Recent zbMATH articles in MSC 35Rhttps://zbmath.org/atom/cc/35R2023-11-13T18:48:18.785376ZWerkzeugLaplacians on infinite graphshttps://zbmath.org/1521.050022023-11-13T18:48:18.785376Z"Kostenko, Aleksey"https://zbmath.org/authors/?q=ai:kostenko.aleksey-s"Nicolussi, Noema"https://zbmath.org/authors/?q=ai:nicolussi.noemaSummary: ``The main focus in this memoir is on Laplacians on both weighted graphs and weighted metric graphs. Let us emphasize that we consider infinite locally finite graphs and do not make any further geometric assumptions. Whereas the existing literature usually treats these two types of Laplacian operators separately, we approach them in a uniform manner in the present work and put particular emphasis on the relationship between them. One of our main conceptual messages is that these two settings should be regarded as complementary (rather than opposite) and exactly their interplay leads to important further insight on both sides. Our central goal is twofold. First of all, we explore the relationships between these two objects by comparing their basic spectral (self-adjointness, spectral gap, etc.), parabolic (Markovian uniqueness, recurrence, stochastic completeness, etc.), and metric (quasi isometries, intrinsic metrics, etc.) properties. In turn, we exploit these connections either to prove new results for Laplacians on metric graphs or to provide new proofs and perspective on the recent progress in weighted graph Laplacians. We also demonstrate our findings by considering several important classes of graphs (Cayley graphs, tessellations, and antitrees).''
Contents: Chapter 1. Introduction; Chapter 2. Laplacians on graphs; Chapter 3. Connections via boundary triplets; Chapter 4. Connections between parabolic properties; Chapter 5. One-dimensional Schrödinger operators with point interactions; Chapter 6. Graph Laplacians as boundary operators; Chapter 7. From continuous to discrete and back; Chapter 8. Examples.
Appendix A: Boundary triplets and Weyl functions; Appendix B: Dirichlet forms; Appendix C: Heat kernel bounds; Appendix D: Glossary of notation.
Reviewer: Shuchao Li (Wuhan)A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional orderhttps://zbmath.org/1521.260152023-11-13T18:48:18.785376Z"Idczak, Dariusz"https://zbmath.org/authors/?q=ai:idczak.dariuszSummary: In the paper, a new Gronwall lemma for functions of two variables with singular integrals is proved. An application to weak relative compactness of the set of solutions to a fractional partial differential equation is given.A note on arclength null quadrature domainshttps://zbmath.org/1521.300172023-11-13T18:48:18.785376Z"Khavinson, Dmitry"https://zbmath.org/authors/?q=ai:khavinson.dmitry-s"Lundberg, Erik"https://zbmath.org/authors/?q=ai:lundberg.erikA domain \(\Omega\subset \mathbb{C}\) is said to be an arclength null quadrature domain if the identity
\[
\int_{\partial \Omega} g(z) ds(z)=0
\]
holds for all functions \(g\) in the Smirnov space \(E^1(\Omega)\). A sufficient condition for a domain \(\Omega\) to be an arclength null quadrature domain is that \(\Omega\) admits a roof function, a positive function \(u\) which is harmonic in \(\Omega\) such that the gradient \(\nabla u\) coincides with the inward-pointing unit normal vector along \(\partial\Omega\).
In this article the authors obtain the existence of a roof function for arclength null quadrature domains having finitely many boundary components.
Reviewer: Marius Ghergu (Dublin)The triharmonic equation on the Heisenberg grouphttps://zbmath.org/1521.310132023-11-13T18:48:18.785376Z"Izadjoo, Majid"https://zbmath.org/authors/?q=ai:izadjoo.majid"Akbari, Mojgan"https://zbmath.org/authors/?q=ai:akbari.mojganSummary: Consider the equation
\[
\begin{cases}
\begin{aligned}
{-}\Delta_H^3 u &= f(\xi,u ,\nabla_Hu,\nabla_H^2u,\nabla_H^3u,\nabla_H^4u,\nabla_H^5u) &\text{ in }\Omega,\\
u &>0 &\text{ in }\Omega,\\
u &= \frac{\partial}{\partial\nu}(\nabla_H^2u)= \frac{\partial}{\partial\nu}(\nabla_H^3u)=0 &\text{on }\partial\Omega,
\end{aligned}
\end{cases}
\]
where \(\Omega\) is a domain of the finite-dimensional space \(\mathbb{H}^n\) and \(f\) is a positive and bounded function. We prove the existence of a solution for the above equation. In addition, we prove the uniqueness and the cylindrical symmetry of the solution.Nonlinear PDE in the presence of singular randomnesshttps://zbmath.org/1521.350032023-11-13T18:48:18.785376Z"Tzvetkov, Nikolay"https://zbmath.org/authors/?q=ai:tzvetkov.nikolaySummary: This paper describes results concerning the construction of lowregularity solutions of nonlinear partial differential equations that depend on a random parameter. The motivations for this study are very varied. However, in the end, the results obtained and the methods used are conceptually very similar.Concentrated solution of Kirchhoff-type equationshttps://zbmath.org/1521.350152023-11-13T18:48:18.785376Z"Lan, Enhao"https://zbmath.org/authors/?q=ai:lan.enhaoSummary: In this paper, we study the Kirchhoff-type equations \[\begin{cases}-h^2A(h^{2-n}\|\nabla u\|^2)\Delta u+V(x)u=u^p \\ 0<u(x)\in H^1(\mathbb{R}^n),\ \lim_{|x|\to\infty}u(x)=0,\end{cases}\] where \(n\ge 1,1<p<2^*-1\), \(h>0\) is a small parameter, \(A\) and \(V\) are continuous functions. Under suitable conditions on \(A\) and \(V\), we show the existence of solution which concentrate at non-degenerate critical point of \(V\).Nonintrusive reduced-order models for parametric partial differential equations via data-driven operator inferencehttps://zbmath.org/1521.350222023-11-13T18:48:18.785376Z"McQuarrie, Shane A."https://zbmath.org/authors/?q=ai:mcquarrie.shane-a"Khodabakhshi, Parisa"https://zbmath.org/authors/?q=ai:khodabakhshi.parisa"Willcox, Karen E."https://zbmath.org/authors/?q=ai:willcox.karen-eSummary: This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning and physics-based modeling. The parametric structure of the governing equations is embedded directly into the reduced-order model, and parameterized reduced-order operators are learned via a data-driven linear regression problem. The result is a reduced-order model that can be solved rapidly to map parameter values to approximate PDE solutions. Such parameterized reduced-order models may be used as physics-based surrogates for uncertainty quantification and inverse problems that require many forward solves of parametric PDEs. Numerical issues such as well-posedness and the need for appropriate regularization in the learning problem are considered, and an algorithm for hyperparameter selection is presented. The method is illustrated for a parametric heat equation and demonstrated for the FitzHugh-Nagumo neuron model.Hopf bifurcation in a reaction-diffusion-advection two species model with nonlocal delay effecthttps://zbmath.org/1521.350262023-11-13T18:48:18.785376Z"Li, Zhenzhen"https://zbmath.org/authors/?q=ai:li.zhenzhen"Dai, Binxiang"https://zbmath.org/authors/?q=ai:dai.binxiang"Han, Renji"https://zbmath.org/authors/?q=ai:han.renjiSummary: The dynamics of a general reaction-diffusion-advection two species model with nonlocal delay effect and Dirichlet boundary condition is investigated in this paper. The existence and stability of the positive spatially nonhomogeneous steady state solution are studied. Then by regarding the time delay \(\tau\) as the bifurcation parameter, we show that Hopf bifurcation occurs near the steady state solution at the critical values \(\tau_n\) (\(n=0,1,2,\dots\)). Moreover, the Hopf bifurcation is forward and the bifurcated periodic solutions are stable on the center manifold. The general results are applied to a Lotka-Volterra competition-diffusion-advection model with nonlocal delay.Asymptotic behavior of nonlocal bistable reaction-diffusion equationshttps://zbmath.org/1521.350312023-11-13T18:48:18.785376Z"Besse, Christophe"https://zbmath.org/authors/?q=ai:besse.christophe"Capel, Alexandre"https://zbmath.org/authors/?q=ai:capel.alexandre"Faye, Grégory"https://zbmath.org/authors/?q=ai:faye.gregory"Fouilhé, Guilhem"https://zbmath.org/authors/?q=ai:fouilhe.guilhemSummary: In this paper, we study the asymptotic behavior of the solutions of nonlocal bistable reaction-diffusion equations starting from compactly supported initial conditions. Depending on the relationship between the nonlinearity, the interaction kernel and the diffusion coefficient, we show that the solutions can either: propagate, go extinct or remain pinned. We especially focus on the latter regime where solutions are pinned by thoroughly studying discontinuous ground state solutions of the problem for a specific interaction kernel serving as a case study. We also present a detailed numerical analysis of the problem.Indirect stabilization of a coupled system by memory effectshttps://zbmath.org/1521.350402023-11-13T18:48:18.785376Z"Tyszka, Guilherme F."https://zbmath.org/authors/?q=ai:tyszka.guilherme-f"Oquendo, Higidio Portillo"https://zbmath.org/authors/?q=ai:portillo-oquendo.higidioSummary: We consider an abstract model of two coupled elastic materials. One of the materials has conservative characteristics, whereas the other one has dissipative properties. The dissipative effect is caused by the presence of a memory term that depends on the fractional stationary operator with exponent \(\theta\in[0,1]\). In this paper, we study the asymptotic behavior of the solutions for this system. We show that the solutions decay polynomially with the rate \(t^{-1/(4-2\theta)}\). For problems with that level of generality, we show that the above rate is the best. We also study the asymptotic behavior when the wave propagation speeds of both materials coincide. For this case, we find that the decay rate is so fast as \(t^{-1/(2-2\theta)}\) for \(\theta\ne 1\). For completeness, we also approach the case \(\theta=1\), where an exponential decay of solutions is obtained.Stability of a viscoelastic Timoshenko system with non-monotonic kernelhttps://zbmath.org/1521.350442023-11-13T18:48:18.785376Z"Zhang, Hai-E."https://zbmath.org/authors/?q=ai:zhang.haie"Xu, Gen-Qi"https://zbmath.org/authors/?q=ai:xu.gen-qi"Chen, Hao"https://zbmath.org/authors/?q=ai:chen.hao.9Summary: In this paper, the stability of a linear Timoshenko beam system involved with infinite memory is considered. Different from the previous results on where the monotony of kernel is always fulfilled, the memory kernel under consideration is assumed to be non-monotonic. The well-posedness of the system is obtained by means of resolvent family theory and the exponential stability is proved under certain conditions. Numerical simulations are also presented to verify the main results.Limiting behavior of random attractors of stochastic supercritical wave equations driven by multiplicative noisehttps://zbmath.org/1521.350472023-11-13T18:48:18.785376Z"Chen, Zhang"https://zbmath.org/authors/?q=ai:chen.zhang"Wang, Bixiang"https://zbmath.org/authors/?q=ai:wang.bixiangSummary: This paper deals with the limiting behavior of random attractors of stochastic wave equations with supercritical drift driven by linear multiplicative white noise defined on unbounded domains. We first establish the uniform Strichartz estimates of the solutions with respect to noise intensity, and then prove the convergence of the solutions of the stochastic equations with respect to initial data as well as noise intensity. To overcome the non-compactness of Sobolev embeddings on unbounded domains, we first utilize the uniform tail-ends estimates to truncate the solutions in a bounded domain and then employ a spectral decomposition to establish the pre-compactness of the collection of all random attractors. We finally prove the upper semicontinuity of random attractor as noise intensity approaches zero.Large time behavior of deterministic and stochastic 3D convective Brinkman-Forchheimer equations in periodic domainshttps://zbmath.org/1521.350492023-11-13T18:48:18.785376Z"Kinra, Kush"https://zbmath.org/authors/?q=ai:kinra.kush"Mohan, Manil T."https://zbmath.org/authors/?q=ai:mohan.manil-tSummary: The large time behavior of deterministic and stochastic three dimensional convective Brinkman-Forchheimer (CBF) equations
\[
\partial_t{\boldsymbol{u}}-\mu \Delta{\boldsymbol{u}}+({\boldsymbol{u}}\cdot \nabla){\boldsymbol{u}}+\alpha{\boldsymbol{u}}+\beta |{\boldsymbol{u}}|^{r-1}{\boldsymbol{u}}+\nabla p={\boldsymbol{f}}, \nabla \cdot{\boldsymbol{u}}=0,
\]
for \(r\ge 3\) (\(\mu,\beta >0\) for \(r>3\) and \(2\beta \mu \ge 1\) for \(r=3\)), in periodic domains is carried out in this work. Our first goal is to prove the existence of global attractors for the 3D deterministic CBF equations. Then, we show the existence of random attractors for the 3D stochastic CBF equations perturbed by small additive smooth noise. Furthermore, we establish the upper semicontinuity of random attractors for the 3D stochastic CBF equations (stability of attractors), when the coefficient of random perturbation approaches to zero. Finally, we address the existence and uniqueness of invariant measures of 3D stochastic CBF equations.Instantaneous blow-up for evolution inequalities of Sobolev type with nonlinear convolution termshttps://zbmath.org/1521.350542023-11-13T18:48:18.785376Z"Alazman, Ibtehal"https://zbmath.org/authors/?q=ai:alazman.ibtehal"Jleli, Mohamed"https://zbmath.org/authors/?q=ai:jleli.mohamed-boussairiSummary: We consider evolution inequalities of Sobolev type involving nonlinearities of the form \(|x|^{\sigma-N}*|u|^p\) and \(|x|^{\sigma-N}*|\nabla u|^p \), where \(*\) is the convolution product in \(\mathbb{R}^N\), \(p>1\) and \(0<\sigma<N\). For each case, we prove the existence of a critical exponent \(p_{cr}(\sigma,N)\in(1,\infty]\) depending on the parameter \(\sigma\) and the dimension \(N\), in the following sense: if \(1<p\le p_{cr}(\sigma,N)\), then there is no local weak solutions; if \(p>p_{cr}(\sigma,N)\), then local weak solutions exist for some initial data.Unstable ground state and blow up result of nonlocal Klein-Gordon equationshttps://zbmath.org/1521.350562023-11-13T18:48:18.785376Z"Carrião, Paulo Cesar"https://zbmath.org/authors/?q=ai:carriao.paulo-cesar"Lehrer, Raquel"https://zbmath.org/authors/?q=ai:lehrer.raquel"Vicente, André"https://zbmath.org/authors/?q=ai:vicente.andreSummary: In this paper we study the behaviour of solutions for a nonlocal hyperbolic equation. We use the Pohozaev manifold combined with a new technique to explicit two invariant regions in the space of initial data. On the first one the solution blows up (in finite or infinite time) and in the second one the solution exists globally. Additionally, we prove that the ground state solution of the elliptic problem associated to the original problem is unstable. The main goal of this paper is to present a new technique which allows us to consider nonlocal problems and to extend the classical result proved by \textit{J. Shatah} [Trans. Am. Math. Soc. 290, 701--710 (1985; Zbl 0617.35072)].Blow-up on metric graphs and Riemannian manifoldshttps://zbmath.org/1521.350582023-11-13T18:48:18.785376Z"Punzo, Fabio"https://zbmath.org/authors/?q=ai:punzo.fabio"Tesei, Alberto"https://zbmath.org/authors/?q=ai:tesei.albertoSummary: We study blow-up versus global existence of solutions to a model semilinear parabolic equation in metric measure spaces. Applications to metric graphs and Riemannian manifolds are considered, pointing out the occurrence of the Fujita phenomenon.Fractional-order operators on nonsmooth domainshttps://zbmath.org/1521.350642023-11-13T18:48:18.785376Z"Abels, Helmut"https://zbmath.org/authors/?q=ai:abels.helmut"Grubb, Gerd"https://zbmath.org/authors/?q=ai:grubb.gerdSummary: The fractional Laplacian \((-\Delta)^a\), \(a\in (0,1)\), and its generalizations to variable-coefficient \(2a\)-order pseudodifferential operators \(P\), are studied in \(L_q\)-Sobolev spaces of Bessel-potential type \(H^s_q\). For a bounded open set \(\Omega \subset \mathbb{R}^n\), consider the homogeneous Dirichlet problem: \(Pu =f\) in \(\Omega\), \(u=0\) in \(\mathbb{R}^n\setminus \Omega\). We find the regularity of solutions and determine the exact Dirichlet domain \(D_{a,s,q}\) (the space of solutions \(u\) with \(f\in H_q^s(\overline{\Omega}))\) in cases where \(\Omega\) has limited smoothness \(C^{1+\tau}\), for \(2a<\tau <\infty\), \(0\leqslant s<\tau -2a\). Earlier, the regularity and Dirichlet domains were determined for smooth \(\Omega\) by the second author, and the regularity was found in low-order Hölder spaces for \(\tau =1\) by Ros-Oton and Serra. The \(H_q^s\)-results obtained now when \(\tau <\infty\) are new, even for \((-\Delta)^a\). In detail, the spaces \(D_{a,s,q}\) are identified as \(a\)-transmission spaces \(H_q^{a(s+2a)}(\overline{\Omega})\), exhibiting estimates in terms of \(\operatorname{dist}(x,\partial \Omega)^a\) near the boundary.
