for the solution of an evolution equation with initial data and a related solution obtained without initial conditions was introduced by G.N. Barenblatt and Ya.B. Zeldovich in the context of extending the concept of strict determinism in statistical physics and quantum mechanics. Here, according to V.P. Maslov, to axiomatize the mathematical theory, we need to know the conditions satisfied by the initial data of the problem. We show that the correct solvability of a problem without initial conditions for fractional differential equations in a Banach space is a necessary, but not sufficient, condition for intermediate asymptotics. Examples of intermediate asymptotics are given.Ground state solutions for fractional Kirchhoff type equations with critical growthhttps://zbmath.org/1536.353592024-07-17T13:47:05.169476Z"Li, Kexue"https://zbmath.org/authors/?q=ai:li.kexueSummary: We study the nonlinear fractional Kirchhoff problem \[\begin{gathered} \Big(a+b\int_{\mathbb{R}^3}|(-\Delta)^{s/2}u|^2dx\Big) (-\Delta)^su+u=f(x,u)+|u|^{2_s^{\ast}-2}u \quad \text{in }\mathbb{R}^3, \\ u\in H^s(\mathbb{R}^3), \end{gathered}\] where \(a, b>0\) are constants, \(s(3/4,1), 2_s^\ast=6/(3-2s), (-\Delta)^s\) is the fractional Laplacian. Under some relaxed assumptions on \(f\), we prove the existence of ground state solutions.Higher-order topological asymptotic formula for the elasticity operator and applicationhttps://zbmath.org/1536.353662024-07-17T13:47:05.169476Z"Barhoumi, Montassar"https://zbmath.org/authors/?q=ai:barhoumi.montassarSummary: This paper is concerned with a geometric inverse problem related to the elasticity equation. We aim to identify an unknown hole from boundary measurements of the displacement field. The Kohn-Vogelius concept is employed for formulating the inverse problem as a topology optimization one. We develop a topological sensitivity analysis based method for detecting the location, size and shape of the unknown hole. We derive a higher-order asymptotic formula describing the variation of a Kohn-Vogelius type functional with respect to the creation of an arbitrary shaped hole inside the computational domain.Hölder stability estimates in determining the time-dependent coefficients of the heat equation from the Cauchy data sethttps://zbmath.org/1536.353832024-07-17T13:47:05.169476Z"Rassas, Imen"https://zbmath.org/authors/?q=ai:rassas.imenSummary: In this paper, we address stability results in determining the time-dependent scalar and vector potentials appearing in the convection-diffusion equation from the knowledge of the Cauchy data set. We prove Hölder-type stability estimates. The key tool used in this work is the geometric optics solution.A stability estimate in the source problem for the radiative transfer equationhttps://zbmath.org/1536.353842024-07-17T13:47:05.169476Z"Romanov, V. G."https://zbmath.org/authors/?q=ai:romanov.vladimir-gSummary: A stability estimate for the solution of a source problem for the stationary radiative transfer equation is given. It is supposed that the source has an isotropic distribution. Earlier, stability estimates for this problem were found in a partial case of the emission tomography problem with a vanishing scattering operator and for the complete transfer equation under additional difficult-to-check conditions imposed on the absorption coefficient and the scattering kernel. In this work, we suggest a new fairly simple approach for obtaining a stability estimate for the problem under the consideration. The transfer equation is considered in a circle of the two-dimension space. In the forward problem, it is assumed that incoming radiation is absent. In the inverse problem of recovering the unknown source, data on solutions of the forward problem related to outgoing radiation are given on a portion of the boundary. The obtained result can be used to estimate the total density of distributed radiation sources.A two-phase free boundary with a logarithmic termhttps://zbmath.org/1536.353872024-07-17T13:47:05.169476Z"Fotouhi, Morteza"https://zbmath.org/authors/?q=ai:fotouhi.morteza"Khademloo, Somayeh"https://zbmath.org/authors/?q=ai:khademloo.somayehSummary: We study minimizers of the energy functional
\[
\int_{\Omega} |\nabla u|^2 +2F(u)dx,
\]
where \(F' (u)\approx |u|^q \log u\) for some \(-1<q<0\). We prove existence, optimal decay, and non-degeneracy of solutions, from free boundary points. Consequently, we derive the porosity property and an estimate on the Hausdorff dimension of the free boundary.On the global \(L^p\) boundedness of multilinear \(h\)-Fourier integral operators with rough amplitudeshttps://zbmath.org/1536.353922024-07-17T13:47:05.169476Z"Aitemrar, Chafika Amel"https://zbmath.org/authors/?q=ai:aitemrar.chafika-amel"Senoussaoui, Abderrahmane"https://zbmath.org/authors/?q=ai:senoussaoui.abderrahmaneSummary: In this paper we investigate the global \(L^p\) boundedness of rough \(h\)-Fourier integral operators defined by generalized rough Hörmander class amplitudes on Banach and quasi-Banach spaces. As consequences we obtain some results of \(L^p\) boundedness of rough bilinear \(h\)-Fourier integral operators.Hölder continuity of exponential pullback attractors for form II Mindlin's strain gradient viscoelastic platehttps://zbmath.org/1536.370722024-07-17T13:47:05.169476Z"Aouadi, Moncef"https://zbmath.org/authors/?q=ai:aouadi.moncefThe author considers the deformations of a viscoelastic plate described by the integro-partial differential equation
\[
u_{tt}-\frac{1}{3}h^{2}\Delta u_{tt}-\alpha \Delta u+\gamma \Delta ^{2}u-\int_{0}^{\infty }m_{1}(s)\Delta \zeta (s)ds\]
\[\qquad
+\int_{0}^{\infty }m_{2}(s)\Delta ^{2}\zeta (s)ds-(p-\delta \left\Vert \nabla u\right\Vert ^{2})\Delta u+f(u)=g(x,t),\]
with \(\zeta _{t}+\zeta _{s}=u_{t}\), \((x,t,s)\in \Omega \times \lbrack \tau ,\infty )\times \mathbb{R}^{+}\). Here \(\Omega \) is a bounded domain in \(\mathbb{R}^{2}\) with smooth boundary \(\partial \Omega \), \(u\) the displacement vector, \(h\) a microstructural parameter, \( \zeta \) the relative displacement memory defined by \(\zeta =\zeta ^{t}(x,s)=u(x,t)-u(x,t-s)\), \(m_{1}\) and \(m_{2}\) non-negative memory kernels which belong to \(C^{1}(\mathbb{R}^{+})\cap L^{1}(\mathbb{R}^{+})\) and such that \[m_{k}^{\prime }(s),m_{k}^{\prime }(s)+\kappa _{k}m_{k}(s)\leq 0\] for some \(\kappa _{k}>0\) on \(\mathbb{R}^{+}\). Furthermore, \(f(u)\) a nonlinear forcing function satisfying \(f(0)=0\), and \[\left\vert f^{\prime }(s)\right\vert \leq C_{f}(1+\left\vert s\right\vert ^{r-1}),\] for all \(s\in \mathbb{R}\), where \( r\geq 1\) and \(C_{f}\) is a positive constant, such that there exist constants \(\beta _{1}\in \lbrack 0,\gamma \lambda _{0}/4)\) and \(\eta _{1}>0\) with \[-\eta _{1}-\beta _{1}s^{2}\leq \widehat{f}(s)=\int_{0}^{s}f(\tau )d\tau \leq f(s)s+\beta _{1}s^{2},\] for all \(s\in \mathbb{R}\). Finally \(g\) is a time-dependent perturbation which belongs to \(L_{loc}^{2}(\mathbb{R} ;L^{2}(\Omega ))\) and satisfies \[\int_{-\infty }^{t}e^{\varsigma s}\left\Vert g(x,s)\right\Vert ^{2}ds<\infty ,\]
for all \(t\in \mathbb{R}\), for an appropriate choice of \(\varsigma >0\). The boundary conditions \( u(x,t)=\Delta u(x,t)=0\), \((x,t)\in \partial \Omega \times \lbrack \tau ,\infty )\), \(\zeta ^{t}(x,s)=\Delta \zeta ^{t}(x,s)=0\), \((x,t,s)\in \partial \Omega \times \lbrack \tau ,\infty )\times \mathbb{R}^{+}\), are imposed, and the solution starts from the initial conditions \(u(x,\tau )=u_{0}^{\tau }(x)=\widehat{u}(x,\tau )\), \(u_{t}(x,\tau )=u_{1}^{\tau }(x)=\widehat{u} _{t}(x,\tau )\), \((x,t)\in \Omega \times \lbrack \tau ,\infty )\), \(\zeta ^{\tau }(x,s)=\widehat{u}(x,\tau )-\widehat{u}(x,\tau -s)=\zeta ^{\tau} _{0}(x,s)\), \((x,s)\in \Omega \times \mathbb{R}^{+}\), where \(\widehat{u}:\Omega \times (-\infty ,\tau ]\rightarrow \mathbb{R}\) is the given past history of \( u\).
