Recent zbMATH articles in MSC 35B https://zbmath.org/atom/cc/35B 2022-06-24T15:10:38.853281Z Werkzeug Symmetry and symmetry breaking for the fractional Caffarelli-Kohn-Nirenberg inequality https://zbmath.org/1485.35008 2022-06-24T15:10:38.853281Z "Ao, Weiwei" https://zbmath.org/authors/?q=ai:ao.weiwei "DelaTorre, Azahara" https://zbmath.org/authors/?q=ai:delatorre.azahara "del Mar González, María" https://zbmath.org/authors/?q=ai:gonzalez.maria-del-mar Summary: In this paper, we will consider the fractional Caffarelli-Kohn-Nirenberg inequality $\Lambda \left( \int_{\mathbb{R}^n} \frac{| u (x) |^p}{|x|^{\beta p}} d x \right)^{\frac{ 2}{ p}} \leq \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{(u(x) - u (y))^2}{ | x - y |^{n + 2 \gamma} | x |^\alpha | y |^\alpha} dy\,dx$ where $$\gamma \in(0, 1), n \geq 2$$, and $$\alpha, \beta \in \mathbb{R}$$ satisfy $\alpha \leq \beta \leq \alpha + \gamma, \quad- 2 \gamma < \alpha < \frac{ n - 2 \gamma}{ 2},$ and the exponent $$p$$ is chosen to be $p = \frac{ 2 n}{ n - 2 \gamma + 2 (\beta - \alpha)},$ such that the inequality is invariant under scaling. We first study the existence and nonexistence of extremal solutions. Our next goal is to show some results on the symmetry and symmetry breaking region for the minimizers; these suggest the existence of a Felli-Schneider type curve separating both regions but, surprisingly, we find a novel behavior as $$\alpha \to - 2 \gamma$$. The main idea in the proofs, as in the classical case, is to reformulate the fractional Caffarelli-Kohn-Nirenberg inequality in cylindrical variables. Then, in order to find the radially symmetric solutions we need to solve a non-local ODE. For this equation we also get uniqueness of minimizers in the radial symmetry class; indeed, we show that the unique continuation argument of \textit{R. L. Frank} and \textit{E. Lenzmann} [Acta Math. 210, No. 2, 261--318 (2013; Zbl 1307.35315)] can be applied to more general operators with good spectral properties. We provide, in addition, a completely new proof of non-degeneracy which works for all critical points. It is based on the variation of constants approach and the non-local Wronskian of the first author et al. [Duke Math. J. 168, No. 17, 3297--3411 (2019; Zbl 1440.35127)]. Randomization improved Strichartz estimates and global well-posedness for supercritical data https://zbmath.org/1485.35009 2022-06-24T15:10:38.853281Z "Burq, Nicolas" https://zbmath.org/authors/?q=ai:burq.nicolas "Krieger, Joachim" https://zbmath.org/authors/?q=ai:krieger.joachim Summary: We introduce a novel data randomisation for the free wave equation which leads to the same range of Strichartz estimates as for radial data, albeit in a non-radial context. We then use these estimates to establish global well-posedness for a wave maps type nonlinear wave equation for certain supercritical data, provided the data are suitably small and randomised. A mean value formula for the iterated Dunkl-Helmoltz operator https://zbmath.org/1485.35012 2022-06-24T15:10:38.853281Z "González Vieli, F. J." https://zbmath.org/authors/?q=ai:gonzalez-vieli.francisco-javier Summary: We establish a spherical mean value formula for the iterated Dunkl-Helmoltz operator, thus generalizing a result of Mejjaoli and Trimèche. We then give an application to distributions with Dunkl transform supported by the unit sphere. Oscillation criteria of certain fractional partial differential equations https://zbmath.org/1485.35013 2022-06-24T15:10:38.853281Z "Xu, Di" https://zbmath.org/authors/?q=ai:xu.di "Meng, Fanwei" https://zbmath.org/authors/?q=ai:meng.fanwei Summary: In this article, we regard the generalized Riccati transformation and Riemann-Liouville fractional derivatives as the principal instrument. In the proof, we take advantage of the fractional derivatives technique with the addition of interval segmentation techniques, which enlarge the manners to demonstrate the sufficient conditions for oscillation criteria of certain fractional partial differential equations. Radial symmetry of solutions to anisotropic and weighted diffusion equations with discontinuous nonlinearities https://zbmath.org/1485.35014 2022-06-24T15:10:38.853281Z "Dipierro, Serena" https://zbmath.org/authors/?q=ai:dipierro.serena "Poggesi, Giorgio" https://zbmath.org/authors/?q=ai:poggesi.giorgio "Valdinoci, Enrico" https://zbmath.org/authors/?q=ai:valdinoci.enrico Summary: For $$1<p < \infty$$, we prove radial symmetry for bounded nonnegative solutions of $\begin{cases} -\operatorname{div}\left\{w(x) H(\nabla u)^{p-1} \nabla_{\xi} H(\nabla u)\right\} = f(u)w(x) &\text{in } \Sigma \cap \Omega, \\ u=0 &\text{on }\Gamma_0, \\ \langle \nabla_\xi H(\nabla u), \nu \rangle = 0 &\text{on } \Gamma_1\setminus\left\{0\right\}, \end{cases}$ where $$\Omega$$ is a Wulff ball, $$\Sigma$$ is a convex cone with vertex at the center of $$\Omega$$, $$\Gamma_0 := \Sigma \cap \partial \Omega$$, $$\Gamma_1 := \partial \Sigma \cap \Omega$$, $$H$$ is a norm, $$w$$ is a given weight and $$f$$ is a possibly discontinuous nonnegative nonlinearity. Given the anisotropic setting that we deal with, the term radial'' is understood in the Finsler framework, that is, the function $$u$$ is radial if there exists a point $$x$$ such that $$u$$ is constant on the Wulff shapes centered at $$x$$. When $$\Sigma = \mathbb{R}^N$$, J. Serra obtained the symmetry result in the isotropic unweighted setting (i.e., when $$H(\xi)\equiv |\xi |$$ and $$w\equiv 1$$). In this case we provide the extension of his result to the anisotropic setting. This provides a generalization to the anisotropic setting of a celebrated result due to Gidas-Ni-Nirenberg and such a generalization is new even for $$p=2$$ whenever $$N>2$$. When $$\Sigma \subsetneq \mathbb{R}^N$$ the results presented are new even in the isotropic and unweighted setting (i.e., when $$H$$ is the Euclidean norm and $$w \equiv 1$$) whenever $$2 \ne p \ne N$$. Even for the previously known case of unweighted isotropic setting with $$p=2$$ and $$\Sigma \subsetneq \mathbb{R}^N$$, the present paper provides an approach to the problem by exploiting integral (in)equalities which is new for $$N>2$$: this complements the corresponding symmetry result obtained via the moving planes method by Berestycki-Pacella. The results obtained in the isotropic and weighted setting (i.e., with $$w \not \equiv 1$$) are new for any $$p$$. Differential invariants, hidden and conditional symmetry https://zbmath.org/1485.35015 2022-06-24T15:10:38.853281Z "Yehorchenko, I. A." https://zbmath.org/authors/?q=ai:yehorchenko.i-a Summary: We present a survey of development of the concept of hidden symmetry in the field of partial differential equations, including a series of results previously obtained by the author. We also add new examples of the classes of equations with hidden symmetry of type II and explain the nature of the earlier established nonclassical symmetry of some equations. We suggest a constructive algorithm for the description of the classes of equations, which have specified conditional or hidden symmetry and/or can be reduced to equations with smaller number of independent variables by using a specific ansatz. We consider reductions existing due to the presence of Lie and conditional symmetry and also of the hidden symmetry of type II. We also discuss relationships between the concepts of hidden and conditional symmetry. It is shown that the hidden symmetry of type II earlier regarded as a separate type of non-Lie symmetry is caused, in fact, by the nontrivial $$Q$$-conditional symmetry of the reduced equations. The proposed approach enables us not only to find hidden symmetry and new reductions of the well-known equations but also to describe a general form of equations with given $$Q$$-conditional and type-II hidden symmetry. As an example, we describe the general classes of equations with hidden and conditional symmetry under rotations in the Lorentz and Euclid groups for which the corresponding hidden and conditional symmetry allows their reduction to radial equations with smaller number of independent variables. Entire solutions of certain fourth order elliptic problems and related inequalities https://zbmath.org/1485.35016 2022-06-24T15:10:38.853281Z "D'Ambrosio, Lorenzo" https://zbmath.org/authors/?q=ai:dambrosio.lorenzo "Mitidieri, Enzo" https://zbmath.org/authors/?q=ai:mitidieri.enzo-luigi Summary: We study distributional solutions of semilinear biharmonic equations of the type $\Delta^2 u + f(u) = 0\quad \text{on }\mathbb{R}^N,$ where $$f$$ is a continuous function satisfying $$f(t) t \geq c|t|^{q+1}$$ for all $$t \in \mathbb{R}$$ with $$c > 0$$ and $$q > 1$$. By using a new approach mainly based on careful choice of suitable weighted test functions and a new version of Hardy-Rellich inequalities, we prove several Liouville theorems independently of the dimension $$N$$ and on the sign of the solutions. Periodic solutions of a phase-field model with hysteresis https://zbmath.org/1485.35017 2022-06-24T15:10:38.853281Z "Bin, Chen" https://zbmath.org/authors/?q=ai:bin.chen "Timoshin, Sergey A." https://zbmath.org/authors/?q=ai:timoshin.sergey-a Summary: In the present paper we consider a partial differential system describing a phase-field model with temperature dependent constraint for the order parameter. The system consists of an energy balance equation with a fairly general nonlinear heat source term and a phase dynamics equation which takes into account the hysteretic character of the process. The existence of a periodic solution for this system is proved under a minimal set of assumptions on the curves defining the corresponding hysteresis region. Periodic entropy solution to a conservation law with nonlocal source arising in radiative gas https://zbmath.org/1485.35018 2022-06-24T15:10:38.853281Z "Li, Ke" https://zbmath.org/authors/?q=ai:li.ke "Ruan, Lizhi" https://zbmath.org/authors/?q=ai:ruan.lizhi "Yang, Anita" https://zbmath.org/authors/?q=ai:yang.anita Summary: We establish the global well-posedness and large-time asymptotic behaviour of spatially periodic