The result has required a new development of methods to handle nonsmooth coordinate changes for pseudodifferential operators, which have not been available before; this constitutes another main contribution of the paper.On the existence and Hölder regularity of solutions to some nonlinear Cauchy-Neumann problemshttps://zbmath.org/1521.350652023-11-13T18:48:18.785376Z"Audrito, Alessandro"https://zbmath.org/authors/?q=ai:audrito.alessandroSummary: We prove \textit{uniform} parabolic Hölder estimates of De Giorgi-Nash-Moser type for sequences of minimizers of the functionals
\[
{\mathcal{E}}_\varepsilon (W) = \int_0^\infty \frac{e^{- t/\varepsilon}}{\varepsilon} \bigg\{\int_{\mathbb{R}_+^{N+1}} y^a \Big(\varepsilon |\partial_t W|^2 + |\nabla W|^2 \Big) \mathrm{d}X + \int_{\mathbb{R}^N \times \{0\}} \Phi (w) \,\mathrm{d}x \bigg\} \,\mathrm{d}t, \qquad \varepsilon \in (0,1)
\]
where \(a \in (-1, 1)\) is a fixed parameter, \(\mathbb{R}_+^{N+1}\) is the upper half-space and \(\mathrm{d}X = \mathrm{d}x \mathrm{d}y\). As a consequence, we deduce the existence and Hölder regularity of weak solutions to a class of weighted nonlinear Cauchy-Neumann problems arising in combustion theory and fractional diffusion.Improved local smoothing estimates for the fractional Schrödinger operatorhttps://zbmath.org/1521.350672023-11-13T18:48:18.785376Z"Gao, Chuanwei"https://zbmath.org/authors/?q=ai:gao.chuanwei"Miao, Changxing"https://zbmath.org/authors/?q=ai:miao.changxing"Zheng, Jiqiang"https://zbmath.org/authors/?q=ai:zheng.jiqiangSummary: In this paper, we consider local smoothing estimates for the fractional Schrödinger operator \(e^{it(-\Delta )^{\alpha /2}}\) with \(\alpha >1\). Using the \(k\)-broad `norm' estimates of \textit{L. Guth} et al. [Acta Math. 223, No. 2, 251--376 (2019; Zbl 1430.42016)], we improve the previously best-known results of local smoothing estimates of \textit{S. Guo} et al. [Anal. PDE 13, No. 5, 1457--1500 (2020; Zbl 1452.42010)] and \textit{K. M. Rogers} and \textit{A. Seeger} [J. Reine Angew. Math. 640, 47--66 (2010; Zbl 1191.35078)].Nonlocal boundary-value problem for an equation with differentiation operator \(z \partial/\partial z\) in a refined Sobolev scalehttps://zbmath.org/1521.350762023-11-13T18:48:18.785376Z"Ilkiv, V. S."https://zbmath.org/authors/?q=ai:ilkiv.volodymyr-stepanovich"Strap, N. I."https://zbmath.org/authors/?q=ai:strap.nataliya-igorivna"Volyanska, I. I."https://zbmath.org/authors/?q=ai:volyanska.iryna-iSummary: We study a nonlocal boundary-value problem for a differential equation with operator of generalized differentiation \(B = z \partial/\partial z\) acting upon the functions of complex variable \(z\). We establish the conditions of solvability of this problem in the scale of Hörmander spaces, which form the refined Sobolev scale of functions of one complex variable. The analyzed problem is ill posed in Hadamard's sense in the case of many operators of generalized differentiation, and its solvability depends on small denominators appearing in the construction of the solution. It is shown that, in the case of one variable, the corresponding denominators are not small and can be estimated from below by certain constants.A linearized integral equation reconstruction method of admittivity distributions using electrical impedance tomographyhttps://zbmath.org/1521.350772023-11-13T18:48:18.785376Z"Sebu, Cristiana"https://zbmath.org/authors/?q=ai:sebu.cristiana"Amaira, Andrei"https://zbmath.org/authors/?q=ai:amaira.andrei"Curmi, Jeremy"https://zbmath.org/authors/?q=ai:curmi.jeremy(no abstract)New structures for closed-form wave solutions for the dynamical equations model related to the ion sound and Langmuir waveshttps://zbmath.org/1521.350782023-11-13T18:48:18.785376Z"Alam, Md Nur"https://zbmath.org/authors/?q=ai:alam.md-nur"Osman, M. S."https://zbmath.org/authors/?q=ai:osman.mohammed-s|osman.mojahid-saeed|osman.m-sh|osman.mohamed-sayed-aliSummary: This treatise analyzes the coupled nonlinear system of the model for the ion sound and Langmuir waves. The modified \((G^\prime/G)\)-expansion procedure is utilized to raise new closed-form wave solutions. Those solutions are investigated through hyperbolic, trigonometric and rational function. The graphical design makes the dynamics of the equations noticeable. It provides the mathematical foundation in diverse sectors of underwater acoustics, electromagnetic wave propagation, design of specific optoelectronic devices and physics quantum mechanics. Herein, we concluded that the studied approach is advanced, meaningful and significant in implementing many solutions of several nonlinear partial differential equations occurring in applied sciences.Boundary control problems for nonlinear reaction-diffusion-convection modelhttps://zbmath.org/1521.350802023-11-13T18:48:18.785376Z"Saritskaya, Zh. Yu."https://zbmath.org/authors/?q=ai:saritskaya.zhanna-yurevna"Brizitskii, R. V."https://zbmath.org/authors/?q=ai:brizitskii.roman-victorovich|brizitskii.roman-viktorovichSummary: The solvability of the boundary control problem for a nonlinear model of mass transfer is proven in the case, when the reaction coefficient depends nonlinearly on concentration of substance and depends on spatial variables. The role of the control is played by the concentration value specified on the entire boundary of the domain.Correction to: ``Infinitely many solutions for a class of fractional Schrödinger equations with sign-changing weight functions''https://zbmath.org/1521.350912023-11-13T18:48:18.785376Z"Chen, Yongpeng"https://zbmath.org/authors/?q=ai:chen.yongpeng"Jin, Baoxia"https://zbmath.org/authors/?q=ai:jin.baoxiaCorrection to the authors' paper [ibid. 2022, Paper No. 86, 13 p. (2022; Zbl 1518.35263)].Heat kernels for a class of hybrid evolution equationshttps://zbmath.org/1521.350992023-11-13T18:48:18.785376Z"Garofalo, Nicola"https://zbmath.org/authors/?q=ai:garofalo.nicola"Tralli, Giulio"https://zbmath.org/authors/?q=ai:tralli.giulioSummary: The aim of this paper is to construct (explicit) heat kernels for some \textit{hybrid} evolution equations which arise in physics, conformal geometry and subelliptic PDEs. Hybrid means that the relevant partial differential operator appears in the form \({\mathscr{L}}_1 + {\mathscr{L}}_2 - \partial_t\), but the variables cannot be decoupled. As a consequence, the relative heat kernel cannot be obtained as the product of the heat kernels of the operators \({\mathscr{L}}_1 - \partial_t\) and \({\mathscr{L}}_2 - \partial_t\). Our approach is new and ultimately rests on the generalised Ornstein-Uhlenbeck operators in the opening of Hörmander's 1967 groundbreaking paper on hypoellipticity.Weak entire solutions of reaction-interface systemshttps://zbmath.org/1521.351042023-11-13T18:48:18.785376Z"Chen, Yan-Yu"https://zbmath.org/authors/?q=ai:chen.yanyu"Ninomiya, Hirokazu"https://zbmath.org/authors/?q=ai:ninomiya.hirokazu"Wu, Chang-Hong"https://zbmath.org/authors/?q=ai:wu.changhongSummary: In this paper, the singular limit problems arising from FitzHugh-Nagumo-type reaction-diffusion systems are studied, which are called reaction-interface systems. All weak entire solutions originating from finitely many excited intervals are completely characterized. For weak entire solutions originating from infinitely many excited intervals, periodic wave trains and time-periodic solutions are discussed. In particular, we study the dispersion relation of periodic wave trains and investigate the dependence of the propagation speed on the period.Spatial segregation of multiple species: a singular limit approachhttps://zbmath.org/1521.351062023-11-13T18:48:18.785376Z"Izuhara, Hirofumi"https://zbmath.org/authors/?q=ai:izuhara.hirofumi"Monobe, Harunori"https://zbmath.org/authors/?q=ai:monobe.harunori"Wu, Chang-Hong"https://zbmath.org/authors/?q=ai:wu.changhongSummary: The spatial segregation of the populations occurs commonly in ecology. One typical way to understand this phenomenon is to consider strong competition in some species. In this paper, we shall consider multiple-species competition-diffusion models. Under the condition that some interspecies competition rates are large, we show that the segregation phenomenon occurs. Furthermore, we derive some two-phase Stefan-like problems appearing as the singular limit, which may provide some modeling interpretation for free boundary problems studied in the literature.Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic \(p\)-Laplace equationshttps://zbmath.org/1521.351132023-11-13T18:48:18.785376Z"Banerjee, Agnid"https://zbmath.org/authors/?q=ai:banerjee.agnid"Garain, Prashanta"https://zbmath.org/authors/?q=ai:garain.prashanta"Kinnunen, Juha"https://zbmath.org/authors/?q=ai:kinnunen.juhaSummary: We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional \(p\)-Laplace equations which includes the fractional parabolic \(p\)-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case, and in the nonlocal elliptic case, to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.Longer lifespan for many solutions of the Kirchhoff equationhttps://zbmath.org/1521.351212023-11-13T18:48:18.785376Z"Baldi, Pietro"https://zbmath.org/authors/?q=ai:baldi.pietro"Haus, Emanuele"https://zbmath.org/authors/?q=ai:haus.emanueleSummary: We consider the Kirchhoff equation \(\partial_{tt} u - \Delta u \big( 1 + \int_{\mathbb{T}^d} |\nabla u|^2 \big) = 0\) on the \(d\)-dimensional torus \(\mathbb{T}^d\), and its Cauchy problem with initial data \(u(0,x), \partial_t u(0,x)\) of size \(\varepsilon\) in the Sobolev class. The effective equation for the dynamics at the quintic order, obtained in previous papers by quasilinear normal form, contains resonances corresponding to nontrivial terms in the energy estimates. Such resonances cannot be avoided by tuning external parameters (simply because the Kirchhoff equation does not contain parameters). In this paper we introduce nonresonance conditions on the initial data of the Cauchy problem and prove a lower bound \(\varepsilon^{-6}\) for the lifespan of the corresponding solutions (the standard local theory gives \(\varepsilon^{-2}\), and the normal form for the cubic terms gives \(\varepsilon^{-4})\). The proof relies on the fact that, under these nonresonance conditions, the growth rate of the ``superactions'' of the effective equations on large time intervals is smaller (by a factor \(\varepsilon^2)\) than its a priori estimate based on the normal form for the cubic terms. The set of initial data satisfying such nonresonance conditions contains several nontrivial examples that are discussed in the paper.Boundary problems for Helmholtz equation and the Cauchy problem for Dirac operatorshttps://zbmath.org/1521.351232023-11-13T18:48:18.785376Z"Shlapunov, Alexander A."https://zbmath.org/authors/?q=ai:shlapunov.alexander-aSummary: Studying an operator equation \(Au=f\) in Hilbert spaces one usually needs the adjoint operator \(A^\star\) for \(A\). Solving the ill-posed Cauchy problem for Dirac type systems in the Lebesgue spaces by an iteration method we propose to construct the corresponding adjoint operator with the use of normally solvable mixed problem for Helmholtz Equation. This leads to the description of necessary and sufficient solvability conditions for the Cauchy Problem and formulae for its exact and approximate solutions.Modification of optimal homotopy asymptotic method for multi-dimensional time-fractional model of Navier-Stokes equationhttps://zbmath.org/1521.351352023-11-13T18:48:18.785376Z"Jan, Himayat Ullah"https://zbmath.org/authors/?q=ai:jan.himayat-ullah"Ullah, Hakeem"https://zbmath.org/authors/?q=ai:ullah.hakeem"Fiza, Mehreen"https://zbmath.org/authors/?q=ai:fiza.mehreen"Khan, Ilyas"https://zbmath.org/authors/?q=ai:khan.ilyas"Mohamed, Abdullah"https://zbmath.org/authors/?q=ai:mohamed.abdullah"Mousa, Abd Allah A."https://zbmath.org/authors/?q=ai:mousa.abd-allah-a(no abstract)Which measure-valued solutions of the monoatomic gas equations are generated by weak solutions?https://zbmath.org/1521.351372023-11-13T18:48:18.785376Z"Gallenmüller, Dennis"https://zbmath.org/authors/?q=ai:gallenmuller.dennis"Wiedemann, Emil"https://zbmath.org/authors/?q=ai:wiedemann.emilSummary: Contrary to the incompressible case, not every measure-valued solution of the compressible Euler equations can be generated by weak solutions or a vanishing viscosity sequence. In the present paper we give sufficient conditions on an admissible measure-valued solution of the isentropic Euler system to be generated by weak solutions. As one of the crucial steps we prove a characterization result for generating \({\mathcal{A}} \)-free Young measures in terms of potential operators including uniform \(L^{\infty } \)-bounds. More concrete versions of our results are presented in the case of a solution consisting of two Dirac measures. We conclude by discussing that are also necessary conditions for generating a measure-valued solution by weak solutions or a vanishing viscosity sequence and will point out that the resulting gap mainly results from obtaining only uniform \(L^p\)-bounds for \(1<p<\infty\) instead of \(p=\infty \).On the motion of a large number of small rigid bodies in a viscous incompressible fluidhttps://zbmath.org/1521.351382023-11-13T18:48:18.785376Z"Feireisl, Eduard"https://zbmath.org/authors/?q=ai:feireisl.eduard"Roy, Arnab"https://zbmath.org/authors/?q=ai:roy.arnab"Zarnescu, Arghir"https://zbmath.org/authors/?q=ai:zarnescu.arghir-daniConsider a viscous incompressible fluid within a domain (in \(\mathbb{R}^2\) or \(\mathbb{R}^3\)). Now, let a finite number of rigid bodies be immersed in the fluid, and allow then eventually collide. The purpose of this paper is to better understand and clarify this phenomena under the conditions stated in its introduction.