The paper extends results previously obtained by the author in [NoDEA, Nonlinear Differ. Equ. Appl. 28, No. 5, Paper No. 52, 39 p. (2021; Zbl 1471.35052)], as it presents results concerning the continuity properties of pullback and exponential pullback attractors in the present context. The author first analyzes the well-posedness of the problem. Introducing \( U=(u,u_{t},\zeta )\), the above problem can be written in abstract form as: \[\frac{dU(t)}{dt}=\mathcal{A}U(t)+\mathfrak{F}(t,U(t)),\] \(t>\tau \), where \(\mathcal{A}:D(\mathcal{A})\subset \mathcal{H}\rightarrow \mathcal{H}\) , with \(\mathcal{H}=V^{2}\times V_{h}\times \mathcal{W}\), is a linear operator and \(\mathfrak{F}\) involves \(F\). A function \(U(t)\in C([\tau ,\infty );\mathcal{H})\) which satisfies the integral equation \[U(t)=e^{ \mathcal{A}(t-\tau )}U(\tau )+\int_{\tau }^{t}e^{\mathcal{A}(t-s)} \mathfrak{F}(s,U(s))ds,\] \(t\geq \tau \), is a mild solution to the problem on the interval \([\tau ,T]\). A continuous function \(U:[\tau ,T)\rightarrow \mathcal{ H}\) with \(U(\tau )=U_{\tau }\) differentiable a.e. on \([\tau ,T)\), and such that for any \(t\in \lbrack \tau ,T)\), \(U(t)\in D(\mathcal{A})\) and the previous abstract equation holds is a strong solution on \([\tau ,T)\) with \( \tau <T\leq \infty \). A function \(U=(u,u_{t},\zeta )\in C([\tau ,\infty ); \mathcal{H})\), such that \(U(\tau )=U_{\tau }=(u_{0}^{\tau },u_{1}^{\tau },\zeta _{\tau })\in \mathcal{H}\), and which satisfies a variational formulation in the sense of distributions is a weak solution to the viscoplastic plate problem. For a strong solution, the total energy is defined as \[\mathcal{E}(t)=E(t)+\int_{\Omega }\widehat{f}(u)dx,\] where \[E(t)= \frac{1}{2}\left\Vert U(t)\right\Vert _{\mathcal{H}}^{2}+\frac{1}{4\delta } (\delta \left\Vert \nabla u\right\Vert ^{2}-p)^{2}.\]
The author proves that the operator \(\mathcal{A}\) generates a semigroup of contractions in \( \mathcal{H}\), whence that for every initial condition \(U(\tau )=U_{\tau }=(u_{0}^{\tau },u_{1}^{\tau },\zeta _{\tau })\in \mathcal{H}\), the viscoplastic plate problem has a unique weak solution satisfying \(U(t)\in C([\tau ,\infty );\mathcal{H})\). If \(U_{\tau }\in D(\mathcal{A})\), this solution is strong and the weak solutions depend continuously on the initial data \(U_{\tau }\) in \(\mathcal{H}\). The author then recalls the basic notations, definitions and abstract results concerning pullback and exponential pullback attractors. He finally proves the existence of pullback attractors for the process generated by the viscoplastic plate problem and their regularity, upper semicontinuity and continuity with respect to the perturbed parameter \(\delta \).
Reviewer: Alain Brillard (Riedisheim)Exponential stability for a flexible structure system with thermodiffusion effects and distributed delayhttps://zbmath.org/1536.370732024-07-17T13:47:05.169476Z"Douib, Madani"https://zbmath.org/authors/?q=ai:douib.madani"Zitouni, Salah"https://zbmath.org/authors/?q=ai:zitouni.salah"Djebabla, Abdelhak"https://zbmath.org/authors/?q=ai:djebabla.abdelhakSummary: In the paper, the well-posedness and asymptotic behavior of solutions to a flexible structure with thermodiffusion effects and distributed delay are studied. Under suitable assumptions on the weight of the damping and the weight of the distributed delay, we prove the existence and the uniqueness of the solution using the semigroup theory. Then, by using the perturbed energy method and constructing some Lyapunov functionals, we obtain the exponential decay of the solution.Exponential attractor for the Cahn-Hilliard-Oono equation in \(\mathbb{R}^N\)https://zbmath.org/1536.370762024-07-17T13:47:05.169476Z"Cholewa, Jan W."https://zbmath.org/authors/?q=ai:cholewa.jan-w"Czaja, Radosław"https://zbmath.org/authors/?q=ai:czaja.radoslawSummary: We consider the Cahn-Hilliard-Oono equation in the whole of \(\mathbb{R}^N\), \(N \leq 3\). We prove the existence of an exponential attractor in \(H^1 \big(\mathbb{R}^N\big)\), which contains a global attractor. We also estimate from above fractal dimension of the attractors.A nonlinear elliptic PDE from atmospheric science: well-posedness and regularity at cloud edgehttps://zbmath.org/1536.490262024-07-17T13:47:05.169476Z"Remond-Tiedrez, Antoine"https://zbmath.org/authors/?q=ai:remond-tiedrez.antoine"Smith, Leslie M."https://zbmath.org/authors/?q=ai:smith.leslie-m"Stechmann, Samuel N."https://zbmath.org/authors/?q=ai:stechmann.samuel-nSummary: The precipitating quasi-geostrophic equations go beyond the (dry) quasi-geostrophic equations by incorporating the effects of moisture. This means that both precipitation and phase changes between a water-vapour phase (outside a cloud) and a water-vapour-plus-liquid phase (inside a cloud) are taken into account. In the dry case, provided that a Laplace equation is inverted, the quasi-geostrophic equations may be formulated as a nonlocal transport equation for a single scalar variable (the potential vorticity). In the case of the precipitating quasi-geostrophic equations, inverting the Laplacian is replaced by a more challenging adversary known as potential-vorticity-and-moisture inversion. The PDE to invert is nonlinear and piecewise elliptic with jumps in its coefficients across the cloud edge. However, its global ellipticity is a priori unclear due to the dependence of the phase boundary on the unknown itself. This is a free boundary problem where the location of the cloud edge is one of the unknowns. Here we present the first rigorous analysis of this PDE, obtaining existence, uniqueness, and regularity results. In particular the regularity results are nearly sharp. This analysis rests on the discovery of a variational formulation of the inversion. This novel formulation is used to answer a key question for applications: which quantities jump across the interface and which quantities remain continuous? Most notably we show that the gradient of the unknown pressure, or equivalently the streamfunction, is Hölder continuous across the cloud edge.Some remarks on almost Hermitian functionalshttps://zbmath.org/1536.530752024-07-17T13:47:05.169476Z"Draghici, Tedi"https://zbmath.org/authors/?q=ai:draghici.tedi-catalin"Sayar, Cem"https://zbmath.org/authors/?q=ai:sayar.cemSummary: We study critical points of natural functionals on various spaces of almost Hermitian structures on a compact manifold \(M^{2n}\). We present a general framework, introducing the notion of gradient of an almost Hermitian functional. As a consequence of the diffeomorphism invariance, we show that a Schur's type theorem still holds for general almost Hermitian functionals, generalizing a known fact for Riemannian functionals. We present two concrete examples, the Gauduchon's functional and a close relative of it. These functionals have been studied previously, but not in the most general setup as we do here, and we make some new observations about their critical points.Quantitative stability