entropy solutions to a scalar conservation law with nonlocal source arising in radiative gas. The global existence is established by the $$L^1$$-contraction and the comparison principle for the vanishing viscosity approximation. Moreover, we show that if the initial data is periodic, the source term induces the solution to decay in $$L^2$$-norm at an exponential rate to the mean value of initial data over one space period. New complex wave structures to the complex Ginzburg-Landau model https://zbmath.org/1485.35019 2022-06-24T15:10:38.853281Z "Wang, Huiqing" https://zbmath.org/authors/?q=ai:wang.huiqing "Alam, Md Nur" https://zbmath.org/authors/?q=ai:alam.md-nur "İlhan, Onur Alp" https://zbmath.org/authors/?q=ai:ilhan.onur-alp "Singh, Gurpreet" https://zbmath.org/authors/?q=ai:singh.gurpreet "Manafian, Jalil" https://zbmath.org/authors/?q=ai:manafian-heris.jalil Summary: We study and analysis the complex Ginzburg-Landau model or CGL model to obtain some new solitary wave structures through the modified $$(G'/G)$$-expansion method. Those solutions can explain through hyperbolic, trigonometric, and rational functions. The graphical design makes the dynamics of the equations noticeable. Herein, we state that the examined method is important, powerful, and significant in performing numerous solitary wave structures of various nonlinear wave models following in physics and engineering as well. Convergence of the Allen-Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to $$90^\circ$$ https://zbmath.org/1485.35020 2022-06-24T15:10:38.853281Z "Abels, Helmut" https://zbmath.org/authors/?q=ai:abels.helmut "Moser, Maximilian" https://zbmath.org/authors/?q=ai:moser.maximilian In this paper, the authors study a parabolic Allen-Cahn equation with a nonlinear Robin condition in a bounded smooth domain in $$\mathbb{R}^2$$. The limiting problem is the mean curvature flow with a contact angle $$\alpha$$ on the boundary. When $$\alpha$$ is close to $$\pi/2$$, by assuming the limiting problem has a local smooth solution, they prove that solutions to the Allen-Cahn equation converges to the limit in a smooth way. This is done by taking an expansion of the solution in $$\varepsilon$$ and then estimating the difference. For this purpose, some linearized estimates are developed, and the restriction on $$\alpha$$ arises from the linearization problem at the boundary contact point. Reviewer: Kelei Wang (Wuhan) Bubbling solutions for a planar exponential nonlinear elliptic equation with a singular source https://zbmath.org/1485.35021 2022-06-24T15:10:38.853281Z "Dong, Jingyi" https://zbmath.org/authors/?q=ai:dong.jingyi "Hu, Jiamei" https://zbmath.org/authors/?q=ai:hu.jiamei "Zhang, Yibin" https://zbmath.org/authors/?q=ai:zhang.yibin Summary: Let $$\Omega$$ be a bounded domain in $$\mathbb{R}^2$$ with smooth boundary, we study the following elliptic Dirichlet problem $\begin{cases} -\Delta\upsilon= e^{\upsilon}-s\phi_1-4\pi\alpha\delta_p-h(x) \quad & \text{ in } \Omega,\\ \upsilon=0 & \text{ on } \partial\Omega, \end{cases}$ where $$s > 0$$ is a large parameter, $$h\in C^{0,\gamma}(\overline{\Omega})$$, $$p\in\Omega$$, $$\alpha \in (-1,+\infty) \backslash \mathbb{N}$$, $$\delta_p$$ denotes the Dirac measure supported at point $$p$$ and $$\phi_1$$ is a positive first eigenfunction of the problem $$-\Delta\phi=\lambda\phi$$ under Dirichlet boundary condition in $$\Omega$$. If $$p$$ is a strict local maximum point of $$\phi_1$$, we show that such a problem has a family of solutions $$\upsilon_s$$ with arbitrary $$m$$ bubbles accumulating to $$p$$, and the quantity $\int_{\Omega} e^{\upsilon_s} \rightarrow 8\pi(m+1+\alpha)\phi_1(p) \text{ as } s\rightarrow+\infty.$ Well-posedness issues for the Prandtl boundary layer equations https://zbmath.org/1485.35022 2022-06-24T15:10:38.853281Z "Gérard-Varet, David" https://zbmath.org/authors/?q=ai:gerard-varet.david "Masmoudi, Nader" https://zbmath.org/authors/?q=ai:masmoudi.nader Summary: These notes are an introduction to the recent paper [the authors, Ann. Sci. Éc. Norm. Supér. (4) 48, No. 6, 1273--1325 (2015; Zbl 1347.35201)], about the well-posedness of the Prandtl equation. The difficulties and main ideas of the paper are described on a simpler linearized model. On the singularly perturbation fractional Kirchhoff equations: critical case https://zbmath.org/1485.35023 2022-06-24T15:10:38.853281Z "Gu, Guangze" https://zbmath.org/authors/?q=ai:gu.guangze "Yang, Zhipeng" https://zbmath.org/authors/?q=ai:yang.zhipeng|yang.zhipeng.1|yang.zhipeng.2 Summary: This article deals with the following fractional Kirchhoff problem with critical exponent $\left( a+b\int\limits_{\mathbb{R}^N} |(-\Delta )^{\frac{s}{2}} u |^2 \mathrm{d}x\right) (-\Delta )^s u=(1+\varepsilon K(x)) u^{2_s^{\ast} -1}, \quad\text{in }\mathbb{R}^N,$ where $$a,b> 0$$ are given constants, $$\varepsilon$$ is a small parameter, $$2_s^{\ast} =\frac{2N}{N-2s}$$ with $$0< s< 1$$ and $$N\geq 4s$$. We first prove the nondegeneracy of positive solutions when $$\varepsilon =0$$. In particular, we prove that uniqueness breaks down for dimensions $$N> 4s$$, i.e., we show that there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singular perturbation problems for $$\varepsilon$$ small. Fast reaction limit and forward-backward diffusion: a Radon-Nikodym approach https://zbmath.org/1485.35024 2022-06-24T15:10:38.853281Z "Skrzeczkowski, Jakub" https://zbmath.org/authors/?q=ai:skrzeczkowski.jakub Summary: We consider two singular limits: a fast reaction limit with a non-monotone nonlinearity and a regularization of the forward-backward diffusion equation. We derive pointwise identities satisfied by the Young measure generated by these problems. As a result, we obtain an explicit formula for the Young measure even without the non-degeneracy assumption used in the previous works. The main new idea is an application of the Radon-Nikodym theorem to decompose the Young measure. Homogenization of a nonlinear monotone problem in a locally periodic domain via unfolding method https://zbmath.org/1485.35025 2022-06-24T15:10:38.853281Z "Aiyappan, Srinivasan" https://zbmath.org/authors/?q=ai:aiyappan.srinivasan "Cardone, Giuseppe" https://zbmath.org/authors/?q=ai:cardone.giuseppe "Perugia, Carmen" https://zbmath.org/authors/?q=ai:perugia.carmen "Prakash, Ravi" https://zbmath.org/authors/?q=ai:prakash.ravi Summary: In this paper, the asymptotic behavior of the solutions of a monotone problem posed in a locally periodic oscillating domain is studied. Nonlinear monotone boundary conditions are imposed on the oscillating part of the boundary whereas the Dirichlet condition is considered on the smooth separate part. Using the unfolding method, under natural hypothesis on the regularity of the domain, we prove the weak $$L^p$$-convergence of the zero-extended solutions of the nonlinear problem and their flows to the solutions of a limit distributional problem. Operator-norm homogenisation estimates for the system of Maxwell equations on periodic singular structures https://zbmath.org/1485.35026 2022-06-24T15:10:38.853281Z "Cherednichenko, Kirill" https://zbmath.org/authors/?q=ai:cherednichenko.kirill-d "D'Onofrio, Serena" https://zbmath.org/authors/?q=ai:donofrio.serena Summary: For arbitrarily small values of $$\varepsilon >0$$, we formulate and analyse the Maxwell system of equations of electromagnetism on $$\varepsilon$$-periodic sets $$S^\varepsilon \subset\mathbb{R}^3$$. Assuming that a family of Borel measures $$\mu^\varepsilon$$, such that $$\mathrm{supp}(\mu^\varepsilon)=S^\varepsilon$$, is obtained by $$\varepsilon$$-contraction of a fixed 1-periodic measure $$\mu$$, and for right-hand sides $$f^\varepsilon \in L^2(\mathbb{R}^3, d\mu^\varepsilon)$$, we prove order-sharp norm-resolvent convergence estimates for the solutions of the system. Our analysis includes the case of periodic singular structures'', when $$\mu$$ is supported by lower-dimensional manifolds. The estimates are obtained by combining several new tools we develop for analysing the Floquet decomposition of an elliptic differential operator on functions from Sobolev spaces with respect to a periodic Borel measure. These tools include a generalisation of the classical Helmholtz decomposition for $$L^2$$ functions, an associated Poincaré-type inequality, uniform with respect to the parameter of the Floquet decomposition, and an appropriate asymptotic expansion inspired by the classical power series. Our technique does not involve any spectral analysis and does not rely on the existing approaches, such as Bloch wave homogenisation or the spectral germ method. Stochastic homogenization of Hamilton-Jacobi equations on a junction https://zbmath.org/1485.35027 2022-06-24T15:10:38.853281Z "Fayad, Rim" https://zbmath.org/authors/?q=ai:fayad.rim "Forcadel, Nicolas" https://zbmath.org/authors/?q=ai:forcadel.nicolas "Ibrahim, Hassan" https://zbmath.org/authors/?q=ai:ibrahim.hassan Summary: We consider the specified stochastic homogenization of first order evolutive Hamilton-Jacobi equations on a very simple junction, i.e. the real line with a junction at the origin. Far from the origin, we assume that the considered hamiltonian is closed to given stationary ergodic hamiltonians (which are different on the left and on the right). Near the origin, there is a perturbation zone which allows are to pass from one hamiltonian to the other. The main result of this paper is a stochastic homogenization as the length of the transition zone goes to zero. More precisely, at the limit we get two deterministic right and left hamiltonians with a deterministic junction condition at the origin. The main difficulty and novelty of the paper come from the fact that the hamiltonian is not stationary ergodic. To the best of our knowledge, this is the first specified stochastic homogenization result. This work is motivated by traffic flow applications. Correlated random walks in heterogeneous landscapes: derivation, homogenization, and invasion fronts https://zbmath.org/1485.35028 2022-06-24T15:10:38.853281Z "Lutscher, Frithjof" https://zbmath.org/authors/?q=ai:lutscher.frithjof "Hillen, Thomas" https://zbmath.org/authors/?q=ai:hillen.thomas Summary: Many models for the movement of particles and individuals are based on the diffusion equation, which, in turn, can be derived from an uncorrelated random walk or a position-jump process. In those models, individuals have a location but no well-defined velocity. An alternative, and sometimes more accurate, model is based on a correlated random walk or a velocity-jump process, where individuals have a well defined location and velocity. The latter approach leads to hyperbolic equations for the density of individuals, rather than parabolic equations that result from the diffusion process. Almost all previous work on correlated random walks considers a homogeneous landscape, whereas diffusion models for uncorrelated walks have been extended to spatially varying environments. In this work, we first derive the equations for a correlated random walk in a one-dimensional spatially varying environment with either smooth variation or piecewise constant variation. Then we show how to derive the so-called parabolic limit from the resulting hyperbolic equations. We develop homogenization theory for the hyperbolic equations, and show that taking the parabolic limit and homogenization are commuting actions. We illustrate our results with two examples from ecology: the persistence and spread of a population in a patchy heterogeneous landscape. Homogenization of the Stokes system in a non-periodically perforated domain https://zbmath.org/1485.35029 2022-06-24T15:10:38.853281Z "Wolf, Sylvain" https://zbmath.org/authors/?q=ai:wolf.sylvain The authors studies the homogenization of the Stokes system posed in a non-periodically perforated domain. Most of the estimates are quantitative. The approach is a direct follow-up of author's work on the Poisson problem and belongs to a larger research initiative involving C. LeBris, X. Blanc, P.-L- Lions and many of their co-authors interested in averaging (homogenizing) PDE problems posed in structured domains with defects. Reviewer: Adrian Muntean (Karlstad) A note on the solution map for the periodic multi-dimensional Camassa-Holm-type system https://zbmath.org/1485.35030 2022-06-24T15:10:38.853281Z "Fu, Ying" https://zbmath.org/authors/?q=ai:fu.ying "Wang, Haiquan" https://zbmath.org/authors/?q=ai:wang.haiquan Summary: Considered in this paper is the initial value problem of periodic multi-dimensional Camassa-Holm-type system. It is shown that the solution map of this problem is not uniformly continuous in Besov spaces $$B^{1+\frac{d}{2}}_{2,1}(\mathbb{T}^d)\times B^{\frac{d}{2}}_{2,1}(\mathbb{T}^d)$$ with $$d\in\mathbb{Z}^+,\,d\ge 1.$$ Based on the local well-posedness results, the method of approximate solutions is utilized. Non-uniform continuity of the Fokas-Olver-Rosenau-Qiao equation in Besov spaces https://zbmath.org/1485.35031 2022-06-24T15:10:38.853281Z "Wu, Xing" https://zbmath.org/authors/?q=ai:wu.xing "Yu, Yanghai" https://zbmath.org/authors/?q=ai:yu.yanghai Summary: In this paper, we consider the solution map of the Cauchy problem to the Fokas-Olver-Rosenau-Qiao equation on the real line and prove that the solution map of this problem is not uniformly continuous on the initial data in Besov spaces. Our result extends the previous results in [\textit{A. Alexandrou Himonas} and \textit{D. Mantzavinos}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 95, 499--529 (2014; Zbl 1282.35328)] and [\textit{J. Li} et al., J. Math. Fluid Mech. 22, No. 4, Paper No. 50, 10 p. (2020; Zbl 1448.35402)]. Nonuniform dependence of solution to the high-order two-component $$b$$-family system https://zbmath.org/1485.35032 2022-06-24T15:10:38.853281Z "Yang, Li" https://zbmath.org/authors/?q=ai:yang.li.3|yang.li|yang.li.2|yang.li.1 "Mu, Chunlai" https://zbmath.org/authors/?q=ai:mu.chunlai "Zhou, Shouming" https://zbmath.org/authors/?q=ai:zhou.shouming Summary: This paper deals with the Cauchy problem of the high-order two-component $$b$$-family system. Based on the local well-posedness results and prior estimates, two sequences of solutions whose distance initially goes to zero but later is bounded below by a positive constant are constructed by the method of approximate solutions; in other words, we prove that the data-to-solution map is not uniformly continuous in Sobolev spaces with the index more than $$\frac{5}{2}$$. On the global bifurcation diagram of the Gelfand problem https://zbmath.org/1485.35033 2022-06-24T15:10:38.853281Z "Bartolucci, Daniele" https://zbmath.org/authors/?q=ai:bartolucci.daniele "Jevnikar, Aleks" https://zbmath.org/authors/?q=ai:jevnikar.aleks Summary: For domains of first kind we describe the qualitative behavior of the global bifurcation diagram of the unbounded branch of solutions of the Gelfand problem crossing the origin. At least to our knowledge this is the first result about the exact monotonicity of the branch of nonminimal solutions which is not just concerned with radial solutions and/or with symmetric domains. Toward our goal we parametrize the branch not by the $$L^{\infty}(\Omega)$$-norm of the solutions but by the energy of the associated mean field problem. The proof relies on a refined spectral analysis of mean-field-type equations and some surprising properties of the quantities triggering the monotonicity of the Gelfand parameter. Diffusion-induced spatio-temporal oscillations in an epidemic model with two delays https://zbmath.org/1485.35034 2022-06-24T15:10:38.853281Z "Du, Yan-fei" https://zbmath.org/authors/?q=ai:du.yanfei "Niu, Ben" https://zbmath.org/authors/?q=ai:niu.ben "Wei, Jun-jie" https://zbmath.org/authors/?q=ai:wei.junjie Summary: We investigate a diffusive, stage-structured epidemic model with the maturation delay and freely-moving delay. Choosing delays and diffusive rates as bifurcation parameters, the only possible way to destabilize the endemic equilibrium is through Hopf bifurcation. The normal forms of Hopf bifurcations on the center manifold are calculated, and explicit formulae determining the criticality of bifurcations are derived. There are two different kinds of stable oscillations near the first bifurcation: on one hand, we theoretically prove that when the diffusion rate of infected immature individuals is sufficiently small or sufficiently large, the first branch of Hopf bifurcating solutions is always spatially homogeneous; on the other, xing this diffusion rate at an appropriate size, stable oscillations with different spatial profiles are observed, and the conditions to guarantee the existence of such solutions are given by calculating the corresponding eigenfunction of the Laplacian at the first Hopf bifurcation point. These bifurcation behaviors indicate that spatial di usion in the epidemic model may lead to spatially inhomogeneous distribution of individuals. Interactions of time delay and spatial diffusion induce the periodic oscillation of the vegetation system https://zbmath.org/1485.35035 2022-06-24T15:10:38.853281Z "Li, Jing" https://zbmath.org/authors/?q=ai:li.jing.13 "Sun, Gui-Quan" https://zbmath.org/authors/?q=ai:sun.guiquan "Jin, Zhen" https://zbmath.org/authors/?q=ai:jin.zhen Summary: Empirical data exhibit a common phenomenon that vegetation biomass fluctuates periodically over time in ecosystem, but the corresponding internal driving mechanism is still unclear. Simultaneously, considering that the conversion of soil water absorbed by roots of the vegetation into vegetation biomass needs a period time, we thus introduce the conversion time into Klausmeier model, then a spatiotemporal vegetation model with time delay is established. Through theoretical analysis, we not only give the occurence conditions of stability switches for system without and with diffusion at the vegetation-existence equilibrium, but also derive the existence conditions of saddle-node-Hopf bifurcation of non-spatial system and Hopf bifurcation of spatial system at the coincidence equilibrium. Our results reveal that the conversion delay induces the interaction between the vegetation and soil water in the form of periodic oscillation when conversion delay increases to the critical value. By comparing the results of system without and with diffusion, we find that the critical value decreases with the increases of spatial diffusion factors, which is more conducive to emergence of periodic oscillation phenomenon, while spatial diffusion factors have no effects on the amplitude of periodic oscillation. These results provide a theoretical basis for understanding the spatiotemporal evolution behaviors of vegetation system. Hankel transforms and weak dispersion https://zbmath.org/1485.35036 2022-06-24T15:10:38.853281Z "Cacciafesta, Federico" https://zbmath.org/authors/?q=ai:cacciafesta.federico "Fanelli, Luca" https://zbmath.org/authors/?q=ai:fanelli.luca Summary: This survey is concerned with a general strategy, based on Hankel transforms and special functions decompositions, to prove weak dispersive estimates for a class of PDE's. Inspired by [\textit{N. Burq} et al., J. Funct. Anal. 203, No. 2, 519--549 (2003; Zbl 1030.35024)], we show how to adapt the method to some scaling critical dispersive models, as the Dirac-Coulomb equation and the fractional Schrödinger and Dirac equation in Aharonov-Bohm field. For the entire collection see [Zbl 1459.37002]. Strichartz estimates for waves in the Friedlander model in dimension 3 https://zbmath.org/1485.35037 2022-06-24T15:10:38.853281Z "Ivanovici, Oana" https://zbmath.org/authors/?q=ai:ivanovici.oana "Lebeau, Gilles" https://zbmath.org/authors/?q=ai:lebeau.gilles "Planchon, Fabrice" https://zbmath.org/authors/?q=ai:planchon.fabrice Summary: On se propose dans cet exposé d'établir des estimations de Strichartz pour l'équation des ondes dans un domaine strictement convexe de $$\mathbb{R}^3$$. Stability for locally coupled wave-type systems with indirect mixed-type dampings https://zbmath.org/1485.35038 2022-06-24T15:10:38.853281Z "Jin, Kun-Peng" https://zbmath.org/authors/?q=ai:jin.kunpeng Summary: In this paper, we investigate the