Reviewer: Igor Leite Freire (São Carlos)\(n\)-soliton, breather, lump solutions and diverse traveling wave solutions of the fractional \((2+1)\)-Dimensional Boussinesq equationhttps://zbmath.org/1521.351422023-11-13T18:48:18.785376Z"Wang, Kang-Jia"https://zbmath.org/authors/?q=ai:wang.kang-jia"Liu, Jing-Hua"https://zbmath.org/authors/?q=ai:liu.jinghua"Si, Jing"https://zbmath.org/authors/?q=ai:si.jing"Shi, Feng"https://zbmath.org/authors/?q=ai:shi.feng"Wang, Guo-Dong"https://zbmath.org/authors/?q=ai:wang.guodong(no abstract)Nonclassical Lie symmetry and conservation laws of the nonlinear time-fractional Korteweg-de Vries equationhttps://zbmath.org/1521.351592023-11-13T18:48:18.785376Z"Hashemi, Mir Sajjad"https://zbmath.org/authors/?q=ai:hashemi.mir-sajjad"Haji-Badali, Ali"https://zbmath.org/authors/?q=ai:haji-badali.ali"Alizadeh, Farzaneh"https://zbmath.org/authors/?q=ai:alizadeh.farzaneh"Inc, Mustafa"https://zbmath.org/authors/?q=ai:inc.mustafaSummary: In this paper, we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann-Liouville concept. In this study, first, we employ the classical and nonclassical Lie symmetries (LS) to acquire similarity reductions of the nonlinear fractional far field Korteweg-de Vries (KdV) equation, and second, we find the related exact solutions for the derived generators. Finally, according to the LS generators acquired, we construct conservation laws for related classical and nonclassical vector fields of the fractional far field KdV equation.Construction of fractal soliton solutions for the fractional evolution equations with conformable derivativehttps://zbmath.org/1521.351622023-11-13T18:48:18.785376Z"Wang, Kangle"https://zbmath.org/authors/?q=ai:wang.kangle(no abstract)New solitary wave solutions of the fractional modified KdV-Kadomtsev-Petviashvili equationhttps://zbmath.org/1521.351632023-11-13T18:48:18.785376Z"Wang, Kang-Le"https://zbmath.org/authors/?q=ai:wang.kangle(no abstract)Totally new soliton phenomena in the fractional Zoomeron model for shallow waterhttps://zbmath.org/1521.351642023-11-13T18:48:18.785376Z"Wang, Kang-Le"https://zbmath.org/authors/?q=ai:wang.kangle(no abstract)The well-posedness for the Camassa-Holm type equations in critical Besov spaces \(B_{p , 1}^{1 + \frac{ 1}{ p}}\) with \(1 \leq p < + \infty \)https://zbmath.org/1521.351662023-11-13T18:48:18.785376Z"Ye, Weikui"https://zbmath.org/authors/?q=ai:ye.weikui"Yin, Zhaoyang"https://zbmath.org/authors/?q=ai:yin.zhaoyang"Guo, Yingying"https://zbmath.org/authors/?q=ai:guo.yingyingIn this paper, the authors prove local well-posedness for the Camassa-Holm (CH) equation
\[
u_t - u_{xxt} + 3uu_x = 2u_xu_{xx} + uu_{xxx}\tag{1}
\]
and the Novikov equation
\[
u_t - u_{xxt} = 3uu_xu_{xx} + u^2u_{xxx} - 4u^2u_x\tag{2}
\]
in the critical Besov spaces \(\mathcal C([0,T]:B_{p,1}^{1+\frac 1p}(\mathbb R))\) with \(1 \leq p < \infty\). In addition, they prove local well-posedness for the two-component Camassa-Holm system
\[
\begin{split}
& u_t + uu_x = -\partial_x(1-\partial_{xx})^{-1}(u^2 + \frac 12 u_x^2 + \frac 12 \rho^2) \\
& \rho_t + u \rho_x = - u_x\rho
\end{split}\tag{3}
\]
in \(\mathcal C([0,T]:B_{p,1}^{1+\frac 1p}) \times \mathcal C([0,T]:B_{p,1}^{\frac 1p})\).
Their results follow by first establishing well-posedness results for the abstract equation
\[
\begin{split}
& u_t + A(u) u_x = F(u) \qquad t >0, x \in \mathbb R \\
& u(0,x) = u_0(x)
\end{split}\tag{4}
\]
where \(A(u)\) is a polynomial and \(F\) is a 'good operator' such that for any \(\phi \in C_0^\infty\) and any \(\epsilon > 0\), the following fact holds.
If \(u_n \phi \rightarrow u\phi\) in \(B_{p,1}^{1+\frac 1p-\epsilon}\), then \(\langle F(u_n),\phi \rangle \rightarrow \langle F(u),\phi \rangle\). In addition, they establish similar well-posedness results for an abstract system related to the two-component Camassa-Holm system. These results enable them to prove local well-posedness for (1), (2), and (3) in the Besov spaces discussed above.
The local well-posedness results for the abstract equation (and abstract system) are obtained by using a Lagrangian coordinate transformation. Specifically, the associated Lagrangian scale of (4) is
\[
\begin{split}
& \frac d{dt}y(t,\xi)=A(u)(t,y(t,\xi)) \qquad t >0, \xi \in \mathbb R \\
& y(0,\xi)=\xi.
\end{split}
\]
Then introducing the new variable \(U(t,\xi) = u(t,y(t,\xi))\), (4) becomes
\[
\begin{split}
& U_t = (F(u))(t,y(t,\xi)) := \widetilde{F}(U,y) \qquad t > 0, \xi \in \mathbb R \\
& U(0,\xi) = U_0(\xi) = u_0(\xi).
\end{split}\tag{5}
\]
Establishing properties of (5) allow the authors to prove well-posedness results for (4) and thus (1) and (5). A similar technique is used to prove local well-posedness for the system (3).
Reviewer: Julie L. Levandosky (Framingham)The dynamics of pulse solutions for reaction diffusion systems on a star shaped metric graph with the Kirchhoff's boundary conditionhttps://zbmath.org/1521.351742023-11-13T18:48:18.785376Z"Ei, Shin-Ichiro"https://zbmath.org/authors/?q=ai:ei.shin-ichiro"Mitsuzono, Ken"https://zbmath.org/authors/?q=ai:mitsuzono.ken"Shimatani, Haruki"https://zbmath.org/authors/?q=ai:shimatani.harukiSummary: In this paper, we consider motions of localized patterns for reaction-diffusion systems of general types on a metric star graph which consists of several half-lines with a common end point called ``the junction point'', where the Kirchhoff boundary condition is imposed. Assuming the existence and the stability of pulse and front like patterns for corresponding 1dimensional problems of reaction-diffusion systems, we rigorously derive ordinary differential equations describing the motions of them on a metric star graph. As the application, the attractive motion of a single pulse solution for the Gray-Scott model toward the junction point is shown. It is also shown that a single front solution of Allen-Cahn equation is repulsive against the junction point. The motion of multi pulse solutions and front solutions are also treated.Existence theorems for a generalized Chern-Simons equation on finite graphshttps://zbmath.org/1521.351752023-11-13T18:48:18.785376Z"Gao, Jia"https://zbmath.org/authors/?q=ai:gao.jia"Hou, Songbo"https://zbmath.org/authors/?q=ai:hou.songboSummary: Consider \(G = (V, E)\) as a finite graph, where \(V\) and \(E\) correspond to the vertices and edges, respectively. We study a generalized Chern-Simons equation \(\Delta u = \lambda\mathrm{e}^u(\mathrm{e}^{bu} - 1) + 4\pi\sum_{j = 1}^N \delta_{p_j}\) on \(G\), where \(\lambda\) and \(b\) are positive constants; \(N\) is a positive integer; \(p_1, p_2, \dots, p_N\) are distinct vertices of \(V\); and \(\delta_{p_j}\) is the Dirac delta mass at \(p_j\). We prove that there exists a critical value \(\lambda_c\) such that the equation has a solution if \(\lambda \geq \lambda_c\) and the equation has no solution if \(\lambda < \lambda_c\). We also prove that if \(\lambda > \lambda_c\), the equation has at least two solutions that include a local minimizer for the corresponding functional and a mountain-pass type solution. Our results extend and complete those of \textit{A. Huang} et al. [Commun. Math. Phys. 377, No. 1, 613--621 (2020; Zbl 1447.35338)] and \textit{S. Hou} and \textit{J. Sun} [Calc. Var. Partial Differ. Equ. 61, No. 4, Paper No. 139, 13 p. (2022; Zbl 1491.35238)].
{\copyright 2023 American Institute of Physics}Optimal Hardy weights on the Euclidean latticehttps://zbmath.org/1521.351762023-11-13T18:48:18.785376Z"Keller, Matthias"https://zbmath.org/authors/?q=ai:keller.matthias"Lemm, Marius"https://zbmath.org/authors/?q=ai:lemm.mariusSummary: We investigate the large-distance asymptotics of optimal Hardy weights on \(\mathbb{Z}^d\), \(d\geq 3\), via the super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is the familiar \(\frac{(d-2)^2}{4}|x|^{-2}\) as \(|x|\to \infty \). We prove that the inverse-square behavior of the optimal Hardy weight is robust for general elliptic coefficients on \(\mathbb{Z}^d\): (1) averages over large sectors have inverse-square scaling, (2) for ergodic coefficients, there is a pointwise inverse-square upper bound on moments, and (3) for i.i.d. coefficients, there is a matching inverse-square lower bound on moments. The results imply \(|x|^{-4}\)-scaling for Rellich weights on \(\mathbb{Z}^d\). Analogous results are also new in the continuum setting. The proofs leverage Green's function estimates rooted in homogenization theory.Blow-up problems for Fujita-type parabolic system involving time-dependent coefficients on graphshttps://zbmath.org/1521.351772023-11-13T18:48:18.785376Z"Wu, Yiting"https://zbmath.org/authors/?q=ai:wu.yiting(no abstract)Liouville-type results for elliptic equations with advection and potential terms on the Heisenberg grouphttps://zbmath.org/1521.351782023-11-13T18:48:18.785376Z"Jleli, Mohamed"https://zbmath.org/authors/?q=ai:jleli.mohamed-boussairi"Kirane, Mokhtar"https://zbmath.org/authors/?q=ai:kirane.mokhtar"Samet, Bessem"https://zbmath.org/authors/?q=ai:samet.bessemSummary: We investigate nonlinear elliptic equations of the form \[-\Delta_Hu(\xi)+A(\xi)\cdot\nabla_Hu(\xi)=V(\xi)f(u),\quad\xi\in\mathbb{H}^n,\] where \(\mathbb{H}^n=(\mathbb{R}^{2n+1},\circ)\) is the \((2n+1)\)-dimensional Heisenberg group, \(\Delta_H\) is the Kohn-Laplacian operator, \(\nabla_H\) is the Heisenberg gradient, \(\cdot\) is the inner product in \(\mathbb{R}^{2n}\), the advection term \(A:\mathbb{H}^n\to\mathbb{R}^{2n}\) is a \(C^1\) vector field satisfying a certain decay condition, the potential function \(V:\mathbb{H}^n\to(0,\infty)\) is continuous, and the nonlinearity \(f(u)\) has the form \(-u^{-p}\), \(p>0\), \(u>0\), or \(e^u\). Namely, we establish Liouville-type results for the class of stable solutions to the considered problems. Next, some special cases of the potential function \(V\) are discussed.Noncoercive diffusion equations with Radon measures as initial datahttps://zbmath.org/1521.351792023-11-13T18:48:18.785376Z"Porzio, Maria Michaela"https://zbmath.org/authors/?q=ai:porzio.maria-michaela"Smarrazzo, Flavia"https://zbmath.org/authors/?q=ai:smarrazzo.flavia"Tesei, Alberto"https://zbmath.org/authors/?q=ai:tesei.albertoSummary: We study Radon measure-valued solutions of the Cauchy-Dirichlet problem for \(\partial_t u = \Delta \phi (u)\) for a continuous, nondecreasing, at most powerlike \(\phi \). We prove well-posedness and regularity results, which depend on whether or not the initial data charge sets of suitable capacity (determined both by the Laplacian and by the growth order of \(\phi \)), and on suitable \textit{compatibility conditions}.Diffusive spatial movement with memory in an advective environmenthttps://zbmath.org/1521.351802023-11-13T18:48:18.785376Z"Zhang, Hua"https://zbmath.org/authors/?q=ai:zhang.hua.5"Wang, Hao"https://zbmath.org/authors/?q=ai:wang.hao.4"Song, Yongli"https://zbmath.org/authors/?q=ai:song.yongli"Wei, Junjie"https://zbmath.org/authors/?q=ai:wei.junjieSummary: The movements of species in a river are driven by random diffusion, unidirectional water flow, and cognitive judgement with spatial memory. In this paper, we formulate a reaction-diffusion-advection model with memory-based diffusion and homogeneous Dirichlet boundary conditions. The existence of a nonconstant positive steady state is proven. We obtain the linear stability of the steady state by analysing the eigenvalues of the associated linear operator: the nonconstant steady state can always be linearly stable regardless of the memory delay, while the model can also possess Hopf bifurcation as the memory delay varies. Moreover, theoretical and numerical results show that large advection annihilates oscillation patterns and drives the species to concentrate downstream.Singular boundary behaviour and large solutions for fractional elliptic equationshttps://zbmath.org/1521.351812023-11-13T18:48:18.785376Z"Abatangelo, Nicola"https://zbmath.org/authors/?q=ai:abatangelo.nicola"Gómez-Castro, David"https://zbmath.org/authors/?q=ai:gomez-castro.david"Vázquez, Juan Luis"https://zbmath.org/authors/?q=ai:vazquez.juan-luisSummary: We perform a unified analysis for the boundary behaviour of solutions to nonlocal fractional equations posed in bounded domains. Based on previous findings for some models of the fractional Laplacian operator, we show how it strongly differs from the boundary behaviour of solutions to elliptic problems modelled upon the Laplace-Poisson equation with zero boundary data.