of harmonic maps from \(\mathbb{R}^2\) to \(\mathbb{S}^2\) with a higher degreehttps://zbmath.org/1536.580082024-07-17T13:47:05.169476Z"Deng, Bin"https://zbmath.org/authors/?q=ai:deng.bin"Sun, Liming"https://zbmath.org/authors/?q=ai:sun.liming"Wei, Jun-cheng"https://zbmath.org/authors/?q=ai:wei.junchengSummary: For degree \(\pm 1\) harmonic maps from \(\mathbb{R}^2\) (or \(\mathbb{S}^2\)) to \(\mathbb{S}^2\), \textit{A. Bernand-Mantel} et al. [Arch. Ration. Mech. Anal. 239, No. 1, 219--299 (2021; Zbl 1466.78007)] recently established a uniformly quantitative stability estimate. Namely, for any map \(u: \mathbb{R}^2\rightarrow\mathbb{S}^2\) with degree \(\pm 1\), the discrepancy of its Dirichlet energy and \(4\pi\) can linearly control the \(\dot{H}^1\)-difference of \(u\) from the set of degree \(\pm 1\) harmonic maps. Whether a similar estimate holds for harmonic maps with a higher degree is unknown. In this paper, we prove that a similar quantitative stability result for a higher degree is true only in a local sense. Namely, given a harmonic map, a similar estimate holds if \(u\) is already sufficiently near to it (modulo Möbius transforms) and the bound in general depends on the given harmonic map. More importantly, we thoroughly investigate an example of the degree 2 case, which shows that it fails to have a uniformly quantitative estimate like the degree \(\pm 1\) case. This phenomenon shows the striking difference between degree \(\pm 1\) ones and higher degree ones. Finally, we also conjecture a new uniformly quantitative stability estimate based on our computation.Singular solutions of semilinear elliptic equations with supercritical growth on Riemannian manifoldshttps://zbmath.org/1536.580112024-07-17T13:47:05.169476Z"Hasegawa, Shoichi"https://zbmath.org/authors/?q=ai:hasegawa.shoichiLet \((M^n,g)\) be an \(n\)-dimensional Riemannian model manifold containing a pole such the expression of the Riemannian metric \(g\) in spherical coordinates around the pole is given by \(ds^2=dr^2+\psi^2(r)d\theta^2,\) with \(r\in (0,R)\) and \(\theta\in \mathbb S^{n-1}\), the \((n-1)\)-dimensional unit sphere, where \(\psi\in C^2([0,R))\), the set of twice continuously differentiable functions on \([0,R)\), such that \(\psi(0)=\psi''(0)=0\) and \(\psi'(0)=1\), let \(f\in C^2([0,\infty))\), such that \(f(u)=-\Delta_gu\) in \((M^n,g)\setminus\{0\}\) and \(\displaystyle F(u)=\int_u^\infty\frac{ds}{f(s)}<\infty\) for \(u\ge u_0\) with some positive \(u_0\), and we assume that the following limit \(q=\displaystyle\lim_{u\to\infty}\left(\frac{[f'(u)]^2}{f(u)f''(u)}\right)\) is finite. Then, for \(r\in (0,r_0]\) for some \(r_0\in (0,R]\), the author states that there is a unique singular solution \(u^*(r)\) of the following ordinary differential equation: \[u''(r)+(n-1)\frac{\psi'(r)}{\psi(r)}u'(r)+f(u)=0\ for\ r\in(0,R).\] Furthermore, when \(r\) tends to zero, \(u^*(r)\) is provided explicitly in terms of \(F,\psi\), and \(q\), see Theorem 1.1 for the full statement.
Reviewer: Mohammed El Aïdi (Bogotá)Bounds for eigenfunctions of the Neumann \(p\)-Laplacian on noncompact Riemannian manifoldshttps://zbmath.org/1536.580202024-07-17T13:47:05.169476Z"Barletta, Giuseppina"https://zbmath.org/authors/?q=ai:barletta.giuseppina"Cianchi, Andrea"https://zbmath.org/authors/?q=ai:cianchi.andrea"Maz'ya, Vladimir"https://zbmath.org/authors/?q=ai:mazya.vladimir-gilelevichThe authors study eigenfunctions properties of the \(p\)-Laplacian operator on \(\Omega\), a connected open set of an \(n\)-dimensional Riemannian manifold \((M^n,g)\) such that \(\mathcal{H}^{n}(\Omega)\), the \(n\)-dimensional Hausdorff measure of \(\Omega\), is finite. To be precise, they look for properties of the solutions associated to the following Neumann boundary value problem:
\[ -\text{div}(|\nabla u|^{p-2}\nabla u)=\gamma|u|^{p-2}u\ in\ \Omega,\tag{1}\]
such that \(\gamma\) is a real constant and \(p>1\). The authors define the perimeter \(P(E,\Omega)\) of a measurable subset \(E\) of \(M^n\) relative to \(\Omega\) which is given as the Hausdorff measure of the intersection of \(\Omega\) by the essential boundary of \(E\), then they define the isoperimetric function as \(\lambda_\Omega\), a positive function on \([0,\frac{\mathcal{H}^n(\Omega)}{2}]\) by \(\lambda_\Omega(s)=\inf\left\{P(E,\Omega):E\subset\Omega,\ s\le \mathcal{H}^n(E)\le \frac{\mathcal{H}^n(\Omega)}{2}\right\}\).
Then, the authors state that a solution of \((1)\) belongs to \(L_q(\Omega)\) (resp. \(L_\infty(\Omega)\)) whenever \(\frac{s}{\lambda_\Omega(s)}\) converges to zero as \(s\) is close to zero (resp. \(\int_0\left(\frac{s}{\lambda_\Omega(s)}\right)^{\frac{p}{p-1}}\frac{ds}{s}<\infty\)), see Theorem 1.1 for the complete statement. In the case when \(\Omega=M^n\), then a solution of \((1)\) does not belong to \(L_q(M^n)\) for \(q>p\) whenever \(\lambda_{M^n}(s)\) is equivalent to \(s\) near zero (Theorem 1.2). Also, the authors state analogous results on (sharpness)bounds for eigenfunctions through the positive \(p\)-isocapacitary function which is given in terms of the infimum of the \(p\)-capacity of \(E\) (Theorems 1.3 and 1.4).
Reviewer: Mohammed El Aïdi (Bogotá)Pairs of complementary transmission conditions for Brownian motionhttps://zbmath.org/1536.600742024-07-17T13:47:05.169476Z"Bobrowski, Adam"https://zbmath.org/authors/?q=ai:bobrowski.adam"Ratajczyk, Elżbieta"https://zbmath.org/authors/?q=ai:ratajczyk.elzbietaSummary: Following our previous work on complementary boundary conditions, we write Cartesian product of two copies of a space of continuous functions on the real line as the direct sum of two subspaces that are invariant under a cosine family of operators underlying Brownian motion. Both these subspaces are formed by pairs of extensions of continuous functions: in the first subspace the form of these extensions is shaped unequivocally by the transmission conditions describing snapping out Brownian motion, in the second, it is shaped by the transmission conditions of skew Brownian motion with certain degree of stickiness. In this sense, the above transmission conditions are complementary to each other.Efficient approximation of solution derivatives for system of singularly perturbed time-dependent convection-diffusion PDEs on Shishkin meshhttps://zbmath.org/1536.650812024-07-17T13:47:05.169476Z"Bose, Sonu"https://zbmath.org/authors/?q=ai:bose.sonu"Mukherjee, Kaushik"https://zbmath.org/authors/?q=ai:mukherjee.kaushikThe paper addresses the challenging task of efficiently capturing the solution and diffusive flux (scaled first-order spatial derivative of the solution) for a coupled system of singularly perturbed convection-diffusion parabolic partial differential equations (PDEs) possessing overlapping boundary layers. As the thickness of the layer shrinks for small diffusion parameters, classical numerical techniques struggle to accurately estimate the solution and diffusive flux on equidistant meshes unless the mesh size is sufficiently large.