stability of a class of wave equations with local coupling effect and indirect mixed-type dampings. Some optimal (so far best) stability results for the energy of the system are established under much weak basic assumptions on the locally coupled terms and the damping terms. In particular, it is worth noting that the local coupling terms here only play a role on a part of the domain but not on entire domain, and the coefficients of the coupling terms may have variable sign. On the other hand, the memory-effect regions do not have to include a part of boundary in present paper, while the condition is a necessity in almost all the previous literature. As will be seen, to overcome the difficulties encountered in obtaining these results, we combine the higher order energy method with positive definite kernel theory through the multiplier technique, which are utilized in our proofs. Fourier transforms and $$L^2$$-stability of diffusion equations https://zbmath.org/1485.35039 2022-06-24T15:10:38.853281Z "Kang, Dongseung" https://zbmath.org/authors/?q=ai:kang.dongseung "Kim, Hoewoon B." https://zbmath.org/authors/?q=ai:kim.hoewoon-b Summary: This paper is concerned with generalized Hyers-Ulam stability of the diffusion equation, $$\displaystyle\frac{\partial u(x,t)}{\partial t} = \triangle u (x, t)$$ with $$u(x,0) = f(x)$$ for $$t > 0$$ and $$x \in \mathbb{R}^n$$. Most of the Hyers-Ulam stability problems of differential equations are involved with $$L^\infty$$-norm or the supremum norm of functions with consideration of either initial conditions or forcing terms. However, an integral method of Fourier transform can be used to obtain the $$L^2$$-estimates for generalized Hyers-Ulam stability of an IVP (initial value problem) of the diffusion equation with a function $$f (x)$$ as an initial condition and we will present the generalized Hyers-Ulam stability of the IVP in the sense of $$L^2$$-norm. Mittag-Leffler stability for a fractional Euler-Bernoulli problem https://zbmath.org/1485.35040 2022-06-24T15:10:38.853281Z "Tatar, Nasser-eddine" https://zbmath.org/authors/?q=ai:tatar.nasser-eddine Summary: We investigate the stability of an Euler-Bernoulli type problem of fractional order. By adding a fractional term of lower-order, namely of order half of the order of the leading fractional derivative, the problem will generalize the well-known telegraph equation. It is shown that this term is capable of stabilizing the system to rest in a Mittag-Leffler manner. Moreover, we consider a much weaker dissipative term consisting of a memory term in the form of a convolution known as viscoelastic term. It is proved that we can still obtain Mittag-Leffler stability under a smallness condition on the involved kernels. The results rely heavily on some established properties of fractional derivatives and some newly introduced functionals. On a two-species chemotaxis-competition system with indirect signal consumption https://zbmath.org/1485.35041 2022-06-24T15:10:38.853281Z "Xiang, Yuting" https://zbmath.org/authors/?q=ai:xiang.yuting "Zheng, Pan" https://zbmath.org/authors/?q=ai:zheng.pan Summary: This paper deals with a two-competing-species chemotaxis system with indirect signal consumption \begin{cases} \begin{aligned} &u_t=d_1\Delta u-\chi_1\nabla \cdot (u\nabla w)+\mu_1u(1-u-a_1v),&(x,t)\in \Omega \times (0,\infty ),\\ &v_t=d_2\Delta v-\chi_2\nabla \cdot (v\nabla w)+\mu_2v(1-v-a_2u),&(x,t)\in \Omega \times (0,\infty ),\\ &w_t=\Delta w-wz,&(x,t)\in \Omega \times (0,\infty ),\\ &z_t=\Delta z-z+u+v,&(x,t)\in \Omega \times (0,\infty ),\\ &\frac{{\partial u}}{{\partial \nu }} = \frac{{\partial v}}{{\partial \nu }} = \frac{{\partial w}}{{\partial \nu }} = \frac{{\partial z}}{{\partial \nu }} = 0,&(x,t)\in \partial \Omega \times (0,\infty ),\\ &\left( {u, v, w, z} \right) \left( {x,0} \right) = \left( {{u_0}\left( x \right) , {v_0}\left( x \right) ,{w_0}\left( x \right) , {z_0}\left( x \right) } \right) ,&x\in \Omega ,\ \end{aligned} \end{cases} under homogeneous Neumann boundary conditions in a smooth bounded domain $$\Omega \subset{\mathbb{R}}^n(n\le 2)$$, with the nonnegative initial data $$\left( {u_0, v_0, w_0, z_0} \right) \in{C^0}\left( {{{\bar{\Omega }}} } \right) \times{C^0}\left( {{{\bar{\Omega }}} } \right) \times{W^{1,\infty }}\left( \Omega \right) \times{W^{1,\infty }}\left( \Omega \right)$$, where $$\chi_i>0, d_i>0, a_i>0$$ and $$\mu_i>0 (i=1,2)$$. It is shown that the system has a global bounded classical solution for arbitrary size of $$\mu_1, \mu_2>0$$. Additionally, we consider the asymptotic stabilization of solutions to the above system as follows: \begin{itemize} \item When $$a_1, a_2 \in (0,1)$$, the global bounded classical solution $$(u, v, w, z)$$ exponentially converges to $$\Big (\frac{1-a_1}{1-a_1 a_2}, \frac{1-a_2}{1-a_1 a_2}, 0, \frac{2-a_1-a_2}{1-a_1 a_2}\Big )$$ in the $$L^{\infty }$$-norm as $$t \rightarrow \infty$$; \item When $$a_1>1>a_2>0$$ and $$a_1a_2<1$$, the global bounded classical solution $$(u, v, w, z)$$ exponentially converges to (0, 1, 0, 1) in the $$L^{\infty }$$-norm as $$t \rightarrow \infty$$; \item When $$a_1=1>a_2>0$$, the global bounded classical solution $$(u, v, w, z)$$ polynomially converges to (0, 1, 0, 1) in the $$L^{\infty }$$-norm as $$t \rightarrow \infty$$. \end{itemize} Effect of a membrane on diffusion-driven Turing instability https://zbmath.org/1485.35042 2022-06-24T15:10:38.853281Z "Ciavolella, Giorgia" https://zbmath.org/authors/?q=ai:ciavolella.giorgia Summary: Biological, physical, medical, and numerical applications involving membrane problems on different scales are numerous. We propose an extension of the standard Turing theory to the case of two domains separated by a permeable membrane. To this aim, we study a reaction-diffusion system with zero-flux boundary conditions on the external boundary and Kedem-Katchalsky membrane conditions on the inner membrane. We use the same approach as in the classical Turing analysis but applied to membrane operators. The introduction of a diagonalization theory for compact and self-adjoint membrane operators is needed. Here, Turing instability is proven with the addition of new constraints, due to the presence of membrane permeability coefficients. We perform an explicit one-dimensional analysis of the eigenvalue problem, combined with numerical simulations, to validate the theoretical results. Finally, we observe the formation of discontinuous patterns in a system which combines diffusion and dissipative membrane conditions, varying both diffusion and membrane permeability coefficients. Stripe patterns orientation resulting from nonuniform forcings and other competitive effects in the Swift-Hohenberg dynamics https://zbmath.org/1485.35043 2022-06-24T15:10:38.853281Z "Coelho, Daniel L." https://zbmath.org/authors/?q=ai:coelho.daniel-l "Vitral, Eduardo" https://zbmath.org/authors/?q=ai:vitral.eduardo "Pontes, José" https://zbmath.org/authors/?q=ai:pontes.jose-pedro "Mangiavacchi, Norberto" https://zbmath.org/authors/?q=ai:mangiavacchi.norberto Summary: Spatio-temporal pattern formation in complex systems presents rich nonlinear dynamics which leads to the emergence of periodic nonequilibrium structures. One of the most prominent equations for the theoretical and numerical study of the evolution of these textures is the Swift-Hohenberg (SH) equation, which presents a bifurcation parameter (forcing) that controls the dynamics by changing the energy landscape of the system, and has been largely employed in phase-field models. Though a large part of the literature on pattern formation addresses uniformly forced systems, nonuniform forcings are also observed in several natural systems, for instance, in developmental biology and in soft matter applications. In these cases, an orientation effect due to forcing gradients is a new factor playing a role in the development of patterns, particularly in the class of stripe patterns, which we investigate through the nonuniformly forced SH dynamics. The present work addresses amplitude instability of stripe textures induced by forcing gradients, and the competition between the orientation effect of the gradient and other bulk, boundary, and geometric effects taking part in the selection of the emerging patterns. A weakly nonlinear analysis suggests that stripes are stable with respect to small amplitude perturbations when aligned with the gradient, and become unstable to such perturbations when when aligned perpendicularly to the gradient. This analysis is vastly complemented by a numerical work that accounts for other effects, confirming that forcing gradients drive stripe alignment, or even reorient them from preexisting conditions. However, we observe that the orientation effect does not always prevail in the face of competing effects, whose hierarchy is suggested to depend on the magnitude of the forcing gradient. Other than the cubic SH equation (SH3), the quadratic-cubic (SH23) and cubic-quintic (SH35) equations are also studied. In the SH23 case, not only do forcing gradients lead to stripe orientation, but also interfere in the transition from hexagonal patterns to stripes. Global solution and spatial patterns for a ratio-dependent predator-prey model with predator-taxis https://zbmath.org/1485.35044 2022-06-24T15:10:38.853281Z "Gao, Xiaoyan" https://zbmath.org/authors/?q=ai:gao.xiaoyan Summary: This paper analyzes the dynamic behavior of a ratio-dependent predator-prey model with predator-taxis, which the prey can move in the opposite direction of predator gradient. The first purpose is to prove rigorously the global existence and boundedness of the classical solution for the model based on the heat operator semigroup theory and some priori estimates. The another purpose is to analyze the stability of positive equilibrium, which the results will be extended to the case that the derivative of prey's functional response with prey is positive, and it will be found that large predator-taxis can stabilize equilibrium even diffusion-driven instability has occurred. Finally, the numerical simulations present that the