In the classical case it is known that, at least in a suitable weak sense, solutions of the homogeneous Dirichlet problem with a forcing term tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary.
Here, we show that, for equations driven by a wide class of nonlocal fractional operators, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Green function.Correction to: ``Solvability and Volterra property of nonlocal problems for mixed fractional-order diffusion-wave equation''https://zbmath.org/1521.351822023-11-13T18:48:18.785376Z"Adil, Nauryzbay"https://zbmath.org/authors/?q=ai:adil.nauryzbay"Berdyshev, Abdumauvlen S."https://zbmath.org/authors/?q=ai:berdyshev.abdumauvlen-suleymanovich"Eshmatov, B. E."https://zbmath.org/authors/?q=ai:eshmatov.bakhodir-e"Baishemirov, Zharasbek D."https://zbmath.org/authors/?q=ai:baishemirov.zharasbek-dFrom the text: Following publication of the original article [ibid. 2023, Paper No. 47, 29 p. (2023; Zbl 1518.35613)], the authors identified an error in the author name Abdumauvlen S. Berdyshev.
The incorrect author name is: Abdumauvlen S. Bersyhev
The correct author name is: Abdumauvlen S. Berdyshev
The author group has been updated above and the original article [loc. cit.] has been corrected.On the critical behavior for time-fractional reaction diffusion problemshttps://zbmath.org/1521.351832023-11-13T18:48:18.785376Z"Aldawish, Ibtisam"https://zbmath.org/authors/?q=ai:aldawish.ibtisam"Samet, Bessem"https://zbmath.org/authors/?q=ai:samet.bessemSummary: We first investigate the existence and nonexistence of weak solutions to the time-fractional reaction diffusion problem
\[
\frac{\partial^\alpha u}{\partial t^\alpha}-\frac{\partial^2 u}{\partial x^2}+u\ge x^{-a}|u|^p, \ t>0, \ x\in(0,1],\quad u(0,x)=u_0(x), \ x\in(0,1]
\]
under the inhomogeneous Dirichlet boundary condition
\[
u(t,1)=\delta,\quad t>0,
\]
where \(u=u(t,x)\), \(0<\alpha<1\), \(\frac{\partial^\alpha}{\partial t^\alpha}\) is the time-Caputo fractional derivative of order \(\alpha\), \(a\ge 0\), \(p>1\) and \(\delta>0\). We show that, if \(a\le 2\), the existence holds for all \(p>1\) while if \(a>2\), then the dividing line with respect to existence or nonexistence is given by the critical exponent \(p^*=a-1\). The proof of the nonexistence result is based on nonlinear capacity estimates specifically adapted to the nonlocal nature of the problem, the modified Helmholtz operator \(-\frac{\partial^2}{\partial x^2}+I\), and the considered boundary condition. The existence part is proved by the construction of explicit solutions. We next extend our study to the case of systems.Spike solutions for a fractional elliptic equation in a compact Riemannian manifoldhttps://zbmath.org/1521.351842023-11-13T18:48:18.785376Z"Bendahou, Imene"https://zbmath.org/authors/?q=ai:bendahou.imene"Khemiri, Zied"https://zbmath.org/authors/?q=ai:khemiri.zied"Mahmoudi, Fethi"https://zbmath.org/authors/?q=ai:mahmoudi.fethiSummary: Given an \(n\)-dimensional compact Riemannian manifold \((M,g)\) without boundary, we consider the nonlocal equation
\[\varepsilon^{2s} P_g^s u + u = u^p \quad \hbox{in }\, (M,g),\]
where \(P_g^s\) stands for the fractional Paneitz operator with principal symbol \((-\Delta_g)^s\), \(s \in (0,1)\), \( p \in (1,2_s^*-1)\) with \(2_s^* := \frac{2n}{n-2s} \), \(n>2s\), represents the critical Sobolev exponent and \(\varepsilon > 0\) is a small real parameter. We construct a family of positive solutions \(u_\varepsilon\) that concentrate, as \(\varepsilon \to 0\) goes to zero, near critical points of the mean curvature \(H\) for \(0 <s< \frac{1}{2}\) and near critical points of a reduced function involving the scalar curvature of the manifold~ \(M\) for \( \frac{1}{2} \leq s < 1\).On the equivalence of classical Helmholtz equation and fractional Helmholtz equation with arbitrary orderhttps://zbmath.org/1521.351852023-11-13T18:48:18.785376Z"Cheng, Xinyu"https://zbmath.org/authors/?q=ai:cheng.xinyu"Li, Dong"https://zbmath.org/authors/?q=ai:li.dong"Yang, Wen"https://zbmath.org/authors/?q=ai:yang.wen.1|yang.wenSummary: We show the equivalence of the classical Helmholtz equation and the fractional Helmholtz equation with arbitrary order. This improves a recent result of \textit{V. Guan} et al. [Commun. Contemp. Math. 25, No. 2, Article ID 2250016, 18 p. (2023; Zbl 1509.35344)].Analysis of fractional differential equations with the help of different operatorshttps://zbmath.org/1521.351862023-11-13T18:48:18.785376Z"Iqbal, Naveed"https://zbmath.org/authors/?q=ai:iqbal.naveed-h"Al Harbi, Moteb Fheed Saad"https://zbmath.org/authors/?q=ai:al-harbi.moteb-fheed-saad"Alshammari, Saleh"https://zbmath.org/authors/?q=ai:alshammari.saleh"Zaland, Shamsullah"https://zbmath.org/authors/?q=ai:zaland.shamsullah(no abstract)Application of Hosoya polynomial to solve a class of time-fractional diffusion equationshttps://zbmath.org/1521.351872023-11-13T18:48:18.785376Z"Jafari, Hossein"https://zbmath.org/authors/?q=ai:jafari.hossein"Ganji, Roghayeh Moallem"https://zbmath.org/authors/?q=ai:ganji.roghayeh-moallem"Narsale, Sonali Mandar"https://zbmath.org/authors/?q=ai:narsale.sonali-mandar"Kgarose, Maluti"https://zbmath.org/authors/?q=ai:kgarose.maluti"Nguyen, Van Thinh"https://zbmath.org/authors/?q=ai:nguyen.van-thinh(no abstract)Uniqueness of the potential in a time-fractional diffusion equationhttps://zbmath.org/1521.351882023-11-13T18:48:18.785376Z"Jing, Xiaohua"https://zbmath.org/authors/?q=ai:jing.xiaohua"Peng, Jigen"https://zbmath.org/authors/?q=ai:peng.jigenSummary: This article concerns the uniqueness of an inverse coefficient problem of identifying a spatially varying potential in a one-dimensional time-fractional diffusion equation. The input sources are given by a complete system in \(L^2 (0,1)\), and measurements are observed at the end point of the spatial interval. Firstly, we provide the positive lower bound of the Green function for the differential operator with different boundary conditions. Then, based on the positive lower bound estimation of the Green function, the relationship between the Green function, the solution of the forward problem, and the potential, such measurements uniquely determine the potential on the entire interval under different boundary conditions.Global solutions of a fractional semilinear pseudo-parabolic equation with nonlocal sourcehttps://zbmath.org/1521.351892023-11-13T18:48:18.785376Z"Li, Na"https://zbmath.org/authors/?q=ai:li.na.4|li.na"Fang, Shaomei"https://zbmath.org/authors/?q=ai:fang.shaomeiSummary: In this paper, the initial boundary value problem for a fractional nonlocal semilinear pseudo-parabolic equation is established. Firstly, we get the local solution by the standard Galerkin method and the priori estimates. Next, by applying potential well argument, the existence and uniqueness of the global solution are proved for initial energy \(J(u_0)\le d\).Global well-posedness of a Cauchy problem for a nonlinear parabolic equation with memoryhttps://zbmath.org/1521.351902023-11-13T18:48:18.785376Z"Nguyen, Anh Tuan"https://zbmath.org/authors/?q=ai:nguyen.anh-tuan"Nghia, Bui Dai"https://zbmath.org/authors/?q=ai:nghia.bui-dai"Nguyen, Van Thinh"https://zbmath.org/authors/?q=ai:nguyen.van-thinh(no abstract)Classification of solutions to mixed order elliptic system with general nonlinearityhttps://zbmath.org/1521.351912023-11-13T18:48:18.785376Z"Peng, Shaolong"https://zbmath.org/authors/?q=ai:peng.shaolongSummary: In this paper, we consider the mixed order elliptic system
\[
\begin{cases} (-\Delta)^{\frac{\alpha}{2}}u(x)=f(u,v), \\
(-\Delta) v(x)=g(u,v), \end{cases}
\]
where \(u\geq 0\), \(\alpha \in (0,2)\), \(v\) may change signs. We aim to study the classification results of solutions to the above semilinear elliptic system in \(\mathbb{R}^2\). We first derive the equivalence between the above PDE system and the corresponding IE (integral equation) system. Then, applying the method of moving spheres in integral form combined with integral inequalities, under certain assumptions, we give a complete classification of the classical solutions to the above system in \(\mathbb{R}^2\) .Terminal value problem for stochastic fractional equation within an operator with exponential kernelhttps://zbmath.org/1521.351922023-11-13T18:48:18.785376Z"Phuong, Nguyen Duc"https://zbmath.org/authors/?q=ai:phuong.nguyen-duc"Hoan, Luu Vu Cam"https://zbmath.org/authors/?q=ai:hoan.luu-vu-cam"Baleanu, Dumitru"https://zbmath.org/authors/?q=ai:baleanu.dumitru-i"Nguyen, Anh Tuan"https://zbmath.org/authors/?q=ai:nguyen.anh-tuan(no abstract)New results on continuity by order of derivative for conformable parabolic equationshttps://zbmath.org/1521.351932023-11-13T18:48:18.785376Z"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Nguyen, Van Tien"https://zbmath.org/authors/?q=ai:nguyen.van-tien"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donal"Can, Nguyen Huu"https://zbmath.org/authors/?q=ai:can.nguyen-huu"Nguyen, Van Thinh"https://zbmath.org/authors/?q=ai:nguyen.van-thinh(no abstract)Space-dependent variable-order time-fractional wave equation: existence and uniqueness of its weak solutionhttps://zbmath.org/1521.351942023-11-13T18:48:18.785376Z"van Bockstal, K."https://zbmath.org/authors/?q=ai:van-bockstal.karel"Hendy, A. S."https://zbmath.org/authors/?q=ai:hendy.ahmed-s"Zaky, M. A."https://zbmath.org/authors/?q=ai:zaky.mahmoud-aSummary: The investigation of an initial-boundary value problem for a fractional wave equation with space-dependent variable-order wherein the coefficients have a dependency on the spatial and time variables is the concern of this work. This type of variable-order fractional differential operator originates in the modelling of viscoelastic materials. The global in time existence of a unique weak solution to the model problem has been proved under appropriate conditions on the data. Rothe's time discretization method is applied to achieve that purpose.On the new exact traveling wave solutions of the time-space fractional strain wave equation in microstructured solids via the variational methodhttps://zbmath.org/1521.351952023-11-13T18:48:18.785376Z"Wang, Kang-Jia"https://zbmath.org/authors/?q=ai:wang.kang-jiaSummary: In this paper, we mainly study the time-space fractional strain wave equation in microstructured solids. He's variational method, combined with the two-scale transform are implemented to seek the solitary and periodic wave solutions of the time-space strain wave equation. The main advantage of the variational method is that it can reduce the order of the differential equation, thus simplifying the equation, making the solving process more intuitive and avoiding the tedious solving process. Finally, the numerical results are shown in the form of 3D and 2D graphs to prove the applicability and effectiveness of the method. The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations in physics.Solitary wave dynamics of the local fractional Bogoyavlensky-Konopelchenko modelhttps://zbmath.org/1521.351962023-11-13T18:48:18.785376Z"Wang, Kangle"https://zbmath.org/authors/?q=ai:wang.kangle(no abstract)Normalized ground states and multiple solutions for nonautonomous fractional Schrödinger equationshttps://zbmath.org/1521.351972023-11-13T18:48:18.785376Z"Yang, Chen"https://zbmath.org/authors/?q=ai:yang.chen.2"Yu, Shu-Bin"https://zbmath.org/authors/?q=ai:yu.shubin"Tang, Chun-Lei"https://zbmath.org/authors/?q=ai:tang.chun-leiSummary: In this paper, we consider the following fractional Schrödinger equations with prescribed \(L^2\)-norm constraint:
\[
\begin{cases}
(-\Delta)^s u = \lambda u + h(\varepsilon x)f(u)\text{ in }\mathbb{R}^N,\\
\int_{\mathbb{R}^N}|u|^2dx = a^2,
\end{cases}
\]
where \(0 < s < 1\), \(N \geq 3\), \(a, \varepsilon > 0\), \(h\in C(\mathbb{R}^N, \mathbb{R^+})\) and \(f\in C(\mathbb{R}, \mathbb{R})\). In the mass subcritical case but under general assumptions on \(f\), we prove the multiplicity of normalized solutions to this problem. Specifically, we show that the number of normalized solutions is at least the number of global maximum points of \(h\) when \(\varepsilon\) is small enough. Before that, without any restrictions on \(\varepsilon\) and the number of global maximum points, the existence of normalized ground states can be determined. In this sense, by studying the relationship between \(h_0 := \inf_{x\in\mathbb{R}^N}h(x)\) and \(h_\infty := \lim_{|x|\rightarrow\infty}h(x)\), we establish new results on the existence of normalized ground states for nonautonomous elliptic equations.Almost periodic solutions of the wave equation with damping and impulsive actionhttps://zbmath.org/1521.351982023-11-13T18:48:18.785376Z"Dvornyk, A. V."https://zbmath.org/authors/?q=ai:dvornyk.a-v"Tkachenko, V. I."https://zbmath.org/authors/?q=ai:tkachenko.victor-iSummary: We obtain sufficient conditions for the existence of piecewise continuous almost periodic solutions to the damped wave equation with impulsive action.An inverse problem of recovering the variable order of the derivative in a fractional diffusion equationhttps://zbmath.org/1521.351992023-11-13T18:48:18.785376Z"Artyushin, A. N."https://zbmath.org/authors/?q=ai:artyushin.aleksandr-nikolaevichSummary: We consider a fractional diffusion equation with variable space-dependent order of the derivative in a bounded multidimensional domain. The initial data are homogeneous and the right-hand side and its time derivative satisfy some monotonicity conditions. Addressing the inverse problem with final overdetermination, we establish the uniqueness of a solution as well as some necessary and sufficient solvability conditions in terms of a certain constructive operator \(A \). Moreover, we give a simple sufficient solvability condition for the inverse problem. The arguments rely on the Birkhoff-Tarski theorem.An identification problem of source function in the system of composite typehttps://zbmath.org/1521.352002023-11-13T18:48:18.785376Z"Belov, Yuriĭ Ya."https://zbmath.org/authors/?q=ai:belov.yurii-yaSummary: An identification problem of source function for the semievolutionary system of two partial differential equations, one of which is parabolic, and the second -- elliptic are investigated. The Cauchy problem and the first boundary-value problem are considered. Initial problems are approximated by problems in which the elliptic equation is replaced with the parabolic equation containing the small parameter \(\varepsilon>0\) at a derivative with respect to time.On the problem of identification of two lower coefficients and the coefficient by the derivative with respect to time in the parabolic equationhttps://zbmath.org/1521.352012023-11-13T18:48:18.785376Z"Datsenko, Anzhelika V."https://zbmath.org/authors/?q=ai:datsenko.anzhelika-v"Polyntseva, Svetlana V."https://zbmath.org/authors/?q=ai:polyntseva.svetlana-vSummary: The theorem of existence and uniqueness of classical solution of identification problem of two lower coefficients and the coefficient by the derivative with respect to time in the class of smooth bounded functions is proved.