To tackle this problem, the authors employ an implicit-Euler method in time and a classical finite difference scheme in space on a layer-adapted Shishkin mesh. They first discuss the parameter-uniform convergence of the numerical solution in \(C^0\)-norm, followed by an error analysis for the scaled discrete space derivative and the discrete time derivative. Subsequently, the parameter-uniform error bound is established in weighted \(C^1\)-norm for global approximation to the solution and the space-time solution derivatives.
The main findings of the manuscript include the derivation of parameter-uniform error estimates for the numerical solution, the scaled discrete space derivative, and the discrete time derivative in \(C^0\)-norm. The authors prove that the error bound of the resulting numerical solution in \(C^0\)-norm is \(O(N^{-1}\ln N + \Delta t)\), where \(N\) is the number of mesh intervals and \(\Delta t\) is the time step. They also establish error bounds for the numerical approximation of the scaled space derivative and the time derivative in \(C^0\)-norm. Finally, they derive the global numerical approximation in an appropriate weighted \(C^1\)-norm and prove that the error bound in \(C^1\)-norm is \(O(N^{-1}(\ln N)^{2})\), assuming \(C_1 N^{-1} \leq \Delta t \leq C_2 N^{-1}\).
The significance of this research lies in its contribution to the development of efficient and accurate numerical methods for solving singularly perturbed PDEs with overlapping boundary layers. The proposed finite difference method on a Shishkin mesh provides a parameter-uniform numerical approximation to the solution and its derivatives, which is crucial for understanding and predicting the behavior of physical phenomena modeled by such PDEs. The error estimates derived in this paper serve as a foundation for further analysis and improvement of numerical methods in this field. Moreover, the numerical experiments presented in the manuscript demonstrate the effectiveness of the proposed method and validate the theoretical findings, highlighting the practical applicability of this research.
In conclusion, the manuscript by Bose and Mukherjee presents a significant advancement in the numerical analysis of singularly perturbed convection-diffusion parabolic PDEs with overlapping boundary layers. Their work contributes to the development of robust and efficient numerical methods for solving such problems, which have wide-ranging applications in various fields of science and engineering, such as fluid dynamics, biology, ecology, and chemical reactor theory.
Reviewer: Denys Dutykh (Le Bourget-du-Lac)A study of distributed-order time fractional diffusion models with continuous distribution weight functionshttps://zbmath.org/1536.650922024-07-17T13:47:05.169476Z"Yu, Qiang"https://zbmath.org/authors/?q=ai:yu.qiang"Turner, Ian"https://zbmath.org/authors/?q=ai:turner.ian-william"Liu, Fawang"https://zbmath.org/authors/?q=ai:liu.fawang"Moroney, Timothy"https://zbmath.org/authors/?q=ai:moroney.timothy-j|moroney.timothy-johnSummary: The distributed-order time fractional diffusion model with Dirichlet nonhomogeneous boundary conditions on a finite domain is considered. Four choices of continuous distribution weight functions with mean \(\mu\) and standard deviation \(\sigma\) are investigated to study their impact on both the short-time and long-time solution behavior. An implicit numerical method implemented on a graded mesh is proposed to solve the model and the stability and convergence analysis are presented. Semi-analytic solutions are also derived for these distributions to assess the accuracy of the scheme. Numerical results highlight that the four continuous distribution weight functions produce a short-time solution behavior that is consistent with those solutions from the classical time fractional partial differential equation with fractional order \(\gamma^* = \mu \). There are however long-time differences in the solution behavior that become more distinguishable as \(\sigma\) increases. In particular, we find a smaller value of \(\sigma\) produces more diffuse profiles and the diffusion rate slows as \(\sigma\) increases. Furthermore, the asymptotic behavior of the solution may be influenced by the time-fractional orders ranging between the smallest nonzero weight order and mean \(\mu\) for the continuous uniform and raised cosine distribution weight functions, respectively. Similar findings are also observed for the truncated normal and beta distributions.
{{\copyright} 2022 Wiley Periodicals LLC.}Finite element methods respecting the discrete maximum principle for convection-diffusion equationshttps://zbmath.org/1536.650992024-07-17T13:47:05.169476Z"Barrenechea, Gabriel R."https://zbmath.org/authors/?q=ai:barrenechea.gabriel-r"John, Volker"https://zbmath.org/authors/?q=ai:john.volker"Knobloch, Petr"https://zbmath.org/authors/?q=ai:knobloch.petrThe paper examines finite element methods applied to convection-diffusion-reaction equations, covering both steady-state and time-dependent problems, from the viewpoint of adherence to the discrete maximum principle. These equations are of great importance because they represent a wide range of phenomena across various fields such as fluid dynamics, chemical engineering, and mathematical finance.
It is well known that the solution of such equations satisfies, under certain assumptions, the maximum principle, which means that the maximum value of a solution within a domain is bounded by the values of the solution on its boundary or, in the case of time-dependent problems, at its initial points. Unfortunately, when these equations are solved numerically, the resulting approximate solution often lacks the same property, known as the discrete maximum principle, and spurious oscillations can occur, especially when the convection term dominates. Hence, in numerous applications, it is preferable for approximate solutions to exhibit similar properties to the exact solutions of the given equation and to satisfy the discrete maximum principle.
The authors study various types of finite element methods for convection-diffusion-reaction equations with dominating convective terms and discuss their adherence to the local and global discrete maximum principles. Their study shows that in the case of steady-state problems, there are only several methods that satisfy the discrete maximum principle and enable the computing of sufficiently accurate solutions. These methods are nonlinear and include algebraically stabilized schemes. For time-dependent problems, the authors state that algebraically stabilized methods, both linear and nonlinear, are currently the only finite element methods satisfying the global discrete maximum principle and enabling the finding of sufficiently accurate solutions.
Reviewer: Dana Černá (Liberec)\(\phi\)-FEM for the heat equation: optimal convergence on unfitted meshes in spacehttps://zbmath.org/1536.651042024-07-17T13:47:05.169476Z"Duprez, Michel"https://zbmath.org/authors/?q=ai:duprez.michel"Lleras, Vanessa"https://zbmath.org/authors/?q=ai:lleras.vanessa"Lozinski, Alexei"https://zbmath.org/authors/?q=ai:lozinski.alexei"Vuillemot, Killian"https://zbmath.org/authors/?q=ai:vuillemot.killianThe paper at hand provides a priori error estimates for finite element approximations of the heat equation using the so-called \(\phi\)-FEM. This method was recently developed as a way to approximate elliptic and parabolic problems on complex domains that are given as level sets. Optimal error estimates in the \(\ell^2(H^1)\)-norm are proved. In addition, error estimates in the \(\ell^\infty(L^2)\)-norm are provided. They show a convergence rate that coincides with what is observed numerically (and is optimal in this sense) but half an order in space lower that what one would expect in terms of approximation theory.
Reviewer: Jan Giesselmann (Darmstadt)Explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral methods for the Dirac equation in the simultaneously massless and nonrelativistic regimehttps://zbmath.org/1536.651182024-07-17T13:47:05.169476Z"Li, Jiyong"https://zbmath.org/authors/?q=ai:li.jiyongThe author proposes and analyzes two explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral methods for the Dirac equation in the simultaneously massless and nonrelativistic regime. These methods are time symmetric, stable under the condition \(\tau \preceq 1\) and preserve modified energy and modified mass in the discrete level. The greatest advantage of the new methods is that while preserving the structure, they are explicit and greatly reduce the computational cost compared to the traditional structure-preserving methods which are usually implicit. The error bounds are proved throughout rigorous analysis. Those bounds indicate the meshing strategies which provides better temporal and spatial resolution capacity compare to the finite difference methods. Numerical results presented in the paper confirm theoretical findings.