pattern formation may arise and predator-taxis is the driving factor for the evolution of spatial inhomogeneity into homogeneity. Oscillations and bifurcation structure of reaction-diffusion model for cell polarity formation https://zbmath.org/1485.35045 2022-06-24T15:10:38.853281Z "Kuwamura, Masataka" https://zbmath.org/authors/?q=ai:kuwamura.masataka "Izuhara, Hirofumi" https://zbmath.org/authors/?q=ai:izuhara.hirofumi "Ei, Shin-ichiro" https://zbmath.org/authors/?q=ai:ei.shin-ichiro Summary: We investigate the oscillatory dynamics and bifurcation structure of a reaction-diffusion system with bistable nonlinearity and mass conservation, which was proposed by \textit{M. Otsuji} et al. [A mass conserved reaction-diffusion system captures properties of cell polarity'', PLoS Comput. Biol. 3, No. 6, e108, 15 p. (2007; \url{doi:10.1371/journal.pcbi.0030108})]. The system is a useful model for understanding cell polarity formation. We show that this model exhibits four different spatiotemporal patterns including two types of oscillatory patterns, which can be regarded as cell polarity oscillations with the reversal and non-reversal of polarity, respectively. The trigger causing these patterns is a diffusion-driven (Turing-like) instability. Moreover, we investigate the effects of extracellular signals on the cell polarity oscillations. Existence of two-solitary waves with logarithmic distance for the nonlinear Klein-Gordon equation https://zbmath.org/1485.35046 2022-06-24T15:10:38.853281Z "Aryan, Shrey" https://zbmath.org/authors/?q=ai:aryan.shrey Quantitative dynamics of irreversible enzyme reaction-diffusion systems https://zbmath.org/1485.35047 2022-06-24T15:10:38.853281Z "Braukhoff, Marcel" https://zbmath.org/authors/?q=ai:braukhoff.marcel "Einav, Amit" https://zbmath.org/authors/?q=ai:einav.amit "Tang, Bao Quoc" https://zbmath.org/authors/?q=ai:tang.bao-quoc Modified Zakharov-Kuznetsov equation posed on a strip https://zbmath.org/1485.35048 2022-06-24T15:10:38.853281Z "Castelli, M." https://zbmath.org/authors/?q=ai:castelli.marco|castelli.marta|castelli.mauro|castelli.m-gabriella "Doronin, G." https://zbmath.org/authors/?q=ai:doronin.gleb-germanovitch Summary: An initial-boundary value problem of Saut-Temam type for the modified Zakharov-Kuznetsov equation posed on a strip is considered. Critical power in nonlinearity has been studied. The results on existence, uniqueness and asymptotic behavior of solutions are presented. Global existence of Timoshenko system with respect to fractional memory operator, spatial fractional thermal effect and distributed delay https://zbmath.org/1485.35049 2022-06-24T15:10:38.853281Z "Choucha, Abdelbaki" https://zbmath.org/authors/?q=ai:choucha.abdelbaki "Boulaaras, Salah" https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud "Ouchenane, Djamel" https://zbmath.org/authors/?q=ai:ouchenane.djamel "Alharbi, Asma" https://zbmath.org/authors/?q=ai:alharbi.asma-olyan-m "Abdalla, Mohamed" https://zbmath.org/authors/?q=ai:abdalla.mohamed Some unexplored questions arising in linear viscoelasticity https://zbmath.org/1485.35050 2022-06-24T15:10:38.853281Z "Conti, Monica" https://zbmath.org/authors/?q=ai:conti.monica-c "Dell'Oro, Filippo" https://zbmath.org/authors/?q=ai:delloro.filippo "Pata, Vittorino" https://zbmath.org/authors/?q=ai:pata.vittorino Summary: We consider the abstract integrodifferential equation $\ddot{u}(t)+A\left[u(t)+\int\limits_0^\infty\mu(s)[u(t)-u(t-s)]ds\right]=0$ modeling the dynamics of linearly viscoelastic solids. The equation is known to generate a semigroup $$S(t)$$ on a certain phase space, whose asymptotic properties have been the object of extensive studies in the last decades. Nevertheless, some relevant questions still remain open, with particular reference to the decay rate of the semigroup compared to the decay of the memory kernel $$\mu$$, and to the structure of the spectrum of the infinitesimal generator of $$S(t)$$. This paper intends to provide some answers. Relation between solutions and initial values for double-nonlinear diffusion equation https://zbmath.org/1485.35051 2022-06-24T15:10:38.853281Z "Deng, Liwei" https://zbmath.org/authors/?q=ai:deng.liwei "Wang, Liangwei" https://zbmath.org/authors/?q=ai:wang.liangwei "Li, Min" https://zbmath.org/authors/?q=ai:li.min.10|li.min.8|li.min.1|li.min.7|li.min.6|li.min.2|li.min.3|li.min.9|li.min.5|li.min.4|li.min "Yin, Jingxue" https://zbmath.org/authors/?q=ai:yin.jingxue Summary: In this paper, we consider the Cauchy problem of the double-nonlinear diffusion equation. We establish the propagation speed estimates and space-time decay estimates for the solutions and study the equivalent relation between the solutions and the initial values. As an application of this relationship, we prove two different asymptotic behaviors for the solutions in the last of this paper. A glioblastoma PDE-ODE model including chemotaxis and vasculature https://zbmath.org/1485.35052 2022-06-24T15:10:38.853281Z "Fernández-Romero, Antonio" https://zbmath.org/authors/?q=ai:fernandez-romero.antonio "Guillén-González, Francisco" https://zbmath.org/authors/?q=ai:guillen-gonzalez.francisco-m "Suárez, Antonio" https://zbmath.org/authors/?q=ai:suarez.antonio Summary: In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes chemotaxis term directed to vasculature. First, we obtain some \textit{a priori} estimates for the (possible) solutions of the model. In particular, under some conditions on the parameters, we obtain that the system does not develop blow-up at finite time. In addition, we design a fully discrete finite element scheme for the model which preserves some pointwise estimates of the continuous problem. Later, we make an adimensional study in order to reduce the number of parameters. Finally, we detect the main parameters determining different width of the ring formed by proliferative and necrotic cells and different regular/irregular behaviour of the tumor surface. New stability estimates of solutions to strong damped wave equation with logarithmic external forces https://zbmath.org/1485.35053 2022-06-24T15:10:38.853281Z "Houma, Nabil" https://zbmath.org/authors/?q=ai:houma.nabil "Zennir, Khaled" https://zbmath.org/authors/?q=ai:zennir.khaled "Beniani, Abderrahmane" https://zbmath.org/authors/?q=ai:beniani.abderrahmane "Djebabela, Abdelhak" https://zbmath.org/authors/?q=ai:djebabela.abdelhak Summary: In this paper, we consider a new stability results of solutions to class of wave equations with weak, strong damping terms and logarithmic source in $$\mathbb{R}^n$$. We prove general stability estimates by introducing suitable Lyapunov functional. Large positive solutions to an elliptic system of competitive type with nonhomogeneous terms https://zbmath.org/1485.35054 2022-06-24T15:10:38.853281Z "Jia, Haohao" https://zbmath.org/authors/?q=ai:jia.haohao "Ma, Feiyao" https://zbmath.org/authors/?q=ai:ma.feiyao "Wo, Weifeng" https://zbmath.org/authors/?q=ai:wo.weifeng Summary: We study the elliptic system of competitive type with nonhomogeneous terms $\Delta u = u^pv^q + h_1(x)$, $\Delta v = u^rv^s + h_2(x)$ in $\Omega$ with two types of boundary conditions: (I) $u = v = +\infty$ and (SF) $u = +\infty$, $v = f$ on $\partial \Omega$, where $f > 0$, $(p - 1)(s - 1) - qr > 0$, and $\Omega\subset \mathbb R ^\mathbb N$ is a smooth bounded domain. The nonhomogeneous terms $h_1(x)$ and $h_2(x)$ may be unbounded near the boundary and may change sign in $\Omega$. First, for a single semilinear elliptic equation with a singular weight and nonhomogeneous term, boundary asymptotic behaviour of large positive solutions is established. Using this asymptotic behaviour, we show existence of large positive solutions for this elliptic system with the boundary condition (SF), existence of maximal solution, boundary asymptotic behaviour and uniqueness of large positive solutions for this elliptic system with (I). Global attractor for 3D Dirac equation with nonlinear point interaction https://zbmath.org/1485.35055 2022-06-24T15:10:38.853281Z "Kopylova, Elena" https://zbmath.org/authors/?q=ai:kopylova.elena-a Summary: We prove global attraction to stationary orbits for 3D Dirac equation with concentrated nonlinearity. We show that each finite energy solution converges as $$t\rightarrow\pm\infty$$ to the set of four-frequency nonlinear eigenfunctions''. The global attraction is caused by nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation. The adaptive biasing force algorithm with non-conservative forces and related topics https://zbmath.org/1485.35056 2022-06-24T15:10:38.853281Z "Lelièvre, Tony" https://zbmath.org/authors/?q=ai:lelievre.tony "Maurin, Lise" https://zbmath.org/authors/?q=ai:maurin.lise "Monmarché, Pierre" https://zbmath.org/authors/?q=ai:monmarche.pierre Summary: We propose a study of the Adaptive Biasing Force method's robustness under generic (possibly non-conservative) forces. We first ensure the flat histogram property is satisfied in all cases. We then introduce a fixed point problem yielding the existence of a stationary state for both the Adaptive Biasing Force and Projected Adapted Biasing Force algorithms, relying on generic bounds on the invariant probability measures of homogeneous diffusions. Using classical entropy techniques, we prove the exponential convergence of both biasing force and law as time goes to infinity, for both the Adaptive Biasing Force and the Projected Adaptive Biasing Force methods. Existence, blow-up and exponential decay estimates for the nonlinear Kirchhoff-Carrier wave equation in an annular with Robin-Dirichlet conditions https://zbmath.org/1485.35057 2022-06-24T15:10:38.853281Z "Ngoc, Le Thi Phuong" https://zbmath.org/authors/?q=ai:le-thi-phuong-ngoc. "Son, Le Huu Ky" https://zbmath.org/authors/?q=ai:son.le-huu-ky "Long, Nguyen Thanh" https://zbmath.org/authors/?q=ai:nguyen-thanh-long. Summary: This paper is devoted to the study of a nonlinear Kirchhoff-Carrier wave equation in an annulus associated with Robin-Dirichlet conditions. At first, by applying the Faedo-Galerkin method, we prove existence and uniqueness