In the proof of the existence and uniqueness of the inverse problem solution using the overdetermination conditions, the original inverse problem is reduced to the direct problem for the loaded (containing traces of unknown functions and their derivatives) equation. The investigation of the correctness of the direct problem is obtained by the method of weak approximation.Identification of the potential coefficient in the wave equation with incomplete data: a sentinel methodhttps://zbmath.org/1521.352022023-11-13T18:48:18.785376Z"Elhamza, Billal"https://zbmath.org/authors/?q=ai:elhamza.billal"Hafdallah, Abdelhak"https://zbmath.org/authors/?q=ai:hafdallah.abdelhakSummary: In this paper, we consider a wave equation with incomplete data, where we do not know the potential coefficient and the initial conditions. From observing the system in the boundary, we want to get information on the potential coefficient independently of the initial conditions. This can be obtained using the sentinel method of Lions, which is a functional insensitive to certain parameters. Shows us through the adjoint system that the existence of the sentinel is equivalent to an optimal control problem. We solve this optimal control problem by using the Hilbert uniqueness method (HUM).An identification problem of the source function of the special form in two-dimensional parabolic equationhttps://zbmath.org/1521.352032023-11-13T18:48:18.785376Z"Frolenkov, Igor' V."https://zbmath.org/authors/?q=ai:frolenkov.igor-v"Kriger, Ekaterina N."https://zbmath.org/authors/?q=ai:kriger.ekaterina-nSummary: The existence, uniqueness and stability of solution by input data of the identification problem for parabolic equation with source function of the special form in the case of Cauchy's data has been proved in this article.An representation of the solution of the inverse problem for a multidimensional parabolic equation with initial data in the form of a producthttps://zbmath.org/1521.352042023-11-13T18:48:18.785376Z"Frolenkov, Igor' V."https://zbmath.org/authors/?q=ai:frolenkov.igor-v"Romanenko, Galina V."https://zbmath.org/authors/?q=ai:romanenko.galina-vSummary: An identification problem of the coefficient at differential operator of second order in multidimensional parabolic equation with Cauchy data was studied in this article. The theorems of existence and uniqueness of the solution for direct and inverse problems has been proved.On reconstruction of small sources from Cauchy data at a fixed frequencyhttps://zbmath.org/1521.352052023-11-13T18:48:18.785376Z"Harris, Isaac"https://zbmath.org/authors/?q=ai:harris.isaac"Le, Thu"https://zbmath.org/authors/?q=ai:le.thu-minh"Nguyen, Dinh-Liem"https://zbmath.org/authors/?q=ai:nguyen.dinh-liemSummary: This short paper is concerned with the numerical reconstruction of small sources from boundary Cauchy data for a single frequency. We study a sampling method to determine the location of small sources in a very fast and robust way. Furthermore, the method can also compute the intensity of point sources provided that the sources are well separated. A simple justification of the method is done using the Green representation formula and an asymptotic expansion of the radiated field for small volume sources. The implementation of the method is non-iterative, computationally cheap, fast, and very simple. Numerical examples are presented to illustrate the performance of the method.A Tikhonov regularization method for Cauchy problem based on a new relaxation modelhttps://zbmath.org/1521.352062023-11-13T18:48:18.785376Z"Huang, Qin"https://zbmath.org/authors/?q=ai:huang.qin"Gong, Rongfang"https://zbmath.org/authors/?q=ai:gong.rongfang"Jin, Qinian"https://zbmath.org/authors/?q=ai:jin.qinian"Zhang, Ye"https://zbmath.org/authors/?q=ai:zhang.yeSummary: In this paper, we consider a Cauchy problem of recovering both missing value and flux on inaccessible boundary from Dirichlet and Neumann data measured on the remaining accessible boundary. Associated with two mixed boundary value problems, a regularized Kohn-Vogelius formulation is proposed. With an introduction of a relaxation parameter, the Dirichlet boundary conditions are approximated by two Robin ones. Compared to the existing work, weaker regularity is required on the Dirichlet data. This makes the proposed model simpler and more efficient in computation. A series of theoretical results are established for the new reconstruction model. Several numerical examples are provided to show feasibility and effectiveness of the proposed method. For simplicity of the statements, we take Poisson equation as the governed equation. However, the proposed method can be applied directly to Cauchy problems governed by more general equations, even other linear or nonlinear inverse problems.Extracting discontinuity using the probe and enclosure methodshttps://zbmath.org/1521.352072023-11-13T18:48:18.785376Z"Ikehata, Masaru"https://zbmath.org/authors/?q=ai:ikehata.masaruSummary: This is a review article on the development of the probe and enclosure methods from past to present, focused on their central ideas together with various applications.Convergence analysis of an alternating direction method of multipliers for the identification of nonsmooth diffusion parameters with total variationhttps://zbmath.org/1521.352082023-11-13T18:48:18.785376Z"Ouakrim, Y."https://zbmath.org/authors/?q=ai:ouakrim.youssef"Boutaayamou, I."https://zbmath.org/authors/?q=ai:boutaayamou.idriss"El Yazidi, Y."https://zbmath.org/authors/?q=ai:el-yazidi.youness"Zafrar, A."https://zbmath.org/authors/?q=ai:zafrar.abderrahimSummary: The paper presents a numerical method for identifying discontinuous conductivities in elliptic equations from boundary observations. The solutions to this inverse problem are obtained through a constrained optimization problem, where the cost functional is a combination of the Kohn-Vogelius and Total Variation functionals. Instead of regularizing the Total Variation stabilization functional, which is commonly used in the literature, we introduce an Alternating Direction Method of Multipliers to preserve the favorable properties of non-smoothness and convexity. The discretization is carried out using a mixed finite element/volume method, while the numerical solutions are iteratively computed using a variant of the Uzawa algorithm. We show the surjectivity of the derivatives of the constraints related to the discrete optimization problem and derive a source condition for the discrete inverse problem. We then investigate the convergence analysis and establish the convergence rate. Finally, we conclude with some numerical experiments to illustrate the efficiency of the proposed method.Lipschitz stability of recovering the conductivity from internal current densitieshttps://zbmath.org/1521.352092023-11-13T18:48:18.785376Z"Qiu, Lingyun"https://zbmath.org/authors/?q=ai:qiu.lingyun"Zheng, Siqin"https://zbmath.org/authors/?q=ai:zheng.siqinSummary: In recent years, coupled physics imaging techniques have been developed to produce clearer images than those produced by electrical impedance tomography. This paper focuses on the inverse problem arising in current density impedance imaging and magneto-acousto-electric tomography. We consider the electrostatic equation \(\nabla\cdot(\sigma\nabla w_b)=0\) in a bounded domain \(\Omega\subset\mathbb{R}^3\) with either the Dirichlet or Neumann boundary condition \(b\), where \(\sigma\) is a scalar conductivity function. The inverse problem is formulated as recovering \(\sigma\) from vector fields \(J_b=\sigma\nabla w_b\) with different boundary conditions \(b\). We provide a local Lipschitz stability, stating that near some known \(\sigma_0\) and under some regularity assumptions, we can find \(b_1\) and \(b_2\) by constructing complex geometrical optics (CGO) solutions such that \(\Vert\ln\sigma^{(1)}-\ln\sigma^{(2)}\Vert_{C^{m,\alpha}(\overline{\Omega})}\leqslant C\sum^2_{j=1}\Vert J^{(1)}_{b_j}-J^{(2)}_{b_j}\Vert_{C^{m,\alpha}(\overline{\Omega})}\). Furthermore, we modify the CGO solutions using the reflection method to make \(b_1\) and \(b_2\) vanish on a portion of a plane, and prove a local Lipschitz stability with partial data.Dynamics of nonlocal and local SIR diffusive epidemic model with free boundarieshttps://zbmath.org/1521.352102023-11-13T18:48:18.785376Z"Li, Chenglin"https://zbmath.org/authors/?q=ai:li.chenglinSummary: This paper is concerned with nonlocal and local diffusive SIR epidemic model with free boundaries including convolution, which is natural extension of reaction diffusion systems with free boundary problems and local diffusions. The existence of unique global solution for this model is considered. Dichotomy of the spreading and vanishing is established. A spreading barrier line is found to determine whether the spreading of disease will fail finally. The spreading of disease will fail when it cannot spread across the spreading barrier line \(l^\ast\), while it will be successful when it transcends over this barrier line. The results show that if the basic reproduction number \(\mathcal{R}_0 < 1\), the spreading of disease will fail eventually, and if \(\mathcal{R}_0 > 1 + \frac{d_2}{\mu_2 + \alpha}\), the spreading of disease will get success finally. We also find that the spreading coefficients play important role in the spreading achievement. When \(1 + \frac{\beta}{\theta\mu_1} < \mathcal{R}_0 < 1 + \frac{d_2}{\mu_2 + \alpha}\), the spreading coefficient decides whether the spreading of disease will be successful. It is shown that the spreading will be successful when the spreading coefficient is relatively big, while the spreading will fail if the spreading coefficient is relatively small.Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficientshttps://zbmath.org/1521.352112023-11-13T18:48:18.785376Z"Nauryz, Targyn"https://zbmath.org/authors/?q=ai:nauryz.targyn-atanbekovich"Briozzo, Adriana C."https://zbmath.org/authors/?q=ai:briozzo.adriana-cSummary: In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modeled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature solution for two phases and free boundaries which describe the position of boiling and melting interfaces. Existence and uniqueness of the similarity type solution is provided by using the fixed point Banach theorem.Fourth-order nonlinear degenerate problem for image decompositionhttps://zbmath.org/1521.352122023-11-13T18:48:18.785376Z"Nokrane, Ahmed"https://zbmath.org/authors/?q=ai:nokrane.ahmed"Alaa, Nour Eddine"https://zbmath.org/authors/?q=ai:alaa.noureddine"Aqel, Fatima"https://zbmath.org/authors/?q=ai:aqel.fatima-al-zahraSummary: The aim of this work is to study a new coupled fourth-order reaction-diffusion system, applied to image decomposition into cartoons and textures. The existence and uniqueness of an entropy solution to the system with initial data \(BH\) are established using Galerkin's method. Then, numerical experiments and comparisons with other models have been performed to show the efficiency of the proposed model in image decomposition.Reaction-diffusion on a time-dependent interval: refining the notion of `critical length'https://zbmath.org/1521.352132023-11-13T18:48:18.785376Z"Allwright, Jane"https://zbmath.org/authors/?q=ai:allwright.janeSummary: A reaction-diffusion equation is studied in a time-dependent interval whose length varies with time. The reaction term is either linear or of KPP type. On a fixed interval, it is well known that if the length is less than a certain critical value then the solution tends to zero. When the domain length may vary with time, we prove conditions under which the solution does and does not converge to zero in long time. We show that, even with the length always strictly less than the `critical length', either outcome may occur. Examples are given. The proof is based on upper and lower estimates for the solution, which are derived in this paper for a general time-dependent interval.Sharp interface limit of stochastic Cahn-Hilliard equation with singular noisehttps://zbmath.org/1521.352142023-11-13T18:48:18.785376Z"Baňas, Ľubomír"https://zbmath.org/authors/?q=ai:banas.lubomir"Yang, Huanyu"https://zbmath.org/authors/?q=ai:yang.huanyu"Zhu, Rongchan"https://zbmath.org/authors/?q=ai:zhu.rongchanSummary: We study the sharp interface limit of the two dimensional stochastic Cahn-Hilliard equation driven by two types of singular noise: a space-time white noise and a space-time singular divergence-type noise. We show that with appropriate scaling of the noise the solutions of the stochastic problems converge to the solutions of the determinisitic Mullins-Sekerka/Hele-Shaw problem.Infinite pinninghttps://zbmath.org/1521.352152023-11-13T18:48:18.785376Z"Dondl, Patrick"https://zbmath.org/authors/?q=ai:dondl.patrick-w"Jesenko, Martin"https://zbmath.org/authors/?q=ai:jesenko.martin"Scheutzow, Michael"https://zbmath.org/authors/?q=ai:scheutzow.michaelSummary: In this work, we address the occurrence of infinite pinning in a random medium. We suppose that an initially flat interface starts to move through the medium due to some constant driving force. The medium is assumed to contain random obstacles. We model their positions by a Poisson point process and their strengths are not bounded. We determine a necessary condition on its distribution so that regardless of the driving force the interface gets pinned.Fractional soliton dynamics of electrical microtubule transmission line model with local \(M\)-derivativehttps://zbmath.org/1521.352162023-11-13T18:48:18.785376Z"Raza, Nauman"https://zbmath.org/authors/?q=ai:raza.nauman"Arshed, Saima"https://zbmath.org/authors/?q=ai:arshed.saima"Khan, Kashif Ali"https://zbmath.org/authors/?q=ai:khan.kashif-ali"Inc, Mustafa"https://zbmath.org/authors/?q=ai:inc.mustafaSummary: In this paper, two integrating strategies namely \(\exp[-\phi(\chi)]\) and \(\frac{G^\prime}{G^2}\)-expansion methods together with the attributes of local-\(M\) derivatives have been acknowledged on the electrical microtubule (MT) model to retrieve soliton solutions. The said model performs a significant role in illustrating the waves propagation in nonlinear systems. MTs are also highly productive in signaling, cell motility, and intracellular transport. The proposed algorithms yielded solutions of bright, dark, singular, and combo fractional soliton type. The significance of the fractional parameters of the fetched results is explained and presented vividly.Equilibria of vortex type Hamiltonians on closed surfaceshttps://zbmath.org/1521.370802023-11-13T18:48:18.785376Z"Ahmedou, Mohameden"https://zbmath.org/authors/?q=ai:ould-ahmedou.mohameden"Bartsch, Thomas"https://zbmath.org/authors/?q=ai:bartsch.thomas.1|bartsch.thomas.2"Fiernkranz, Tim"https://zbmath.org/authors/?q=ai:fiernkranz.timThis paper is concerned with the proof of existence of critical points of a certain type of functionals on a closed Riemannian surface. These manifolds are assumed to be not homeomorphic to the sphere or the projection plane.