Reviewer: Ljiljana Teofanov (Novi Sad)A new positivity-preserving technique for high-order schemes to solve extreme problems of Euler equations on structured mesheshttps://zbmath.org/1536.651292024-07-17T13:47:05.169476Z"Tan, Yan"https://zbmath.org/authors/?q=ai:tan.yan"Zhang, Qiang"https://zbmath.org/authors/?q=ai:zhang.qiang.4|zhang.qiang.2|zhang.qiang.3|zhang.qiang.8|zhang.qiang.1|zhang.qiang.13|zhang.qiang|zhang.qiang.18"Zhu, Jun"https://zbmath.org/authors/?q=ai:zhu.jun.4|zhu.junThe paper presents a significant advancement in computational fluid dynamics, specifically addressing challenges in solving extreme problems of compressible Euler equations using high-order schemes. The primary contribution of this paper is the introduction of a new positivity-preserving (new PP) technique that is applied to fifth-order finite volume unequal-sized weighted essentially non-oscillatory (WENO) schemes on structured meshes. This technique significantly enhances the computation of problems involving low density or pressure, which are common in fields such as aerospace, meteorology, and oceanography.
The classical problem addressed in this paper revolves around the numerical difficulties encountered in solving the Euler equations for compressible fluids, especially under extreme conditions such as high Mach numbers. Traditional high-order numerical schemes often fail in these scenarios due to the emergence of negative density or pressure, which undermines the physical realism and stability of the simulations. To address this challenge, the authors develop a new PP technique that ensures the positivity of density and pressure across the computational domain, thereby preserving the physical integrity of the simulation.
The methodology employed involves a sophisticated detective process during the spatial reconstruction phase of the time-marching algorithm. This process involves examining the positivity of density and pressure at specific checking points within each cell of the mesh. If negative values are detected, a novel compression limiter is activated to modify the polynomial representations of density and pressure, ensuring their positivity throughout the target cell. This approach is distinct from previous techniques due to its ability to overestimate the minimum and maximum values of the polynomials without extensive scanning, thereby improving computational efficiency and robustness.
The manuscript provides detailed algorithmic descriptions for implementing the new PP technique in both one and two-dimensional contexts, along with theoretical discussions to underpin its validity. Extensive numerical experiments demonstrate the superiority of the new technique over classical PP methods, particularly in terms of computational efficiency and the ability to handle extreme problems without sacrificing accuracy or stability.
The significance of this research lies in its potential to expand the applicability of high-order numerical schemes to a wider range of challenging fluid dynamics problems. By ensuring the positivity of essential physical quantities, the new PP technique facilitates more accurate and stable simulations under conditions that were previously intractable. This advancement not only has implications for the development of more reliable computational tools in fluid dynamics but also opens new avenues for exploration in various applied sciences where extreme fluid behaviour is of interest.
In conclusion, the paper makes a substantial contribution to the field of computational fluid dynamics by addressing a longstanding challenge in the numerical simulation of compressible flows. The new positivity-preserving technique introduced in this study represents a significant step forward in the development of high-order schemes capable of tackling extreme problems with improved reliability and efficiency. This work not only enhances our computational capabilities but also broadens our understanding of fluid behaviour under extreme conditions, with potential applications across a range of scientific and engineering disciplines.
Reviewer: Denys Dutykh (Le Bourget-du-Lac)Low-order fictitious domain method with enhanced mass conservation for an interface Stokes problemhttps://zbmath.org/1536.651382024-07-17T13:47:05.169476Z"Corti, Daniele C."https://zbmath.org/authors/?q=ai:corti.daniele-c"Delay, Guillaume"https://zbmath.org/authors/?q=ai:delay.guillaume"Fernández, Miguel A."https://zbmath.org/authors/?q=ai:fernandez.miguel-angel"Vergnet, Fabien"https://zbmath.org/authors/?q=ai:vergnet.fabien"Vidrascu, Marina"https://zbmath.org/authors/?q=ai:vidrascu.marinaThe simulation of incompressible flows with immersed moving interfaces plays a fundamental role in various fields from the biomechanics of heart valves to the aeroelasticity of parachutes. This type of problems is approximated either by fitted or unfitted mesh numerical methods. Standard fictitious domain methods for incompressible flows do not conserve the mass across the interface, and therefore lead to numerical inaccuracies. Motivated by this, a new fictitious domain method is proposed for a Stokes problem with a Dirichlet constraint on an immersed interface. The authors show, adding a single global velocity constraint enforced on one side of the interface via a scalar Lagrange multiplier leads to a mass conserving method. A priori numerical analysis of the method is provided. The numerical results show that the symmetric variant of the proposed method provides similar or superior accuracy to alternative fictitious domain methods, without the need of resorting to penalty terms which worsen the conditioning of the resulting system matrix. The proposed method is further applied to simulate fluid-structure interaction of heart valves.
Reviewer: Bülent Karasözen (Ankara)High order Morley elements for biharmonic equations on polytopal partitionshttps://zbmath.org/1536.651472024-07-17T13:47:05.169476Z"Li, Dan"https://zbmath.org/authors/?q=ai:li.dan.4"Wang, Chunmei"https://zbmath.org/authors/?q=ai:wang.chunmei"Wang, Junping"https://zbmath.org/authors/?q=ai:wang.junping"Zhang, Shangyou"https://zbmath.org/authors/?q=ai:zhang.shangyouSummary: This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular elements, the extension leverages weak Galerkin finite element methods to accommodate higher degrees of polynomials and the flexibility of general polytopal elements. By utilizing the Schur complement of the weak Galerkin method, the extension allows for fewest local degrees of freedom while maintaining sufficient accuracy and stability for the numerical solutions. The numerical scheme incorporates locally constructed weak tangential derivatives and weak second order partial derivatives, resulting in an accurate approximation of the biharmonic equation. Optimal order error estimates in both a discrete \(H^2\) norm and the usual \(L^2\) norm are established to assess the accuracy of the numerical approximation. Additionally, numerical results are presented to validate the developed theory and demonstrate the effectiveness of the proposed extension.Multiscale topology optimization of electromagnetic metamaterials using a high-contrast homogenization methodhttps://zbmath.org/1536.741922024-07-17T13:47:05.169476Z"Murai, Naoki"https://zbmath.org/authors/?q=ai:murai.naoki"Noguchi, Yuki"https://zbmath.org/authors/?q=ai:noguchi.yuki"Matsushima, Kei"https://zbmath.org/authors/?q=ai:matsushima.kei"Yamada, Takayuki"https://zbmath.org/authors/?q=ai:yamada.takayukiSummary: This study proposes a multiscale topology optimization method for electromagnetic metamaterials using a level set-based topology optimization method that incorporates a high-contrast homogenization method. The high-contrast homogenization method can express wave propagation behavior in metamaterials for various frequencies. It can also capture unusual properties caused by local resonances, which cannot be estimated by conventional homogenization approaches. We formulated multiscale topology optimization problems where objective functions are defined by the macroscopic wave propagation behavior, and microstructures forming a metamaterial are set as design variables. Sensitivity analysis was conducted based on the concepts of shape and topological derivatives. As numerical examples, we offer optimized designs of metamaterials composed of multiple unit cell structures working as a demultiplexer based on negative permeability. The mechanism of the obtained metamaterials is discussed based on homogenized coefficients.A new blowup criterion of strong solutions to the two-dimensional equations of compressible nematic liquid crystal flowshttps://zbmath.org/1536.760082024-07-17T13:47:05.169476Z"Liu, Yang"https://zbmath.org/authors/?q=ai:liu.yang.29"Guo, Renying"https://zbmath.org/authors/?q=ai:guo.renying"Zhao, Weiwei"https://zbmath.org/authors/?q=ai:zhao.weiweiSummary: In this paper, we concern the Cauchy problem of two-dimensional (2D) compressible nematic liquid crystal flows with vacuum as far-field density. Under a geometric condition for the initial orientation field, we establish a blowup criterion in terms of the integrability of the density for strong solutions to the compressible nematic liquid crystal flows. This criterion generalizes previous results of compressible nematic liquid crystal flows with vacuum, which concludes the initial boundary problem and Cauchy problem.