results. Then, by constructing a Lyapunov functional, we prove a blow up result for solutions with a negative initial energy and establish a sufficient condition to obtain the exponential decay of weak solutions. On asymptotic behavior of solution to a nonlinear wave equation with space-time speed of propagation and damping terms https://zbmath.org/1485.35058 2022-06-24T15:10:38.853281Z "Ogbiyele, Paul A." https://zbmath.org/authors/?q=ai:ogbiyele.paul-adewale "Arawomo, Peter O." https://zbmath.org/authors/?q=ai:arawomo.peter-olutola Summary: In this paper, we consider the asymptotic behavior of solution to the nonlinear damped wave equation $u_{tt} -\operatorname{div} \left( a(t,x)\nabla u \right) +b(t,x)u_t = -|u|^{p-1} u \quad t \in [0,\infty), \quad x \in \mathbb{R}^n$ $u(0,x) = u_0(x), \qquad u_t(0,x) = u_1(x) \quad x \in \mathbb{R}^n$ with space-time speed of propagation and damping potential. We obtained $$L^2$$ decay estimates via the weighted energy method and under certain suitable assumptions on the functions $$a(t, x)$$ and $$b(t, x)$$. The technique follows that of \textit{J. Lin} et al. [J. Differ. Equations 248, No. 2, 403--422 (2010; Zbl 1184.35213)] with modification to the region of consideration in $$\mathbf{R}^n$$. These decay result extends the results in the literature. Growth of solutions for a coupled nonlinear Klein-Gordon system with strong damping, source, and distributed delay terms https://zbmath.org/1485.35059 2022-06-24T15:10:38.853281Z "Rahmoune, Abdelaziz" https://zbmath.org/authors/?q=ai:rahmoune.abdelaziz "Ouchenane, Djamel" https://zbmath.org/authors/?q=ai:ouchenane.djamel "Boulaaras, Salah" https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud "Agarwal, Praveen" https://zbmath.org/authors/?q=ai:agarwal.praveen Summary: In this work, the exponential growth of solutions for a coupled nonlinear Klein-Gordon system with distributed delay, strong damping, and source terms is proved. Take into consideration some suitable assumptions. Boundedness and stabilization in the 3D minimal attraction-repulsion chemotaxis model with logistic source https://zbmath.org/1485.35060 2022-06-24T15:10:38.853281Z "Ren, Guoqiang" https://zbmath.org/authors/?q=ai:ren.guoqiang "Liu, Bin" https://zbmath.org/authors/?q=ai:liu.bin.5|liu.bin.4|liu.bin.8|liu.bin.1|liu.bin.2|liu.bin.6|liu.bin|liu.bin.7|liu.bin.3|liu.bin.9 Summary: In this paper, we consider the fully parabolic attraction-repulsion chemotaxis system with logistic source in a three-dimensional bounded domain with smooth boundary. We first derive an explicit formula $$\mu_*=\mu_*(3,d_1,d_2,d_3,\beta_1,\beta_2,\chi ,\xi )$$ for the logistic damping rate $$\mu$$ such that the system has no blowups whenever $$\mu >\mu_*$$. In addition, the asymptotic behavior of the solutions is discussed; we obtain the other explicit formula $$\mu^*=\mu^*(d_1,d_2,d_3,\alpha_1,\alpha_2,\beta_1,\beta_2,\chi ,\xi ,\lambda )$$ for the logistic damping rate so that the convergence rate is expressed explicitly whenever $$\mu >\mu^*$$. Our results generalize and improve partial previously known ones. Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $$\mathbb{R}^N$$. III: Transition fronts https://zbmath.org/1485.35061 2022-06-24T15:10:38.853281Z "Salako, Rachidi B." https://zbmath.org/authors/?q=ai:salako.rachidi-bolaji "Shen, Wenxian" https://zbmath.org/authors/?q=ai:shen.wenxian Summary: The current work is the third of a series of three papers devoted to the study of asymptotic dynamics in the following parabolic-elliptic chemotaxis system with space and time dependent logistic source, $\begin{cases} \partial_tu=\Delta u -\chi \nabla \cdot (u\nabla v)+u(a(x,t)-b(x,t)u),&{}\quad x\in{\mathbb{R}}^N,\\ 0=\Delta v-\lambda v+\mu u ,&{}\quad x\in{\mathbb{R}}^N, \end{cases}\tag{1}$ where $$N\ge 1$$ is a positive integer, $$\chi$$, $$\lambda$$ and $$\mu$$ are positive constants, and the functions $$a(x, t)$$ and $$b(x, t)$$ are positive and bounded. In the first of the series [the authors, Math. Models Methods Appl. Sci. 28, No. 11, 2237--2273 (2018; Zbl 1426.35034)], we studied the phenomena of pointwise and uniform persistence for solutions with strictly positive initial data, and the asymptotic spreading for solutions with compactly supported or front like initial data. In the second of the series [the authors, J. Math. Anal. Appl. 464, No. 1, 883--910 (2018; Zbl 1390.35379)], we investigate the existence, uniqueness and stability of strictly positive entire solutions of (1). In particular, in the case of space homogeneous logistic source (i.e. $$a(x,t)\equiv a(t)$$ and $$b(x,t)\equiv b(t))$$, we proved in [Zbl 1390.35379, loc. cit.] that the unique spatially homogeneous strictly positive entire solution $$(u^*(t),v^*(t))$$ of (1) is uniformly and exponentially stable with respect to strictly positive perturbations when $$0<2\chi \mu <\inf_{t\in{\mathbb{R}}}b(t)$$. In the current part of the series, we discuss the existence of transition front solutions of (1) connecting (0, 0) and $$(u^*(t),v^*(t))$$ in the case of space homogeneous logistic source. We show that for every $$\chi >0$$ with $$\chi \mu \big (1+\frac{\sup_{t\in{\mathbb{R}}}a(t)}{\inf_{t\in{\mathbb{R}}}a(t)}\big )<\inf_{t\in{\mathbb{R}}}b(t)$$, there is a positive constant $${c}^*_\chi$$ such that for every $$\underline{c}> {c}^*_{\chi }$$ and every unit vector $$\xi$$, (1) has a transition front solution of the form $$(u(x,t),v(x,t))=(U(x\cdot \xi -C(t),t),V(x\cdot \xi -C(t),t))$$ satisfying that $$C'(t)=\frac{a(t)+\kappa^2}{\kappa }$$ for some positive number $$\kappa , \liminf_{t-s\rightarrow \infty }\frac{C(t)-C(s)}{t-s}=\underline{c}$$, and $\lim_{x\rightarrow -\infty }\sup_{t\in{\mathbb{R}}}|U(x,t)-u^*(t)|=0 \quad \text{and}\quad \lim_{x\rightarrow \infty }\sup_{t\in{\mathbb{R}}}|\frac{U(x,t)}{e^{-\kappa x}}-1|=0.$ Furthermore, we prove that there is no transition front solution $$(u(x,t),v(x,t))=(U(x\cdot \xi -C(t),t),V(x\cdot \xi -C(t),t))$$ of (1) connecting (0, 0) and $$(u^*(t),v^*(t))$$ with least mean speed less than $$2\sqrt{\underline{a}}$$, where $$\underline{a}=\liminf_{t-s\rightarrow \infty }\frac{1}{t-s}\int_s^ta(\tau )d\tau$$. Large-time behavior of solutions to the Cauchy problem for degenerate parabolic system https://zbmath.org/1485.35062 2022-06-24T15:10:38.853281Z "Tedeev, Anatoli F." https://zbmath.org/authors/?q=ai:tedeev.anatolii-fedorovich Summary: We consider nonnegative solutions to the Cauchy problem for a degenerate parabolic system of the form \begin{aligned} u_t &=\mathrm{div}(v^{\alpha_1} \nabla u^{m_1}, \quad (x,t) \in S_T = \mathbb{R}^N \times (0,T)\\ v_t &=\mathrm{div}(u^{\alpha_2} \nabla v^{m_2}, \quad (x,t) \in S_T\\ u(x,0) &=u_0(x) \geq 0, \quad v(x,0)=v_0(x) \geq 0, \quad x \in \mathbb{R}^N, \quad N \geq 1. \end{aligned} Under the suitable assumptions on the parameters of nonlinearities and initial data, we obtained optimal decay estimates of a solution for a large time. Moreover, the phenomena of extinction in finite time was established. Provided that initial data are compactly supported, we proved the property of finite speed of propagation. Optimal decay rate for higher-order derivatives of solution to the 3D compressible quantum magnetohydrodynamic model https://zbmath.org/1485.35063 2022-06-24T15:10:38.853281Z "Wang, Juan" https://zbmath.org/authors/?q=ai:wang.juan "Zhang, Yinghui" https://zbmath.org/authors/?q=ai:zhang.yinghui Summary: We investigate optimal decay rates for higher-order spatial derivatives of strong solutions to the 3D Cauchy problem of the compressible viscous quantum magnetohydrodynamic model in the $$H^5 \times H^4 \times H^4$$ framework, and the main novelty of this work is three-fold: First, we show that fourth order spatial derivative of the solution converges to zero at the $$L^2$$-rate $$(1+t)^{-\frac{11}{4}}$$, which is same as one of the heat equation, and particularly faster than the $$L^2$$-rate $$(1+t)^{-\frac{5}{4}}$$ in [\textit{X. Pu} and \textit{X. Xu}, Z. Angew. Math. Phys. 68, No. 1, Paper No. 18, 17 p. (2017; Zbl 1369.35068)] and the $$L^2$$-rate $$(1+t)^{-\frac{9}{4}}$$ in [\textit{X. Xi} et al., Z. Angew. Math. Phys. 70, No. 1, Paper No. 7, 16 p. (2019; Zbl 1408.35160)]. Second, we prove that fifth-order spatial derivative of density $$\rho$$ converges to zero at the $$L^2$$-rate $$(1+t)^{-\frac{13}{4}}$$, which is same as that of the heat equation, and particularly faster than ones of [\textit{X. Pu} and \textit{X. Xu}, Z. Angew. Math. Phys. 68, No. 1, Paper No. 18, 17 p. (2017; Zbl 1369.35068)] and [\textit{X. Xi} et al., Z. Angew. Math. Phys. 70, No. 1, Paper No. 7, 16 p. (2019; Zbl 1408.35160)]. Third, we show that the high-frequency part of the fourth order spatial derivatives of the velocity $$u$$ and magnetic $$B$$ converge to zero at the $$L^2$$-rate $$(1+t)^{-\frac{13}{4}}$$, which are faster than ones of themselves, and totally new as compared to [\textit{X. Pu} and \textit{X. Xu}, Z. Angew. Math. Phys. 68, No. 1, Paper No. 18, 17 p. (2017; Zbl 1369.35068)] and [\textit{X. Xi} et al., Z. Angew. Math. Phys. 70, No. 1, Paper No. 7, 16 p. (2019; Zbl 1408.35160)]. Asymptotic behavior for solutions to an oncolytic virotherapy model involving triply haptotactic terms https://zbmath.org/1485.35064 2022-06-24T15:10:38.853281Z "Wei, Ya-nan" https://zbmath.org/authors/?q=ai:wei.yanan "Wang, Yifu" https://zbmath.org/authors/?q=ai:wang.yifu "Li, Jing" https://zbmath.org/authors/?q=ai:li.jing.13 Summary: In this paper, based on $$L^p-L^q$$ estimate for the Neumann heat semigroup, we investigate the asymptotic behavior for solutions to an oncolytic virotherapy model given by $\begin{cases} u_t=\Delta u-\xi_u\nabla \cdot \left( u\nabla v \right) -\rho_uuz, &x\in \Omega ,t> 0,\\ w_t=\Delta w-\xi_w\nabla \cdot \left( w\nabla v \right) -\delta_ww+\rho_wuz, &x\in \Omega ,t> 0,\\ v_t=-(\alpha_uu+\alpha_ww)v-\delta_v v, &x\in \Omega ,t> 0,\\ z_t=\Delta z-\xi_z\nabla \cdot \left( z\nabla v \right) -\delta_zz-\rho_zuz+\beta w, &x\in \Omega ,t> 0, \end{cases} \tag{0.1}$ where $$u, w, v$$ and $$z$$ denote the density of uninfected cancer cells, oncolytic