The functionals under consideration (vortex-type Hamiltonians) involve the Green function of the Laplace-Beltrami operator and a certain \(C^1\) function. In particular, under a suitable choice of this function one gets the Kirchhoff-Routh Hamiltonian of fluid dynamics.
The main results are stated in Theorems 1.1 and 1.3.
Reviewer: Igor Leite Freire (São Carlos)Quasi-periodic solutions for quintic completely resonant derivative beam equations on \(\mathbb{T}^2\)https://zbmath.org/1521.370842023-11-13T18:48:18.785376Z"Ge, Chuanfang"https://zbmath.org/authors/?q=ai:ge.chuanfang"Geng, Jiansheng"https://zbmath.org/authors/?q=ai:geng.jianshengSummary: In the present paper, we consider two dimensional completely resonant, derivative, quintic nonlinear beam equations with reversible structure. Because of this reversible system without external parameters or potentials, Birkhoff normal form reduction is necessary before applying Kolmogorov-Arnold-Moser (KAM) theorem. As application of KAM theorem, the existence of partially hyperbolic, small amplitude, quasi-periodic solutions of the reversible system is proved in this paper.
{\copyright 2023 American Institute of Physics}Well-posedness of fractional stochastic complex Ginzburg-Landau equations driven by regular additive noisehttps://zbmath.org/1521.370922023-11-13T18:48:18.785376Z"Liu, Aili"https://zbmath.org/authors/?q=ai:liu.aili"Zou, Yanyan"https://zbmath.org/authors/?q=ai:zou.yanyan"Ren, Die"https://zbmath.org/authors/?q=ai:ren.die"Shu, Ji"https://zbmath.org/authors/?q=ai:shu.jiSummary: This paper deals with the well-posedness of the solutions of the fractional complex Ginzburg-Landau equation driven by locally Lipschitz nonlinear diffusion terms defined on \(R^n\). We first give the pathwise uniform estimates and uniform estimates on average. Then we prove the existence, uniqueness and measurability of solutions for the equation.Self-similarity in homogeneous stationary and evolution problemshttps://zbmath.org/1521.470732023-11-13T18:48:18.785376Z"Cholewa, Jan W."https://zbmath.org/authors/?q=ai:cholewa.jan-w"Rodriguez-Bernal, Anibal"https://zbmath.org/authors/?q=ai:rodriguez-bernal.anibalThe authors of this paper investigate the self-similarity properties related to linear elliptic and evolutionary problems involving homogeneous operators in several spaces including measures. They employ these techniques to analyse in particular \(2m\)th-order diffusion equations and the associated fractional problems. Homogenous operators are those that interact in a special form with the dilations of functions in \(\mathbb{R}^N\), defined by
\[
\phi_R(x)=\phi(Rx),\quad x\in\mathbb{R}^N,\ R>0.
\]
An operator \(L\) with domain \(\mathrm{D}(L)\) in a Banach space \(X\) of functions or distributions which is invariant by rescaling is homogeneous of degree \(\sigma \in \mathbb{R}\) if
\[
L(\phi_R)=R^\sigma(L\phi)_R,\quad \phi\in\mathrm{D}(L),\ R>0.
\]
In fact, the operator semigroup \((S(t))_{t\geq0}\) serving as the solution of the equation \(u_t+Lu=0\), \(t\geq0\), is homogenous of degree \(\sigma\), i.e., for all \(t > 0\), \(R > 0\) and \(\phi \in X\), one has
\[
S(t)\phi_R=(S(R^\sigma t)\phi)_R.
\]
The authors analyse these types of semigroups and resolvent operators. Among others, the authors investigate \(2m\)-order parabolic equations as well as Ornstein-Uhlenbeck-type semigroups.
Reviewer: Christian Budde (Bloemfontein)A uniform Sobolev inequality for ancient Ricci flows with bounded Nash entropyhttps://zbmath.org/1521.530702023-11-13T18:48:18.785376Z"Chan, Pak-Yeung"https://zbmath.org/authors/?q=ai:chan.pak-yeung"Ma, Zilu"https://zbmath.org/authors/?q=ai:ma.zilu"Zhang, Yongjia"https://zbmath.org/authors/?q=ai:zhang.yongjiaThe authors show that an ancient Ricci flow with uniformly bounded Nash entropy has uniformly bounded \(\nu\)-functional. Consequently, on such an ancient solution, there are uniform logarithmic Sobolev and Sobolev inequalities. As an application of the Sobolev inequality, they also prove a certain volume growth lower bound for steady gradient Ricci solitons without any curvature positivity. Under the additional condition that the scalar curvature attains its maximum, the authors show that the steady gradient Ricci soliton must have quadratic volume growth without imposing any non-collapsed condition.
Reviewer: Abimbola Abolarinwa (Lagos)Ricci flow on manifolds with boundary with arbitrary initial metrichttps://zbmath.org/1521.530712023-11-13T18:48:18.785376Z"Chow, Tsz-Kiu Aaron"https://zbmath.org/authors/?q=ai:chow.tsz-kiu-aaronThis paper is concerned with investigating the short time existence and uniqueness of the Ricci flow on a compact manifold \((M,g)\) with boundary \(\partial M\). That is, finding a solution to the following PDE with arbitrary smooth initial data \(g_0\):
\[
\begin{cases} \frac{\partial}{\partial t} g(t) = -2\mathrm{Ric}(g(t))&\text{on }M \times (0,T],\\
A_{g(t)} = 0 &\text{on } \partial M \times (0,T],\\
g(0) = g_0, \end{cases}
\]
where \(A_{g(t)}\) denotes the second fundamental form of the boundary \(\partial M\) with respect to the metric \(g(t)\). Past work on this problem required, for instance, that the boundary of the initial data \(g_0\) be totally geodesic. However, the author is able to remove this assumption.
The strategy of the proof is to consider the double \(\widetilde{M}\) of the compact manifold \(M\) with boundary. Since \(\widetilde{M}\) is a closed manifold, one would hope that standard results and techniques for Ricci flow could be used. However, the initial metric on \(\widetilde{M}\) is only Hölder continuous. In order to prove short-time existence and uniqueness for a Ricci flow starting at such rough initial data, the author uses a by now standard and powerful strategy of considering the Ricci-De Turck flow and then transferring the existence and uniqueness of such a flow to the desired Ricci flow. That each of these two flows induces a solution to the other additionally requires knowledge of a particular solution to the harmonic map heat flow. Given the roughness of the initial data in the aforementioned situations, a lot of detailed analysis, which includes dealing with suitable weighted parabolic Hölder spaces, is conducted by the author. This all culminates in uses of the Banach fixed point theorem to prove the desired results.
As a consequence of the work mentioned above, the author is able to prove that various curvature conditions are preserved along the Ricci flow, provided the boundary \(\partial M\) of the initial data satisfies certain curvature conditions. The list of these preserved conditions is the following:
(1) If \((M,g_0)\) has a convex boundary, then having positive curvature operator, being PIC\(1\), and being PIC\(2\) are all preserved under the flow;
(2) If \((M,g_0)\) has a two-convex boundary, then having PIC is preserved under the flow;
(3) If \((M,g_0)\) has mean convex boundary, then positive scalar curvature is preserved along the flow.
Reviewer: Louis Yudowitz (Stockholm)Partial regularity for harmonic maps into spheres at a singular or degenerate free boundaryhttps://zbmath.org/1521.580062023-11-13T18:48:18.785376Z"Moser, Roger"https://zbmath.org/authors/?q=ai:moser.roger"Roberts, James"https://zbmath.org/authors/?q=ai:roberts.james-w|roberts.james-a|roberts.james-lMotivated by the study of fractional harmonic mappings, the authors provide regularity results for a class of free boundary harmonic mappings on certain domains exhibiting degenerate features.
Reviewer: Dumitru Motreanu (Perpignan)On the boundary complex of the \(k\)-Cauchy-Fueter complexhttps://zbmath.org/1521.580102023-11-13T18:48:18.785376Z"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.18Summary: The \(k\)-Cauchy-Fueter complex, \(k=0,1,\ldots\), in quaternionic analysis are the counterpart of the Dolbeault complex in the theory of several complex variables. In this paper, we construct explicitly boundary complexes of these complexes on boundaries of domains, corresponding to the tangential Cauchy-Riemann complex in complex analysis. They are only known boundary complexes outside of complex analysis that have interesting applications to the function theory. As an application, we establish the Hartogs-Bochner extension for \(k\)-regular functions, the quaternionic counterpart of holomorphic functions. These boundary complexes have a very simple form on a kind of quadratic hypersurfaces, which have the structure of right-type nilpotent Lie groups of step two. They allow us to introduce the quaternionic Monge-Ampère operator and open the door to investigate pluripotential theory on such groups. We also apply abstract duality theorem to boundary complexes to obtain the generalization of Malgrange's vanishing theorem and the Hartogs-Bochner extension for \(k\)-CF functions, the quaternionic counterpart of CR functions, on this kind of groups.The Allen-Cahn equation with generic initial datumhttps://zbmath.org/1521.600272023-11-13T18:48:18.785376Z"Hairer, Martin"https://zbmath.org/authors/?q=ai:hairer.martin"Lê, Khoa"https://zbmath.org/authors/?q=ai:le.khoa|le.khoa-n"Rosati, Tommaso"https://zbmath.org/authors/?q=ai:rosati.tommaso-cornelisThe authors consider the Allen-Cahn equation \(\partial_{t}u- \Delta u = u- u^{3}\) with a rapidly mixing Gaussian field as an initial condition. They study the scaling limit of such an equation and show that after a suitably long time the dynamics are well approximated by a certain class of Gaussian nodal sets which evolve under a mean curvature flow. While ideally one would like to consider the white noise initial data, this is not possible in this setting since the scaling exponent of the white noise is below a critical exponent for which one might expect any form of a local well-posedness for the equation. The authors show that if the amplitude of the initial condition is not too large, the equation generates fronts described by nodal sets of the Bargmann-Fock Gaussian field, which then evolve according to a mean curvature flow.
Reviewer: Maria Gordina (Storrs)Stability analysis for a class of stochastic delay nonlinear systems driven by G-Lévy processhttps://zbmath.org/1521.600292023-11-13T18:48:18.785376Z"Ma, Li"https://zbmath.org/authors/?q=ai:ma.li.3"Li, Yujing"https://zbmath.org/authors/?q=ai:li.yujing"Zhu, Quanxin"https://zbmath.org/authors/?q=ai:zhu.quanxinThe authors study regularity and stability of solutions to a class of stochastic delay differential equations driven by G-Lévy processes. The notion of such processes was introduced about 15 years ago as an extension of G-Brownian motion with a sublinear expectation. One of the first results in the article is a Burkholder-Davis-Gundy (BDG) inequality for the jump measure. Then the BDG inequality is used to show the existence and uniqueness of solutions under non-Lipschitz condition. Under local Lipschitz and one-sided polynomial growth conditions, the authors in addition establish quasi-sure exponential stability and the \(p\)th moment exponential stability of the solution.
Reviewer: Maria Gordina (Storrs)Invariant measure for 2D stochastic Cahn-Hilliard-Navier-Stokes equationshttps://zbmath.org/1521.600302023-11-13T18:48:18.785376Z"Qiu, Zhaoyang"https://zbmath.org/authors/?q=ai:qiu.zhaoyang"Wang, Huaqiao"https://zbmath.org/authors/?q=ai:wang.huaqiao"Huang, Daiwen"https://zbmath.org/authors/?q=ai:huang.daiwenThis paper focuses on the invariant measure for stochastic Cahn-Hilliard-Navier-Stokes equations in two-dimensional spaces. Applying the Maslowski-Seidler method, the authors establish the existence of invariant measure in state space \(L^2_x\times H^1\) with the weak topology. Furthermore, they prove the existence of global pathwise solutions using the stochastic compactness argument.
This paper is innovative and interesting. It provides important theoretical tools and methods for studying the dynamic behavior of the stochastic Cahn-Hilliard-Navier-Stokes equations.
Reviewer: Guanggan Chen (Chengdu)SPDE bridges with observation noise and their spatial approximationhttps://zbmath.org/1521.600342023-11-13T18:48:18.785376Z"di Nunno, Giulia"https://zbmath.org/authors/?q=ai:di-nunno.giulia"Ortiz-Latorre, Salvador"https://zbmath.org/authors/?q=ai:ortiz-latorre.salvador"Petersson, Andreas"https://zbmath.org/authors/?q=ai:petersson.andreasThis paper is devoted to introduce SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. In this article, the SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. A general framework for the spatial discretization of these bridge processes is introduced. Explicit convergence rates are derived for a spectral and a finite element based method. It is shown that for sufficiently rough observation noise, the rates are essentially the same as those of the corresponding discretization of the original SPDE.