{\copyright} 2023 John Wiley \& Sons, Ltd.Homogenization of a semilinear elliptic problem in a thin composite domain with an imperfect interfacehttps://zbmath.org/1536.761042024-07-17T13:47:05.169476Z"Ma, Hongru"https://zbmath.org/authors/?q=ai:ma.hongru"Tang, Yanbin"https://zbmath.org/authors/?q=ai:tang.yanbinSummary: In this paper, we consider the asymptotic behavior of a semilinear elliptic problem in a thin two-composite domain with an imperfect interface, where the flux is discontinuous. For this thin domain, both the height and the period are of order \(\epsilon\). We first use Minty-Browder theorem to prove the well-posedness of the problem and then apply the periodic unfolding method to obtain the limit problems and some corrector results for three cases of a real parameter \(\gamma=-1\), \(\gamma \in (-1,1)\) and \(\gamma <-1\), respectively. To deal with the semilinear terms, the extension operator and the averaged function are used.
{\copyright} 2023 John Wiley \& Sons Ltd.Effective heat transfer between a porous medium and a fluid layer: homogenization and simulationhttps://zbmath.org/1536.800022024-07-17T13:47:05.169476Z"Eden, Michael"https://zbmath.org/authors/?q=ai:eden.michael"Freudenberg, Tom"https://zbmath.org/authors/?q=ai:freudenberg.tomSummary: We investigate the effective heat transfer in complex systems involving porous media and surrounding fluid layers in the context of mathematical homogenization. We differentiate between two fundamentally different cases: Case (a), where the solid part of the porous media consists of disconnected inclusions, and Case (b), where the solid matrix is connected. For both scenarios, we consider a heat equation with convection where a small scale parameter \(\varepsilon>0\) characterizes the heterogeneity of the porous medium and conducts a limit process \(\varepsilon\to 0\) via two-scale convergence for the solutions of the \(\varepsilon\)-problems. In Case (a), we arrive at a one-temperature problem exhibiting a memory term and in Case (b) at a two-phase mixture model. We compare and discuss these two limit models with several simulation studies both with and without convection.Convergence of the self-dual \(U(1)\)-Yang-Mills-Higgs energies to the \((n-2)\)-area functionalhttps://zbmath.org/1536.810072024-07-17T13:47:05.169476Z"Parise, Davide"https://zbmath.org/authors/?q=ai:parise.davide"Pigati, Alessandro"https://zbmath.org/authors/?q=ai:pigati.alessandro"Stern, Daniel"https://zbmath.org/authors/?q=ai:stern.daniel-lSummary: Given a hermitian line bundle \(L\rightarrow M\) on a closed Riemannian manifold \((M^n, g)\), the self-dual Yang-Mills-Higgs energies are a natural family of functionals
\[
E_{\epsilon} (u,\nabla):=\int_M \left(|\nabla u|^2 +\epsilon^2 |F_{\nabla}|^2 +\frac{(1-|u|^2 )^2}{4\epsilon^2}\right)
\]
defined for couples \((u,\nabla)\) consisting of a section \(u\in \Gamma (L)\) and a hermitian connection \(\nabla\) with curvature \(F_{\nabla}\). While the critical points of these functionals have been well-studied in dimension two by the gauge theory community, it was shown in [52] that critical points in higher dimension converge as \(\epsilon \rightarrow 0\) (in an appropriate sense) to minimal submanifolds of codimension two, with strong parallels to the correspondence between the Allen-Cahn equations and minimal hypersurfaces. In this paper, we complement this idea by showing the \(\Gamma\)-convergence of \(E_{\epsilon}\) to \((2\pi\) times) the codimension two area: more precisely, given a family of couples \((u_{\epsilon},\nabla_{\epsilon})\) with \(\sup_{\epsilon} E_{\epsilon} (u_{\epsilon}, \nabla_{\epsilon})<\infty\), we prove that a suitable gauge invariant Jacobian \(J(u_{\epsilon},\nabla_{\epsilon})\) converges to an integral \((n-2)\)-cycle \(\Gamma\), in the homology class dual to the Euler class \(c_1 (L)\), with mass \(2 \pi \mathbb{M}(\Gamma) \leq \liminf_{\epsilon \rightarrow 0}E_{\epsilon} (u_{\epsilon},\nabla_{\epsilon})\). We also obtain a recovery sequence, for any integral cycle in this homology class. Finally, we apply these techniques to compare min-max values for the \((n-2)\)-area from the Almgren-Pitts theory with those obtained from the Yang-Mills-Higgs framework, showing that the former values always provide a lower bound for the latter. As an ingredient, we also establish a Huisken-type monotonicity result along the gradient flow of \(E_{\epsilon}\).
{\copyright} 2023 Wiley Periodicals LLC.Unified analysis of finite-size error for periodic Hartree-Fock and second order Møller-Plesset perturbation theoryhttps://zbmath.org/1536.810412024-07-17T13:47:05.169476Z"Xing, Xin"https://zbmath.org/authors/?q=ai:xing.xin"Li, Xiaoxu"https://zbmath.org/authors/?q=ai:li.xiaoxu"Lin, Lin"https://zbmath.org/authors/?q=ai:lin.lin.1Summary: Despite decades of practice, finite-size errors in many widely used electronic structure theories for periodic systems remain poorly understood. For periodic systems using a general Monkhorst-Pack grid, there has been no comprehensive and rigorous analysis of the finite-size error in the Hartree-Fock theory (HF) and the second order Møller-Plesset perturbation theory (MP2), which are the simplest wavefunction based method, and the simplest post-Hartree-Fock method, respectively. Such calculations can be viewed as a multi-dimensional integral discretized with certain trapezoidal rules. Due to the Coulomb singularity, the integrand has many points of discontinuity in general, and standard error analysis based on the Euler-Maclaurin formula gives overly pessimistic results. The lack of analytic understanding of finite-size errors also impedes the development of effective finite-size correction schemes. We propose a unified analysis to obtain sharp convergence rates of finite-size errors for the periodic HF and MP2 theories. Our main technical advancement is a generalization of the result of \textit{J. N. Lyness} [Math. Comput. 30, 1--23 (1976; Zbl 0343.65007)] for obtaining sharp convergence rates of the trapezoidal rule for a class of non-smooth integrands. Our result is applicable to three-dimensional bulk systems as well as low dimensional systems (such as nanowires and 2D materials). Our unified analysis also allows us to prove the effectiveness of the Madelung-constant correction to the Fock exchange energy, and the effectiveness of a recently proposed staggered mesh method for periodic MP2 calculations (see [\textit{X. Xing} et al., J. Chem. Theory Comput. 