viruses infected cancer cells, extracellular matrix and oncolytic virus particles, respectively. It is showed that when suitably regular initial data satisfy a certain small condition, infected cancer cells and virus particle populations will both become extinct asymptotically. The rates of convergence for the chemotaxis-Navier-Stokes equations in a strip domain https://zbmath.org/1485.35065 2022-06-24T15:10:38.853281Z "Wu, Jie" https://zbmath.org/authors/?q=ai:wu.jie.4|wu.jie.2|wu.jie.3|wu.jie|wu.jie.6|wu.jie.1|wu.jie.5 "Lin, Hongxia" https://zbmath.org/authors/?q=ai:lin.hongxia Summary: In this paper, we study the long-time behavior of the chemotaxis-Navier-Stokes system \begin{aligned} &\partial_t n + \boldsymbol{u} \cdot \nabla n = \lambda \Delta n - \nabla \cdot (\chi (c)n \nabla c),\\ &\partial_t c + \boldsymbol{u} \cdot \nabla c = \mu \Delta c - f(c)n,\\ &\partial_t \boldsymbol{u} + \boldsymbol{u} \cdot \nabla \boldsymbol{u} = \zeta \Delta \boldsymbol{u} - n\nabla \phi,\\ &\nabla \cdot \boldsymbol{u} =0, \quad t>0, x \in \Omega \end{aligned} posed in a strip domain $$\Omega := \mathbb{R}^2 \times [0,1] \subset \mathbb{R}^3$$. In [\textit{Y. Peng} and \textit{Z. Xiang}, Math. Models Methods Appl. Sci. 28, No. 5, 869--920 (2018; Zbl 1391.35206)], the authors have established the global existence of strong solutions to this system with non-flux boundary conditions for $$n$$ and $$c$$ and non-slip boundary conditions for $$\boldsymbol{u}$$. Our main purpose is to establish the time-decay rates for such solutions. This will be done by using the anisotropic $$L^p$$ interpolation and the iterative techniques. Optimal convergence rates to diffusion waves for solutions of $$p$$-system with damping on quadrant https://zbmath.org/1485.35066 2022-06-24T15:10:38.853281Z "Zhang, Nangao" https://zbmath.org/authors/?q=ai:zhang.nangao Summary: This paper is concerned with the asymptotic behavior of solutions to the initial-boundary value problem for the $$p$$-system with linear damping. We show that the solutions to this system globally exist and converge time-asymptotically to nonlinear diffusion wave whose profile is self-similar solution to the corresponding parabolic equation governed by the classical Darcy's law. Compared with the results obtained by \textit{K. Nishihara} and \textit{T. Yang} [J. Differ. Equations 156, No. 2, 439--458 (1999; Zbl 0933.35121)], the better convergence rates are obtained. The proof is based on time-weighted energy estimates together with Green's function method. Eventual smoothness and stabilization in a three-dimensional Keller-Segel-Navier-Stokes system with rotational flux https://zbmath.org/1485.35067 2022-06-24T15:10:38.853281Z "Zheng, Jiashan" https://zbmath.org/authors/?q=ai:zheng.jiashan Summary: We consider the spatially 3-D version of the following Keller-Segel-Navier-Stokes system with rotational flux $\begin{cases} n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nS(x, n, c)\nabla c),\quad x\in \Omega,\; t>0,\\ c_t+u\cdot \nabla c=\Delta c-c+n,\quad x\in \Omega,\; t>0,\\ u_t+\kappa (u \cdot \nabla )u+\nabla P=\Delta u+n\nabla \phi,\quad x\in \Omega,\; t>0,\\ \nabla \cdot u=0,\quad x\in \Omega,\; t>0 \end{cases}\tag{$$\ast$$}$ under no-flux boundary conditions in a bounded domain $$\Omega \subseteq \mathbb{R}^3$$ with smooth boundary, where $$\phi \in W^{2, \infty}(\Omega)$$ and $$\kappa \in \mathbb{R}$$ represent the prescribed gravitational potential and the strength of nonlinear fluid convection, respectively. Here the matrix-valued function $$S(x, n, c)\in C^2(\bar{\Omega}\times [0, \infty)^2; \mathbb{R}^{3\times 3})$$ denotes the rotational effect which satisfies $$|S(x, n, c)|\le C_S(1 + n)^{-\alpha}$$ with some $$C_S > 0$$ and $$\alpha \ge 0$$. Compared with the signal consumption case as in chemotaxis-(Navier-)Stokes system, the quantity $$c$$ of system ($$\ast$$) is no longer a priori bounded by its initial norm in $$L^\infty$$, which means that we have less regularity information on $$c$$. Moreover, the tensor-valued sensitivity functions result in new mathematical difficulties, mainly linked to the fact that a chemotaxis system with such rotational fluxes thereby loses an energy-like structure. In this paper, under an explicit condition on the size of $$C_S$$ relative to $$C_N$$, we can prove that the \textit{weak} solution $$(n, c, u)$$ becomes smooth ultimately, and that it approaches the unique spatially homogeneous steady state $$(\bar{n}_0, \bar{n}_0, 0)$$, where $$\bar{n}_0=\frac{1}{|\Omega|}\int_\Omega n_0$$ and $$C_N$$ is the best Poincaré constant. To the best of our knowledge, there are the first results on asymptotic behavior of the system. Exponential attractors for the 3D fractional-order Bardina turbulence model with memory and horizontal filtering https://zbmath.org/1485.35068 2022-06-24T15:10:38.853281Z "Annese, Michele" https://zbmath.org/authors/?q=ai:annese.michele "Bisconti, Luca" https://zbmath.org/authors/?q=ai:bisconti.luca "Catania, Davide" https://zbmath.org/authors/?q=ai:catania.davide Summary: We consider the 3D simplified Bardina turbulence model with horizontal filtering, fractional dissipation, and the presence of a memory term incorporating hereditary effects. We analyze the regularity properties and the dissipative nature of the considered system and, in our main result, we show the existence of a global exponential attractor in a suitable phase space. Existence and characterization of attractors for a nonlocal reaction-diffusion equation with an energy functional https://zbmath.org/1485.35069 2022-06-24T15:10:38.853281Z "Caballero, R." https://zbmath.org/authors/?q=ai:caballero.ruben "Marín-Rubio, P." https://zbmath.org/authors/?q=ai:marin-rubio.pedro "Valero, José" https://zbmath.org/authors/?q=ai:valero.jose Summary: In this paper we study a nonlocal reaction-diffusion equation in which the diffusion depends on the gradient of the solution. Firstly, we prove the existence and uniqueness of regular and strong solutions. Secondly, we obtain the existence of global attractors in both situations under rather weak assumptions by defining a multivalued semiflow (which is a semigroup in the particular situation when uniqueness of the Cauchy problem is satisfied). Thirdly, we characterize the attractor either as the unstable manifold of the set of stationary points or as the stable one when we consider solutions only in the set of bounded complete trajectories. Global attractor of the fractional damping wave equation on $$\mathbb{R}^3$$ https://zbmath.org/1485.35070 2022-06-24T15:10:38.853281Z "Guo, Yantao" https://zbmath.org/authors/?q=ai:guo.yantao "Ding, Pengyan" https://zbmath.org/authors/?q=ai:ding.pengyan Summary: The paper studies the well-posedness and the longtime behaviors of the fractional damping wave equation on $$\mathbb{R}^3$$ with critical nonlinearity, we firstly prove the well-posedness of weak solutions via an established Strichartz estimate; Secondly, by the uniformly tail estimate and the energy identity, we prove that the corresponding semigroup possesses a global attractor in the energy space $$H^1(\mathbb{R}^3) \times L^2(\mathbb{R}^3)$$. Finite time blowup of 2D Boussinesq and 3D Euler equations with $$C^{1, \alpha}$$ velocity and boundary https://zbmath.org/1485.35071 2022-06-24T15:10:38.853281Z "Chen, Jiajie" https://zbmath.org/authors/?q=ai:chen.jiajie "Hou, Thomas Yizhao" https://zbmath.org/authors/?q=ai:hou.thomas-yizhao The authors study simulatenously two partial cases of Euler equations: axisymmetrical Euler system and Boussinesq system (2D due to the Boussinesq approximation). A finite-time blow-up is proved for the latter, the finite time singularity formation is proved for the former under additional conditions. Initial data are of $$C^{1,\alpha}$$. The Boussinesq system is in the $$\mathbb{R}_+^2$$ with the potential for the velocity. The Euler equations are in the cylinder. In both cases no-flow boundary condition is used. Reviewer: Ilya A. Chernov (Petrozavodsk) Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic-elliptic attraction-repulsion chemotaxis system https://zbmath.org/1485.35072 2022-06-24T15:10:38.853281Z "Chiyo, Yutaro" https://zbmath.org/authors/?q=ai:chiyo.yutaro "Yokota, Tomomi" https://zbmath.org/authors/?q=ai:yokota.tomomi Summary: This paper deals with the quasilinear attraction-repulsion chemotaxis system $\begin{cases} u_t=\nabla \cdot \big ((u+1)^{m-1}\nabla u -\chi u(u+1)^{p-2}\nabla v +\xi u(u+1)^{q-2}\nabla w\big ) +f(u), \\ 0=\Delta v+\alpha u-\beta v, \\ 0=\Delta w+\gamma u-\delta w \end{cases}$ in a bounded domain $$\Omega \subset{\mathbb{R}}^n (n \in{\mathbb{N}})$$ with smooth boundary $$\partial \Omega$$, where $$m, p, q \in{\mathbb{R}}, \chi , \xi , \alpha , \beta , \gamma , \delta >0$$ are constants, and $$f$$ is a function of logistic type such as $$f(u)=\lambda u-\mu u^{\kappa }$$ with $$\lambda , \mu >0$$ and $$\kappa \ge 1$$, provided that the case $$f(u) \equiv 0$$ is included in the study of boundedness, whereas $$\kappa$$ is sufficiently close to 1 in considering blow-up in the radially symmetric setting. In the case that $$\xi =0$$ and $$f(u) \equiv 0$$, global existence and boundedness have already been proved under the condition $$p<m+\frac{2}{n}$$. Also, in the case that $$m=1, p=q=2$$ and $$f$$ is a function of logistic type, finite-time blow-up has already been established by assuming $$\chi \alpha -\xi \gamma >0$$. This paper classifies boundedness and blow-up into the cases $$p<q$$ and $$p>q$$ without any condition for the sign of $$\chi \alpha -\xi \gamma$$ and the case $$p=q$$ with $$\chi \alpha -\xi \gamma <0$$ or $$\chi \alpha -\xi \gamma >0$$. Blow-up of result in a nonlinear wave equation with delay and source term https://zbmath.org/1485.35073 2022-06-24T15:10:38.853281Z "Lakroumbe, Tayeb" https://zbmath.org/authors/?q=ai:lakroumbe.tayeb "Abdelli, Mama" https://zbmath.org/authors/?q=ai:abdelli.mama "Beniani, Abderrahmane" https://zbmath.org/authors/?q=ai:beniani.abderrahmane Summary: In this paper we consider the initial boundary value problem for a nonlinear damping and