Reviewer: Hossam A. Ghany (al-Qāhira)Multilevel quasi-Monte Carlo for optimization under uncertaintyhttps://zbmath.org/1521.650072023-11-13T18:48:18.785376Z"Guth, Philipp A."https://zbmath.org/authors/?q=ai:guth.philipp-a"Van Barel, Andreas"https://zbmath.org/authors/?q=ai:van-barel.andreasSummary: This paper considers the problem of optimizing the average tracking error for an elliptic partial differential equation with an uncertain lognormal diffusion coefficient. In particular, the application of the multilevel quasi-Monte Carlo (MLQMC) method to the estimation of the gradient is investigated, with a circulant embedding method used to sample the stochastic field. A novel regularity analysis of the adjoint variable is essential for the MLQMC estimation of the gradient in combination with the samples generated using the circulant embedding method. A rigorous cost and error analysis shows that a randomly shifted quasi-Monte Carlo method leads to a faster rate of decay in the root mean square error of the gradient than the ordinary Monte Carlo method, while considering multiple levels substantially reduces the computational effort. Numerical experiments confirm the improved rate of convergence and show that the MLQMC method outperforms the multilevel Monte Carlo method and single level quasi-Monte Carlo method.Approximation of SPDE covariance operators by finite elements: a semigroup approachhttps://zbmath.org/1521.650102023-11-13T18:48:18.785376Z"Kovács, Mihály"https://zbmath.org/authors/?q=ai:kovacs.mihaly"Lang, Annika"https://zbmath.org/authors/?q=ai:lang.annika"Petersson, Andreas"https://zbmath.org/authors/?q=ai:petersson.andreasSummary: The problem of approximating the covariance operator of the mild solution to a linear stochastic partial differential equation is considered. An integral equation involving the semigroup of the mild solution is derived and a general error decomposition is proven. This formula is applied to approximations of the covariance operator of a stochastic advection-diffusion equation and a stochastic wave equation, both on bounded domains. The approximations are based on finite element discretizations in space and rational approximations of the exponential function in time. Convergence rates are derived in the trace class and Hilbert-Schmidt norms with numerical simulations illustrating the results.Analytical and numerical investigation on the tempered time-fractional operator with application to the Bloch equation and the two-layered problemhttps://zbmath.org/1521.650732023-11-13T18:48:18.785376Z"Feng, Libo"https://zbmath.org/authors/?q=ai:feng.libo"Liu, Fawang"https://zbmath.org/authors/?q=ai:liu.fawang"Anh, Vo V."https://zbmath.org/authors/?q=ai:anh.vo-v"Qin, Shanlin"https://zbmath.org/authors/?q=ai:qin.shanlin(no abstract)Generalized finite difference method with irregular mesh for a class of three-dimensional variable-order time-fractional advection-diffusion equationshttps://zbmath.org/1521.650772023-11-13T18:48:18.785376Z"Wang, Zhaoyang"https://zbmath.org/authors/?q=ai:wang.zhaoyang"Sun, HongGuang"https://zbmath.org/authors/?q=ai:sun.hongguangSummary: Fractional advection-diffusion equation, as a generalization of classical advection-diffusion equation, has been always mentioned to simulate anomalous diffusion in porous media. This work introduces a meshless generalized finite difference method (GFDM) to solve a class of three-dimensional variable-order time fractional advection-diffusion equation (TFADE) in finite domains. Three examples with known analytic solutions in different domains are given to demonstrate that the method is accurate and stable. To reduce computational and storage cost, we discretize the time derivative terms of TFADE by a fast finite difference method (FFDM) based on sum-of-exponentials (SOE) approximation. Meanwhile, discretizing space derivative terms, GFDM generates a linear equation set including function values of neighboring nodes with various weight coefficients. Then the partial derivatives of TFADE are indicated as the linear system above. Also, this paper investigates the irregular mesh in the finite spatial domain, which is more closely meets the description of practice problems. Numerical results indicate that models with irregular mesh can also be simulated by GFDM which maintains high accuracy. Furthermore, the method is stable and accurate in solving three-dimensional irregular domain problems, where the relative errors can be less than 0.01\%. This paper shows that FFDM based on SOE approximation can improve computational efficiency, and GFDM can flexibly and efficiently solve three-dimensional variable-order and variable-coefficient TFADE.SUSHI for a non-linear time fractional diffusion equation with a time independent delayhttps://zbmath.org/1521.650812023-11-13T18:48:18.785376Z"Benkhaldoun, Fayssal"https://zbmath.org/authors/?q=ai:benkhaldoun.fayssal"Bradji, Abdallah"https://zbmath.org/authors/?q=ai:bradji.abdallahSummary: We establish a linear implicit finite volume scheme for a non-linear time fractional diffusion equation with a time independent delay in any space dimension. The fractional order derivative is given in the Caputo sense. The discretization in space is performed using the SUSHI ((Scheme Using stabilized Hybrid Interfaces) developed in [\textit{R. Eymard} et al., IMA J. Numer. Anal. 30, No. 4, 1009--1043 (2010; Zbl 1202.65144)], whereas the discretization in time is given by a constrained time step-size. The approximation of the fractional order derivative is given by \(L1\)-formula.
We prove rigorously new convergence results in \(L^\infty (L^2)\) and \(L^2(H^1_0)\)-discrete norms. The order is proved to be optimal in space and it is \(k^{2-\alpha }\) in time, with \(k\) is the constant time step and \(\alpha\) is the fractional order of the Caputo derivative.
This paper is a continuation of some of our previous works which dealt either with only the linear fractional PDEs (Partial Differential Equations) without delays, e.g. [\textit{A. Bradji}, Springer Proc. Math. Stat. 323, 305--314 (2020; Zbl 1454.65074); Comput. Math. Appl. 79, No. 2, 500--520 (2020; Zbl 1443.65268); C. R., Math., Acad. Sci. Paris 356, No. 4, 439--448 (2018; Zbl 1447.65071); \textit{A. Bradji} and \textit{J. Fuhrmann}, Lect. Notes Comput. Sci. 10187, 33--45 (2017; Zbl 1371.65090)], or with only time dependent PDEs (the time derivative is given in the usual sense) with delays, e.g. [\textit{F. Benkhaldoun} et al., Lect. Notes Comput. Sci. 13127, 498--506 (2022; Zbl 1487.65127); \textit{F. Benkhaldoun} and \textit{A. Bradji}, Springer Proc. Math. Stat. 323, 315--324 (2020; Zbl 1454.65071); \textit{A. Bradji} and \textit{T. Ghoudi}, Lect. Notes Comput. Sci. 11189, 351--359 (2019; Zbl 1416.65290)].
For the entire collection see [Zbl 1511.65004].Numerical solution of an inverse source problem for a time-fractional PDE via direct meshless local Petrov-Galerkin methodhttps://zbmath.org/1521.650872023-11-13T18:48:18.785376Z"Molaee, Tahereh"https://zbmath.org/authors/?q=ai:molaee.tahereh"Shahrezaee, Alimardan"https://zbmath.org/authors/?q=ai:shahrezaee.alimardan(no abstract)A meshless method to solve the variable-order fractional diffusion problems with fourth-order derivative termhttps://zbmath.org/1521.650882023-11-13T18:48:18.785376Z"Safari, Farzaneh"https://zbmath.org/authors/?q=ai:safari.farzaneh"Jing, Li"https://zbmath.org/authors/?q=ai:jing.li"Lu, Jun"https://zbmath.org/authors/?q=ai:lu.jun"Chen, Wen"https://zbmath.org/authors/?q=ai:chen.wen(no abstract)The direct meshless local Petrov-Galerkin technique with its error estimate for distributed-order time fractional cable equationhttps://zbmath.org/1521.650922023-11-13T18:48:18.785376Z"Habibirad, Ali"https://zbmath.org/authors/?q=ai:habibirad.ali"Hesameddini, Esmail"https://zbmath.org/authors/?q=ai:hesameddini.esmail"Azin, Hadis"https://zbmath.org/authors/?q=ai:azin.hadis"Heydari, Mohammad Hossein"https://zbmath.org/authors/?q=ai:heydari.mohammadhosseinSummary: Distributed-order fractional calculus is a quickly growing concept of the more general area of fractional calculus that has significant and extensive usage for designing complex systems. This work is used the direct meshless local Petrov-Galerkin (DMLPG) technique for the numerical solution of the distributed-order time fractional Cable equation. DMLPG implements a generalized moving least square (GMLS) process to discretize the equation in space variables. By using this scheme, the test function is approximated via the values at nodes, directly. Thus, this algorithm passes integration with the MLS shape functions substituting with a more inexpensive integration than polynomials. Here, the distributed integral is discretized by the \(M\)-point Gauss-Legendre quadrature rule. Then, the finite difference scheme is applied to approximate the fractional derivative discretization. Also, the unconditionally stability and rate of convergence \(O ( \tau^{2 - \max \{ \alpha , \beta \}} )\) of the time-discrete technique are demonstrated. Moreover, the current method converts the problem into a system of linear algebraic equations. To demonstrate the capability and flexibility of our scheme, some examples with different geometric domains are supposed in two-dimensional cases.On the application of a hierarchically semi-separable compression for space-fractional parabolic problems with varying time stepshttps://zbmath.org/1521.650942023-11-13T18:48:18.785376Z"Slavchev, Dimitar"https://zbmath.org/authors/?q=ai:slavchev.dimitar"Margenov, Svetozar"https://zbmath.org/authors/?q=ai:margenov.svetozar-dSummary: Anomalous (fractional) diffusion is observed when the Brownian motion hypotheses are violated. It is modeled with the fractional Laplace operator, which can be defined in several ways. In this work we use the integral definition with the Riesz potential. For the discretization in space we apply the finite element method and for the discretization in time -- a backward Euler scheme with varying time steps. The fractional Laplacian is a non-local operator and the arising stiffness matrix is dense. The time dependent problem is reduced to solving a sequence of linear systems whose matrices are constructed from the stiffness matrix, lumped mass matrix and the time step. When the time step changes we must refactorize the matrix before solving the current system. If the time step doesn't change we can solve with the matrix factorized on a previous time step change. When utilizing the generic method using a block LU factorization, the computational complexity of the forward elimination is \(O(n^3)\) and \(O(n^2)\) of the backward substitution. In this work we develop an alternative method based on the hierarchically semi-separable (HSS) compression. With this method we compress the matrix at the beginning only. The HSS compression has a computational complexity \(O(n^2r)\). Then, when the time step changes we need to apply ULV-like factorization with computational complexity of \(O(nr^2)\). The solution step with the factorized matrix at each time step has computational complexity of \(O(nr)\). Here, \(r\) is the maximum off-diagonal rank of the approximate matrix, which is computed during the compression process. For suitable problems \(r\) is much smaller than the number of unknowns \(n\). The numerical experiments presented show the advantages of the developed HSS compression based solution method.
For the entire collection see [Zbl 1511.65004].An RBF-FD method for the time-fractional advection-dispersion equation with nonlinear source termhttps://zbmath.org/1521.651012023-11-13T18:48:18.785376Z"Londoño, Mauricio A."https://zbmath.org/authors/?q=ai:londono.mauricio-a"Giraldo, Ramón"https://zbmath.org/authors/?q=ai:giraldo.ramon"Rodríguez-Cortés, Francisco J."https://zbmath.org/authors/?q=ai:rodriguez-cortes.francisco-jSummary: Fractional advection-dispersion equations have proved to be useful for modeling a wide range of problems in environmental and engineering sciences. In this work, we adapt a Radial Basis Function-generated Finite Difference (RBF-FD) method to obtain approximated numerical solutions of the initial-boundary value problem of the time-fractional advection-dispersion equation with variable coefficients and nonlinear source. We use a strategy of minimization of the local truncation error in approximating the initial condition to find appropriate local shape parameters for the Gaussian RBF. For discretizing the fractional time derivative, in the Caputo's sense, we use a scheme of \(( 4 - \alpha )\) th-order, where \(\alpha \in (0,1)\) is the order of the fractional derivative. We evaluate the performance of the RBF-FD method for different 2-dimensional problems on domains with complex geometries in which a high precision is exhibited of the solutions found. Particularly, we test our method for approximating solutions of Fisher's equation with fractional time derivative, where the source that depends on the solution, is approximated by a linearization with respect to time, and we obtain a rate of convergence of second order in time.Coupled Kansa and hybrid optimization methodological approach for Kolmogorov-Feller equationshttps://zbmath.org/1521.651062023-11-13T18:48:18.785376Z"Salleh, Ihsane"https://zbmath.org/authors/?q=ai:salleh.ihsane"Belkourchia, Yassin"https://zbmath.org/authors/?q=ai:belkourchia.yassin"Azrar, Lahcen"https://zbmath.org/authors/?q=ai:azrar.lahcen(no abstract)Numerical approximation of time-dependent fractional convection-diffusion-wave equation by RBF-FD methodhttps://zbmath.org/1521.651082023-11-13T18:48:18.785376Z"Zhang, Xindong"https://zbmath.org/authors/?q=ai:zhang.xindong"Yao, Lin"https://zbmath.org/authors/?q=ai:yao.linSummary: In this paper, a method based on radial basis function finite difference (RBF-FD) is developed for solving the time fractional convection-diffusion-wave equation (TFCDWE). We first approximate the equation by a scheme of order \(O(\tau+h^2)\), where \(\tau,h\) are the time step size and spatial step size, respectively. We prove the stability and convergence of the discrete scheme, then the multiquadric RBF-FD approach is used to approximate the spatial derivatives. The aim of this paper is to show that the RBF-FD method is useful for solving our mentioned equation when the shape parameter selection is appropriate. The proposed method can be applied to complex domain, and has the advantages of mesh-free and simple procedure. Finally, numerical examples are proposed to verify the correctness of our previous theoretical analysis and to demonstrate the superiority of the RBF-FD method.Continuous time limit of the stochastic ensemble Kalman inversion: strong convergence analysishttps://zbmath.org/1521.651172023-11-13T18:48:18.785376Z"Blömker, Dirk"https://zbmath.org/authors/?q=ai:blomker.dirk"Schillings, Claudia"https://zbmath.org/authors/?q=ai:schillings.claudia"Wacker, Philipp"https://zbmath.org/authors/?q=ai:wacker.philipp"Weissmann, Simon"https://zbmath.org/authors/?q=ai:weissmann.simonThe ensemble Kalman filter (EnKF) for inverse problems, also known as ensemble Kalman inversion (EKI) is considered. The main focus of this work is to theoretically verify the convergence of the discrete EKI method to its continuous time formulation. The article is organized as follows. Section 1 is an Introduction. A brief overview of the existing literature, Mathematical setup, EKI: The EnKF applied to inverse problems, and an Outline of the paper are given in Subsections 1.1--1.4. In Section 2, the authors' general numerical approximation results for stochastic differential equations are presented, which are then applied to the solution of general nonlinear inverse problems with the EKI method in Section 3. The application to linear inverse problems is presented in Section 4. Some conclusions are presented in the last Section 5. Discussing possible further directions to take are given. Most of the authors' proofs are shifted to the appendix in order to keep the focus on the key contribution presented in this article.