17, No. 8, 4733--4745 (2021; \url{doi:10.1021/acs.jctc.1c00207})]). Our analysis connects the effectiveness of the staggered mesh method with integrands with removable singularities, and suggests a new staggered mesh method for reducing finite-size errors of periodic HF calculations.Resonances in a single-lead reflection from a disordered medium: \(\sigma\)-model approachhttps://zbmath.org/1536.810742024-07-17T13:47:05.169476Z"Fyodorov, Yan V."https://zbmath.org/authors/?q=ai:fyodorov.yan-v"Skvortsov, Mikhail A."https://zbmath.org/authors/?q=ai:skvortsov.mikhail-andreevich"Tikhonov, Konstantin S."https://zbmath.org/authors/?q=ai:tikhonov.konstantin-sergeevichSummary: Using the framework of supersymmetric non-linear \(\sigma\)-model we develop a general non-perturbative characterization of universal features of the density \(\rho(\Gamma)\) of the imaginary parts (``width'') for \(S\)-matrix poles (``resonances'') describing waves incident and reflected from a disordered medium via \(M\)-channel waveguide/lead. Explicit expressions for \(\rho(\Gamma)\) are derived for several instances of systems with broken time-reversal invariance, in particular for quasi-1D and 3D media. In the case of perfectly coupled lead with a few channels \((M\sim 1)\) the most salient features are tails \(\rho(\Gamma)\sim\Gamma^{- 1}\) for narrow resonances reflecting exponential localization and \(\rho(\Gamma)\sim\Gamma^{-2}\) for broad resonances reflecting states located in the vicinity of the attached wire. For multimode quasi 1D wires with \(M\gg 1\), an intermediate asymptotics \(\rho(\Gamma)\sim\Gamma^{-3/2}\) is shown to emerge reflecting diffusive nature of decay into wide enough contacts.Multisoliton complex systems with explicit superpotential interactionshttps://zbmath.org/1536.810752024-07-17T13:47:05.169476Z"Lohe, M. A."https://zbmath.org/authors/?q=ai:lohe.m-aSummary: We consider scalar field theories in \(1+1\) dimensions with \(N\) fields \(\varphi_1, \dots\varphi_N\) which interact through a potential \(V = V(\varphi_1, \dots\varphi_N)\), which is defined in terms of an explicit superpotential \(W\). We construct \(W\) for any \(N\) in terms of a known superpotential \(w\) for a single-scalar model, such as that for the sine-Gordon equation or the \(\varphi^4\) model, leading to an expression for \(V\) which has multiple minima that supports solitons. Static solitons which minimize the total energy in each soliton sector appear as solutions of first-order Bogomolny equations, which have a gradient structure. These are identical in form to equations which arise in the context of synchronization phenomena in complex systems, with the space and time variables interchanged. The sine-Gordon superpotential, for example, leads to an explicit periodic superpotential \(W\) for \(N\) scalar fields, with associated Bogomolny equations that are equivalent to the well-known Kuramoto equations which describe the synchronization of identical phase oscillators on the unit circle. The known asymptotic properties of the Kuramoto system, for both positive and negative coupling constants, ensure that finite-energy solitons exist for any given set of intermediate values imposed at the origin. Besides the models derived from the sine-Gordon equation, we investigate \(\varphi^4\) and \(\varphi^6\) models with \(N\) scalar fields and show numerically that solitons again exist over a wide range of parameters. We also derive general properties of the elementary meson excitations of the system, in particular we show that meson-soliton bound states exist over a restricted range of mass parameters with respect to an exact solution of the \(\varphi^6\) system for \(N = 3\).
{{\copyright} 2023 The Author(s). Published by IOP Publishing Ltd}Time-independent, paraxial and time-dependent Madelung trajectories near zeroshttps://zbmath.org/1536.810772024-07-17T13:47:05.169476Z"Berry, Michael"https://zbmath.org/authors/?q=ai:berry.michael-ii|berry.michael-victor|berry.michael-j-ii|berry.michael-wSummary: The Madelung trajectories associated with a wavefunction are the integral curves (streamlines) of its phase gradient, interpretable in terms of the local velocity (momentum) vector field. The pattern of trajectories provides an immediately visualisable representation of the wave. The patterns can be completely different when the same wave equation describes different physical contexts. For the time-independent Schrödinger or Helmholtz equation, trajectories circulate around the phase singularities (zeros) of the wavefunction; and in the paraxially approximate wave, streamlines spiral slowly in or out of the zeros as well as circulating. But if the paraxial wave equation is reinterpreted as the time-dependent Schrödinger equation, its Madelung trajectories do not circulate around the zeros in spacetime: they undulate while avoiding them, except for isolated trajectories that encounter each zero in a cusp singularity. The different local trajectory geometries are illustrated with two examples; a local model explains the spacetime cusps.
{{\copyright} 2023 The Author(s). Published by IOP Publishing Ltd}Symmetries of the squeeze-driven Kerr oscillatorhttps://zbmath.org/1536.810922024-07-17T13:47:05.169476Z"Iachello, Francesco"https://zbmath.org/authors/?q=ai:iachello.francesco"Cortiñas, Rodrigo G."https://zbmath.org/authors/?q=ai:cortinas.rodrigo-g"Pérez-Bernal, Francisco"https://zbmath.org/authors/?q=ai:perez-bernal.francisco"Santos, Lea F."https://zbmath.org/authors/?q=ai:santos.lea-fSummary: We study the symmetries of the static effective Hamiltonian of a driven superconducting nonlinear oscillator, the so-called squeeze-driven Kerr Hamiltonian, and discover a remarkable quasi-spin symmetry \(su(2)\) at integer values of the ratio \(\eta = \Delta/K\) of the detuning parameter \(\Delta\) to the Kerr coefficient \(K\). We investigate the stability of this newly discovered symmetry to high-order perturbations arising from the static effective expansion of the driven Hamiltonian. Our finding may find applications in the generation and stabilization of states useful for quantum computing. Finally, we discuss other Hamiltonians with similar properties and within reach of current technologies.
{{\copyright} 2023 IOP Publishing Ltd}One-loop beta-functions of quartic enhanced tensor field theorieshttps://zbmath.org/1536.811572024-07-17T13:47:05.169476Z"Ben Geloun, Joseph"https://zbmath.org/authors/?q=ai:ben-geloun.joseph"Toriumi, Reiko"https://zbmath.org/authors/?q=ai:toriumi.reikoSummary: Enhanced tensor field theories (eTFTs) have dominant graphs that differ from the melonic diagrams of conventional tensor field theories. They therefore describe pertinent candidates to escape the so-called branched polymer phase, the universal geometry found for tensor models. For generic order \(d\) of the tensor field, we compute the perturbative \(\beta\)-functions at one-loop of two just-renormalizable quartic eTFT coined by \(+\) or \(\times\), depending on their vertex weights. The models \(+\) has two quartic coupling constants \((\lambda, \lambda_+)\), and two 2-point couplings (mass, \(Z_a\)). Meanwhile, the model \(\times\) has two quartic coupling constants \((\lambda, \lambda_\times)\) and three 2-point couplings (mass, \(Z_a\), \(Z_{2a}\)). At all orders, both models have a constant wave function renormalization: \(Z = 1\) and therefore no anomalous dimension. Despite such peculiar behavior, both models acquire nontrivial radiative corrections for the coupling constants. The RG flow of the model \(+\) exhibits a particular asymptotic safety: \(\lambda_+\) is marginal without corrections thus is a fixed point of arbitrary constant value. All remaining couplings determine relevant directions and get suppressed in the UV. Concerning the model \(\times\), \(\lambda_\times\) is marginal and again a fixed point (arbitrary constant value), \(\lambda\), \(\mu\) and \(Z_a\) are all relevant couplings and flow to 0. Meanwhile \(Z_{2a}\) is a marginal coupling and becomes a linear function of the time scale. This model can neither be called asymptotically safe or free.