Reviewer: Temur A. Jangveladze (Tbilisi)Robust BPX preconditioner for fractional Laplacians on bounded Lipschitz domainshttps://zbmath.org/1521.651232023-11-13T18:48:18.785376Z"Borthagaray, Juan Pablo"https://zbmath.org/authors/?q=ai:borthagaray.juan-pablo"Nochetto, Ricardo H."https://zbmath.org/authors/?q=ai:nochetto.ricardo-h"Wu, Shuonan"https://zbmath.org/authors/?q=ai:wu.shuonan"Xu, Jinchao"https://zbmath.org/authors/?q=ai:xu.jinchaoSummary: We propose and analyze a robust Bramble-Pasciak-Xu (BPX) preconditioner for the integral fractional Laplacian of order \(s \in (0, 1)\) on bounded Lipschitz domains. Compared with the standard BPX preconditioner, an additional scaling factor \(1-\widetilde{\gamma}^s\), for some fixed \(\widetilde{\gamma} \in (0, 1)\), is incorporated to the coarse levels. For either quasi-uniform grids or graded bisection grids, we show that the condition numbers of the resulting systems remain uniformly bounded with respect to both the number of levels and the fractional power.Elastic transmission eigenvalues and their computation via the method of fundamental solutionshttps://zbmath.org/1521.651362023-11-13T18:48:18.785376Z"Kleefeld, A."https://zbmath.org/authors/?q=ai:kleefeld.andreas"Pieronek, L."https://zbmath.org/authors/?q=ai:pieronek.lukasSummary: A stabilized version of the fundamental solution method to catch ill-conditioning effects is investigated with focus on the computation of complex-valued elastic interior transmission eigenvalues in two dimensions for homogeneous and isotropic media. Its algorithm can be implemented very shortly and adopts to many similar partial differential equation-based eigenproblems as long as the underlying fundamental solution function can be easily generated. We develop a corroborative approximation analysis which also implicates new basic results for transmission eigenfunctions and present some numerical examples which together prove successful feasibility of our eigenvalue recovery approach.A finite point method for the fractional cable equation using meshless smoothed gradientshttps://zbmath.org/1521.742312023-11-13T18:48:18.785376Z"Li, Xiaolin"https://zbmath.org/authors/?q=ai:li.xiaolin.1"Li, Shuling"https://zbmath.org/authors/?q=ai:li.shulingSummary: This paper presents a meshless finite point method (FPM) for the numerical analysis of the fractional cable equation. A second-order time discrete scheme is proposed to approximate both integer-order and fractional-order time derivatives. Then, based on the stabilized moving least squares approximation and the meshless smoothed gradients, a new implementation of the FPM is provided to enhance the accuracy and convergence rate in space. Theoretical error of the FPM is analyzed. Numerical results verify the efficiency of the method and show that the method can gain second-order accuracy in time and fourth-order accuracy in space.Boundary element modeling of fractional nonlinear generalized photothermal stress wave propagation problems in FG anisotropic smart semiconductorshttps://zbmath.org/1521.743172023-11-13T18:48:18.785376Z"Fahmy, Mohamed Abdelsabour"https://zbmath.org/authors/?q=ai:fahmy.mohamed-abdelsabourSummary: The main aim of this article is to develop an efficient boundary element method (BEM) modeling of the fractional nonlinear generalized photo thermal stress wave propagation problems in the context of functionally graded (FG) anisotropic smart semiconductors. Due to nonlinearity, fractional order heat conduction and strongly anisotropy of mechanical properties, the governing equations system of such problems is often very difficult to solve using classical analytical methods. Therefore, a reliable and efficient coupling scheme based on BEM was proposed to address this challenge, where, the Cartesian transformation method (CTM) has been implemented to calculate the domain integrals, and the generalized modified shift-splitting (GMSS) has been implemented for solving the linear systems arising from BEM. The calculation findings are depicted in graphical forms to display the impacts of temperature-dependent, anisotropy, piezoelectric, graded parameter and fractional parameter on the nonlinear photo thermal stress wave propagation in the considered structure. The numerical findings confirm the consistency and efficacy of the developed modeling methodology.A meshless collocation method for solving the inverse Cauchy problem associated with the variable-order fractional heat conduction model under functionally graded materialshttps://zbmath.org/1521.744162023-11-13T18:48:18.785376Z"Hu, Wen"https://zbmath.org/authors/?q=ai:hu.wen"Fu, Zhuojia"https://zbmath.org/authors/?q=ai:fu.zhuojia"Tang, Zhuochao"https://zbmath.org/authors/?q=ai:tang.zhuochao"Gu, Yan"https://zbmath.org/authors/?q=ai:gu.yan(no abstract)Direct and inverse problems on the joint movement of the three viscous liquids in the flat layershttps://zbmath.org/1521.760902023-11-13T18:48:18.785376Z"Lemeshkova, Elena N."https://zbmath.org/authors/?q=ai:lemeshkova.elena-nikolaevnaSummary: The exact stationary decision of the problem about the joint movement of the three viscous liquids in the flat layers has been found. The decision of the direct and inverse non-stationary problem has been given in the form of the final analytical formulas using the method of Laplas transformation. The following statement has been proved: if a gradient of the pressure in one liquid has a final limit, then the decision is located on a stationary mode. Also for a problem about the ``the flooded layer'' movement it has been shown that velocities converge to the different constants with the time growth.On the process of filtration of fractional viscoelastic liquid foodhttps://zbmath.org/1521.768362023-11-13T18:48:18.785376Z"Meng, Yahui"https://zbmath.org/authors/?q=ai:meng.yahui"Li, Botong"https://zbmath.org/authors/?q=ai:li.botong"Si, Xinhui"https://zbmath.org/authors/?q=ai:si.xinhui"Chen, Xuehui"https://zbmath.org/authors/?q=ai:chen.xuehui"Liu, Fawang"https://zbmath.org/authors/?q=ai:liu.fawangSummary: In the process of filtration, fluid impurities precipitate/accumulate; this results in an uneven inner wall of the filter, consequently leading to non-uniform suction/injection. The Riemannian-Liouville fractional derivative model is used to investigate viscoelastic incompressible liquid food flowing through a permeable plate and to generalize Fick's law. Moreover, we consider steady-state mass balance during ultrafiltration on a plate surface, and a fractional-order concentration boundary condition is established, thereby rendering the problem real and complex. The governing equation is numerically solved using the finite difference algorithm. The effects of the fractional constitutive models, generalized Reynolds number, generalized Schmidt number, and permeability parameter on the velocity and concentration fields are compared. The results show that an increase in fractional-order \(\alpha\) in the momentum equation leads to a decrease in the horizontal velocity. Anomalous diffusion described by the fractional derivative model weakens the mass transfer; therefore, the concentration decreases with increasing fractional derivative \(\gamma\) in the concentration equation.An unconditionally stable fast high order method for thermal phase change modelshttps://zbmath.org/1521.800342023-11-13T18:48:18.785376Z"Wang, Weiwen"https://zbmath.org/authors/?q=ai:wang.weiwen"Azaiez, Mejdi"https://zbmath.org/authors/?q=ai:azaiez.mejdi"Xu, Chuanju"https://zbmath.org/authors/?q=ai:xu.chuanjuSummary: Thermal phase change problems arise in a large number of applications. In this paper, we consider a phase field model instead of the classical Stefan model to describe phenomena, which may appear in some complex phase change problems such as dendritic crystal growth, phase transformations in metallic alloys, etc. Our aim is to propose efficient and accurate schemes for the model, which is the coupling of a heat transfer equation and a phase field equation. The schemes are constructed based on an auxiliary variable approach for the phase field equation and semi-implicit treatment for the heat transfer equation. The main novelty of the paper consists in: (i) construction of the efficient schemes, which only requires solving several second-order elliptic problems with constant coefficients; (ii) proof of the unconditional stability of the schemes; (iii) fast high order solver for the resulting equations at each time step. A series of numerical examples are presented to verify the theoretical claims and to illustrate the efficiency of our method. As far as we know, it seems this is the first attempt made for the thermal phase change model of this type.Laterally loaded piles and pile groups partially embedded in transversely isotropic fractional viscoelastic saturated soilshttps://zbmath.org/1521.860362023-11-13T18:48:18.785376Z"Ai, Zhi Yong"https://zbmath.org/authors/?q=ai:ai.zhiyong"Wang, Da Shan"https://zbmath.org/authors/?q=ai:wang.da-shan"Zhao, Yong Zhi"https://zbmath.org/authors/?q=ai:zhao.yongzhi"Li, Pan Cong"https://zbmath.org/authors/?q=ai:li.pan-cong(no abstract)Numerical approximation of the first-passage time distribution of time-varying diffusion decision models: a mesh-free approachhttps://zbmath.org/1521.913042023-11-13T18:48:18.785376Z"Rasanan, Amir Hosein Hadian"https://zbmath.org/authors/?q=ai:rasanan.amir-hosein-hadian"Evans, Nathan J."https://zbmath.org/authors/?q=ai:evans.nathan-j"Rieskamp, Jörg"https://zbmath.org/authors/?q=ai:rieskamp.jorg"Rad, Jamal Amani"https://zbmath.org/authors/?q=ai:rad.jamal-amani(no abstract)Pricing options under time-fractional model using Adomian decompositionhttps://zbmath.org/1521.913872023-11-13T18:48:18.785376Z"Kharrat, Mohamed"https://zbmath.org/authors/?q=ai:kharrat.mohamedSummary: In this chapter, we define and describe some models to price European and American options. As the dynamics of volatility is intrinsic in terms of the hedging and the pricing of options, we shall present in this chapter both cases: the constant volatility [\textit{F. Black} and \textit{M. Scholes}, J. Polit. Econ. 81, No. 3, 637--654 (1973; Zbl 1092.91524)] and the stochastic volatility model [\textit{S. L. Heston}, Rev. Financ. Stud. 6, No. 2, 327--343 (1993; Zbl 1384.35131)]. At the beginning, we display a closed-form solution of a European option generated by the fractional Heston stochastic volatility model. Subsequently, we provide the analytical solution to the fractional linear complement problem related to the evaluation of American put option generated by the fractional Black and Scholes model. In the closing of this chapter, we attempt to set forward the solution of the fractional linear complementarity problem related to the evaluation of American put option generated by the fractional Heston stochastic volatility model. Investing the Adomian decomposition, a numerical investigation is undertaken to corroborate the theoretical results. The results of this chapter are published in [\textit{M. Kharrat}, Nonlinear Dyn. Syst. Theory 18, No. 2, 191--195 (2018; Zbl 1416.91377); ``Pricing American put option under fractional Heston model'', Pramana J. Phys. 95, No. 1, Article No. 3, 7 p. (2021; \url{doi:10.1007/s12043-020-02039-z}); ``Pricing American put option under fractional model'', Filomat 35, No. 10, 3433--3441 (2021; \url{doi:10.2298/FIL2110433K})].
For the entire collection see [Zbl 1497.37001].Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation datahttps://zbmath.org/1521.920072023-11-13T18:48:18.785376Z"Lee, Seungjoon"https://zbmath.org/authors/?q=ai:lee.seungjoon"Psarellis, Yorgos M."https://zbmath.org/authors/?q=ai:psarellis.yorgos-m"Siettos, Constantinos I."https://zbmath.org/authors/?q=ai:siettos.constantinos-i"Kevrekidis, Ioannis G."https://zbmath.org/authors/?q=ai:kevrekidis.ioannis-gSummary: We propose a machine learning framework for the data-driven discovery of macroscopic chemotactic Partial Differential Equations (PDEs) -- and the closures that lead to them- from high-fidelity, individual-based stochastic simulations of \textit{Escherichia coli} bacterial motility. The fine scale, chemomechanical, hybrid (continuum -- Monte Carlo) simulation model embodies the underlying biophysics, and its parameters are informed from experimental observations of individual cells. Using a parsimonious set of collective observables, we learn effective, coarse-grained ``Keller-Segel class'' chemotactic PDEs using machine learning regressors: (a) (shallow) feedforward neural networks and (b) Gaussian Processes. The learned laws can be \textit{black-box} (when no prior knowledge about the PDE law structure is assumed) or \textit{gray-box} when parts of the equation (e.g. the pure diffusion part) is known and ``hardwired'' in the regression process. More importantly, we discuss data-driven \textit{corrections} (both additive and functional), to analytically known, \textit{approximate} closures.Approximate controllability results in \(\alpha \)-norm for some partial functional integrodifferential equations with nonlocal initial conditions in Banach spaceshttps://zbmath.org/1521.930172023-11-13T18:48:18.785376Z"Ndambomve, Patrice"https://zbmath.org/authors/?q=ai:ndambomve.patrice"Kpoumie, Moussa El-Khalil"https://zbmath.org/authors/?q=ai:kpoumie.moussa-el-khalil"Ezzinbi, Khalil"https://zbmath.org/authors/?q=ai:ezzinbi.khalilIn this work the authors have discussed the approximate controllability of some nonlinear partial functional integrodifferential equations with nonlocal initial condition in Hilbert spaces under the assumption that the corresponding linear part is approximately controllable. The results are obtained by using the fractional power theory and \(\alpha\)-norm, the measure of noncompactness and the Mönch fixed-point theorem, and the theory of analytic resolvent operators for integral equations. This paper is a generalization of the work of \textit{N. I. Mahmudov} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 3, 536--546 (2008; Zbl 1129.93004)] and established without the assumption of compactness of the resolvent operator. An example is provided to illustrate the main results.
Reviewer: Krishnan Balachandran (Coimbatore)Lyapunov stability analysis for incommensurate nabla fractional order systemshttps://zbmath.org/1521.931492023-11-13T18:48:18.785376Z"Wei, Yiheng"https://zbmath.org/authors/?q=ai:wei.yiheng"Zhao, Xuan"https://zbmath.org/authors/?q=ai:zhao.xuan|zhao.xu-an"Wei, Yingdong"https://zbmath.org/authors/?q=ai:wei.yingdong"Chen, Yangquan"https://zbmath.org/authors/?q=ai:chen.yangquanSummary: This paper investigates the problem of stability analysis for a class of incommensurate nabla fractional order systems. In particular, both Caputo definition and Riemann-Liouville definition are under consideration. With the convex assumption, several elementary fractional difference inequalities on Lyapunov functions are developed. According to the essential features of nabla fractional calculus, the sufficient conditions are given first to guarantee the asymptotic stability for the incommensurate system by using the direct Lyapunov method. To substantiate the efficacy and effectiveness of the theoretical results, four examples are elaborated.