{{\copyright} 2023 IOP Publishing Ltd}Potential scatterings in the \(L^2\) space: (2) rigorous scattering probability of wave packetshttps://zbmath.org/1536.812292024-07-17T13:47:05.169476Z"Ishikawa, Kenzo"https://zbmath.org/authors/?q=ai:ishikawa.kenzoSummary: In this study, potential scatterings are formulated in experimental setups with Gaussian wave packets in accordance with a probability principle and associativity of products. A breaking of an associativity is observed in scalar products with stationary scattering states in a majority of short-range potentials. Due to the breaking, states of different energies are not orthogonal and their superposition is not suitable for representing a normalized isolate state. Free wave packets in perturbative expansions in coupling strengths keep the associativity, and give a rigorous amplitude that preserves manifest unitarity and other principles of the quantum mechanics. An absolute probability is finite and comprises cross sections and new terms of unique properties. The results also demonstrate an interference term displaying unique behavior at an extreme forward direction.Upper bounds of local electronic densities in moleculeshttps://zbmath.org/1536.812592024-07-17T13:47:05.169476Z"Ashida, Sohei"https://zbmath.org/authors/?q=ai:ashida.soheiSummary: The eigenfunctions of electronic Hamiltonians determine the stable structures and dynamics of molecules through the local distributions of their densities. In this paper an a priori upper bound for such local distributions of the densities is given. The bound means that concentration of electrons is prohibited due to the repulsion between the electrons. A relation between one-electron and two-electron densities resulting from the antisymmetry of the eigenfunctions plays a crucial role in the proof.Complex dynamic analysis of a reaction-diffusion network information propagation model with non-smooth controlhttps://zbmath.org/1536.912632024-07-17T13:47:05.169476Z"Ma, Xuerong"https://zbmath.org/authors/?q=ai:ma.xuerong"Shen, Shuling"https://zbmath.org/authors/?q=ai:shen.shuling"Zhu, Linhe"https://zbmath.org/authors/?q=ai:zhu.linheSummary: Rumors do the social's perception seal, leaving people untouched with the true thing. Since the harmfulness of rumors is well known, in order to cut off the net of rumors and make the society run in order, it is necessary to have a deeper understanding of the spread of rumors. In this paper, the non-smooth reaction-diffusion system with considering the encouraging effect of secondary propagation of Internet platform on rumor propagation is used to study the dynamic system of rumor propagation. Firstly, the existence of the non-negative solution is proved by using the theory of upper and lower solutions for mixed monotone system. Secondly, the value of the basic reproduction number is calculated and the existence of positive equilibrium are discussed. Thirdly, the backward bifurcation and the saddle-node bifurcation are investigated. Fourthly, the stability of rumor propagation equilibrium and Hopf bifurcation are analyzed theoretically. Finally, some numerical simulations which proves the validity of the above theory are provided.Depleting the signal: analysis of chemotaxis-consumption models -- a surveyhttps://zbmath.org/1536.920172024-07-17T13:47:05.169476Z"Lankeit, Johannes"https://zbmath.org/authors/?q=ai:lankeit.johannes"Winkler, Michael"https://zbmath.org/authors/?q=ai:winkler.michaelThis is a review of results on the chemotaxis-consumption models and their modifications. The simplest model consists of two parabolic equations
\begin{align*}
u_t&=\Delta u-\nabla\cdot(u\nabla v),\\
v_t&=\Delta v-uv,
\end{align*}
considered usually in two- or three-dimensional bounded domains, with the homogeneous Neumann boundary conditions. Existence of solutions, long time behavior and spatial structures formation in certain cases are discussed. Modifications and generalizations of the simplest model include nonlinear diffusion, singular sensitivities and other biologically or physically relevant boundary conditions.
Reviewer: Piotr Biler (Wrocław)Global dynamics for a two-species chemotaxis system with loophttps://zbmath.org/1536.920212024-07-17T13:47:05.169476Z"Zhou, Xing"https://zbmath.org/authors/?q=ai:zhou.xing"Ren, Guoqiang"https://zbmath.org/authors/?q=ai:ren.guoqiangSummary: In this paper, we are concerned with the two-species and two-stimuli chemotaxis system with loop in a bounded domain with smooth boundary. Under appropriate regularity assumptions of the initial data, we show that the system possesses a unique and global bounded classical solution. In addition, the asymptotic behavior of the solutions is discussed. Our results generalize and improve partial previously known ones.Bifurcation analysis on a river population model with varying boundary conditionshttps://zbmath.org/1536.920972024-07-17T13:47:05.169476Z"Niu, Ben"https://zbmath.org/authors/?q=ai:niu.ben"Zhang, Hua"https://zbmath.org/authors/?q=ai:zhang.hua.6"Wei, Junjie"https://zbmath.org/authors/?q=ai:wei.junjieSummary: A delayed reaction-diffusion-advection equation subject to constant-flux and free-flow boundary conditions is considered, which models single population dynamics in a river. At first, we show the existence of a nonconstant steady state induced by the change of constant flux value. Then by analyzing the distribution of eigenvalues, the stability of the constant and nonconstant steady states and the existence of Hopf bifurcations are obtained. And an algorithm for determining the direction and stability of Hopf bifurcations is derived by applying the center manifold theory and normal form method for PFDEs. Finally, the effects of advection and downstream boundary condition on periodic oscillations are discussed theoretically and numerically.Modeling the role of seasonal variability on the dynamics of mosquito-borne diseaseshttps://zbmath.org/1536.921472024-07-17T13:47:05.169476Z"Sisodiya, Omprakash Singh"https://zbmath.org/authors/?q=ai:sisodiya.omprakash-singh"Misra, O. P."https://zbmath.org/authors/?q=ai:misra.om-prakash"Dhar, Joydip"https://zbmath.org/authors/?q=ai:dhar.joydipSummary: In this article, we have proposed an non-autonomous mathematical model to describe the dynamics of mosquito-borne diseases taking into account seasonal variation. In the proposed model, the disease transmission rate and the growth rate of aquatic mosquito populations are considered seasonally. The non-autonomous model is shown to have a disease-free, globally asymptotically stable cyclic state whenever the time-dependent reproduction number \(R_C(t)\) is less than unity. From the model analysis, we find that a unique positive endemic periodic solution of a non-autonomous system exists only when \(R_C(t) > 1\). The persistence and severity of an epidemic can be described by a time-dependent periodic reproduction number \(R_C(t)\). Furthermore, it is shown that if \(R_C(t) <1\), the disease will not spread and may eventually disappear. We also propose an optimal control problem applied to control the disease with two other parameters namely insecticide and spraying. It has been shown that a control strategy consisting of insecticides and combined spraying can have a synergistic effect in reducing the incidence of mosquito-borne diseases. Finally, numerical simulations are performed to illustrate the results of our analysis.The role of memory-based movements in the formation of animal home rangeshttps://zbmath.org/1536.921762024-07-17T13:47:05.169476Z"Ranc, Nathan"https://zbmath.org/authors/?q=ai:ranc.nathan"Cain, John W."https://zbmath.org/authors/?q=ai:cain.john-w"Cagnacci, Francesca"https://zbmath.org/authors/?q=ai:cagnacci.francesca"Moorcroft, Paul R."https://zbmath.org/authors/?q=ai:moorcroft.paul-rSummary: Most animals live in spatially-constrained home ranges. The prevalence of this space-use pattern in nature suggests that general biological mechanisms are likely to be responsible for their occurrence. Individual-based models of animal movement in both theoretical and empirical settings have demonstrated that the revisitation of familiar areas through memory can lead to the formation of stable home ranges. Here, we formulate a deterministic, mechanistic home range model that includes the interplay between a bi-component memory and resource preference, and evaluate resulting patterns of space-use. We show that a bi-component memory process can lead to the formation of stable home ranges and control its size, with greater spatial memory capabilities being associated with larger home range size. The interplay between memory and resource preferences gives rise to a continuum of space-use patterns-from spatially-restricted movements into a home range that is influenced by local resource heterogeneity, to diffusive-like movements dependent on larger-scale resource distributions, such as in nomadism. Future work could take advantage of this model formulation to evaluate the role of memory in shaping individual performance in response to varying spatio-temporal resource patterns.Pattern bifurcation in a nonlocal erosion equationhttps://zbmath.org/1536.933812024-07-17T13:47:05.169476Z"Kulikov, D. A."https://zbmath.org/authors/?q=ai:kulikov.dmitrii-anatolevichSummary: This paper considers a periodic boundary value problem for a nonlinear partial differential equation with a deviating spatial variable. It is called the nonlocal erosion equation and was proposed as a model for the formation of dynamic patterns on the semiconductor surface. As is demonstrated below, the formation of a spatially inhomogeneous relief is a self-organization process. An inhomogeneous relief appears due to local bifurcations in the neighborhood of homogeneous equilibria when they change their stability. The analysis of this problem is based on modern methods of the theory of infinite-dimensional dynamic systems, including such branches as the theory of invariant manifolds, the apparatus of normal forms, and asymptotic methods for studying dynamic systems.