a delay term of the form: $|u_t|^l u_{tt} - \Delta u(x,t) - \Delta u_{tt}+ \mu_1 |u_t|^{m-2} u_t+ \mu_2|u_t(t-\tau)|^{m-2} u_t(t-\tau) = b|u|^{p-2} u,$ with initial conditions and Dirichlet boundary conditions. Under appropriate conditions on $$\mu_1$$, $$\mu_2$$, we prove that there are solutions with negative initial energy that blow-up finite time if $$p \geq \max\{l+2,m\}$$. Blowup time estimates for the heat equation with a nonlocal boundary condition https://zbmath.org/1485.35074 2022-06-24T15:10:38.853281Z "Lu, Heqian" https://zbmath.org/authors/?q=ai:lu.heqian "Hu, Bei" https://zbmath.org/authors/?q=ai:hu.bei "Zhang, Zhengce" https://zbmath.org/authors/?q=ai:zhang.zhengce Summary: We study the blowup time for the heat equation $$u_t=\Delta u$$ in a bounded domain $$\Omega \subset{\mathbb{R}}^n(n\geqslant 2)$$ with the nonlocal boundary condition, where the normal derivative $$\partial u/\partial \vec{\eta}=\int \limits_{\Omega }u^p\text{d}z$$ on one part of boundary $$\Gamma_1\subseteq \partial \Omega$$ for some $$p>1$$, while $$\partial u/\partial \vec{\eta}=0$$ on the rest part of the boundary. By constructing suitable auxiliary functions and analyzing the representation formula of $$u$$, we establish the finite time blowup of the solution and get both upper and lower bounds for the blowup time in terms of the parameter $$p$$, the initial value $$u_0(x)$$ and the volume of $$\Gamma_1$$. In many other studies, they require the convexity of the domain $$\Omega$$ and only deal with the case $$\Gamma_1=\partial \Omega$$. In this article, we remove the convexity assumption and consider the problem with $$\Gamma_1\subseteq \partial \Omega$$. Non global solutions for a class of Klein-Gordon equations https://zbmath.org/1485.35075 2022-06-24T15:10:38.853281Z "Saanouni, T." https://zbmath.org/authors/?q=ai:saanouni.tarek Summary: Under sufficient conditions on the data, solutions to a class of Klein-Gordon equations with exponential type non-linearity and arbitrary positive energy blow-up in finite time. Semilinear viscous Moore-Gibson-Thompson equation with the derivative-type nonlinearity: global existence versus blow-up https://zbmath.org/1485.35076 2022-06-24T15:10:38.853281Z "Shi, Jincheng" https://zbmath.org/authors/?q=ai:shi.jincheng "Zhang, Yan" https://zbmath.org/authors/?q=ai:zhang.yan.5|zhang.yan.4 "Cai, Zihan" https://zbmath.org/authors/?q=ai:cai.zihan "Liu, Yan" https://zbmath.org/authors/?q=ai:liu.yan Summary: In this paper, we study global existence and blow-up of solutions to the viscous Moore-Gibson-Thompson (MGT) equation with the nonlinearity of derivative-type $$|u_t|^p$$. We demonstrate global existence of small data solutions if $$p > 1+4/n ( n\leq 6 )$$ or $$p\geq 2-2/n ( n\geq 7 )$$, and blow-up of nontrivial weak solutions if $$1 <p\leq 1+1/n$$. Deeply, we provide estimates of solutions to the nonlinear problem. These results complete the recent works for semilinear MGT equations by \textit{W. Chen} and \textit{R. Ikehata} [J. Differ. Equations 292, 176--219 (2021; Zbl 1466.35259)]. Blow-up of weak solutions for a porous elastic system with nonlinear damping and source terms https://zbmath.org/1485.35077 2022-06-24T15:10:38.853281Z "Tran, Quang-Minh" https://zbmath.org/authors/?q=ai:tran.quang-minh "Vu, Thi-Thi" https://zbmath.org/authors/?q=ai:vu.thi-thi "Freitas, Mirelson M." https://zbmath.org/authors/?q=ai:freitas.mirelson-m Summary: This paper deals with blow-up solutions of the initial boundary value problems for a porous elastic system with nonlinear damping and source terms at arbitrary initial energy level. We estimate the lower bound and upper bound of the lifespan of the blow-up solution, and also estimate the blow-up rate by considering both linear and nonlinear weak damping terms. Blow-up and peakons for a higher-order $$\mu$$-Camassa-Holm equation https://zbmath.org/1485.35078 2022-06-24T15:10:38.853281Z "Wang, Hao" https://zbmath.org/authors/?q=ai:wang.hao.12|wang.hao.7|wang.hao.13|wang.hao.2|wang.hao.10|wang.hao.9|wang.hao.5|wang.hao.6|wang.hao.3|wang.hao.1|wang.hao.11|wang.hao.4 "Luo, Ting" https://zbmath.org/authors/?q=ai:luo.ting "Fu, Ying" https://zbmath.org/authors/?q=ai:fu.ying "Qu, Changzheng" https://zbmath.org/authors/?q=ai:qu.changzheng Summary: This paper proposes a higher-order $$\mu$$-Camassa-Holm equation, which is regarded as a higher-order extension of the $$\mu$$-Camassa-Holm equation, and preserves some properties of the $$\mu$$-Camassa-Holm equation. We first show that the equation admits the peaked traveling wave solution, which is given by a Green function of the momentum operator. Local well-posedness of the Cauchy problem in the suitable Sobolev space is established. Finally, the blow-up criterion and wave breaking mechanism for solutions with certain initial profiles are studied. It turns out that all the nonlinearities even the first-order nonlinearity may have the effect on the blow up. Global existence and blow-up phenomena for the Hunter-Saxton equation on the line https://zbmath.org/1485.35079 2022-06-24T15:10:38.853281Z "Ye, Weikui" https://zbmath.org/authors/?q=ai:ye.weikui "Yin, Zhaoyang" https://zbmath.org/authors/?q=ai:yin.zhaoyang Summary: In this paper, we consider the Cauchy problem for the Hunter-Saxton (HS) equation on the line. Firstly, we establish the local well-posedness for the integral form of the (HS) equation in some special spaces $$E^s_{p, r}$$, which mix Lebesgue spaces and homogeneous Besov spaces. Then we present a global existence result and provide a sufficient condition for strong solutions to blow up in finite time for the equation. The global existence is a new result comparing to the circle case in [the second author, SIAM J. Math. Anal. 36, No. 1, 272--283 (2004; Zbl 1151.35321)], since Yin [loc. cit.] proved that all solutions of (HS) equation with initial data that are not constant functions blow up in finite time. Finally, we give the ill-posedness and the unique continuation of the (HS) equation. Maximal integrability for general elliptic problems with diffusive measures https://zbmath.org/1485.35080 2022-06-24T15:10:38.853281Z "Byun, Sun-Sig" https://zbmath.org/authors/?q=ai:byun.sun-sig "Song, Kyeong" https://zbmath.org/authors/?q=ai:song.kyeong Summary: We consider a nonlinear elliptic equation with Orlicz growth having a diffusive measure on the right-hand side. A maximal gradient integrability for such a measure data problem is established in the scale of Marcinkiewicz-Morrey spaces. Gradient estimates for singular parabolic $$p$$-Laplace type equations with measure data https://zbmath.org/1485.35081 2022-06-24T15:10:38.853281Z "Dong, Hongjie" https://zbmath.org/authors/?q=ai:dong.hongjie "Zhu, Hanye" https://zbmath.org/authors/?q=ai:zhu.hanye Summary: We are concerned with gradient estimates for solutions to a class of singular quasilinear parabolic equations with measure data, whose prototype is given by the parabolic $$p$$-Laplace equation $$u_t-\Delta_p u=\mu$$ with $$p\in (1, 2)$$. The case when $$p\in\big (2-\frac{1}{n+1},2\big)$$ were studied in [\textit{T. Kuusi} and \textit{G. Mingione}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 12, No. 4, 755--822 (2013; Zbl 1288.35145)]. In this paper, we extend the results in Kuusi and Mingione [loc. cit.] to the open case when $$p\in\big(\frac{2n}{n+1},2-\frac{1}{n+1}\big]$$ if $$n\ge 2$$ and $$p\in (\frac{5}{4}, \frac{3}{2}]$$ if $$n=1$$. More specifically, in a more singular range of $$p$$ as above, we establish pointwise gradient estimates via linear parabolic Riesz potential and gradient continuity results via certain assumptions on parabolic Riesz potential. Stability results for backward heat equations with time-dependent coefficient in the Banach space $$L_p (\mathbb{R})$$ https://zbmath.org/1485.35082 2022-06-24T15:10:38.853281Z "Duc, Nguyen Van" https://zbmath.org/authors/?q=ai:duc.nguyen-van "Muoi, Pham Quy" https://zbmath.org/authors/?q=ai:muoi.pham-quy "Anh, Nguyen Thi Van" https://zbmath.org/authors/?q=ai:anh.nguyen-thi-van Summary: In this paper, we investigate the problem of backward heat equations with time-dependent coefficient in the Banach space $$L_p (\mathbb{R}),\; (1 < p < \infty)$$. For this problem, we first prove the stability estimates of Hölder type. After that the Tikhonov-type regularization is applied to solve the problem. A priori and a posteriori parameter choice rules are investigated, which yield error estimates of Hölder type. Numerical implementations are presented to show the validity of the proposed scheme. Harnack estimate for positive solutions to a nonlinear equation under geometric flow https://zbmath.org/1485.35083 2022-06-24T15:10:38.853281Z "Fasihi-Ramandi, Ghodratallah" https://zbmath.org/authors/?q=ai:ramandi.ghodratallah-fasihi "Azami, Shahroud" https://zbmath.org/authors/?q=ai:azami.shahroud Summary: In the present paper, we obtain gradient estimates for positive solutions to the following nonlinear parabolic equation under general geometric flow on complete noncompact manifolds $\frac{ \partial u}{ \partial t} = \triangle u + a (x, t) u^p + b (x, t) u^q$ where, $$0 < p, q < 1$$ are real constants and $$a(x, t)$$ and $$b(x, t)$$ are functions which are $$C^2$$ in the $$x$$-variable and $$C^1$$ in the $$t$$-variable. We shall get an interesting Harnack inequality as an application. Carleson estimates for the singular parabolic $$p$$-Laplacian in time-dependent domains https://zbmath.org/1485.35084 2022-06-24T15:10:38.853281Z "Gianazza, Ugo" https://zbmath.org/authors/?q=ai:gianazza.ugo-pietro Summary: We deal with the parabolic $$p$$-Laplacian in the so-called singular super-critical range $$\frac{2N}{N+1}< p < 2$$, and we prove Carleson estimates for non-negative solutions in suitable non-cylindrical domains $$\Omega\subset\mathbb{R}^{N+1}$$. The sets $$\Omega$$ satisfy a proper NTA condition, tailored on the parabolic $$p$$-Laplacian. As an intermediate step, we show that in these domains non-negative solutions which vanish at the boundary, are Hölder continuous up to the same boundary. Nonlinear parabolic equations with Robin boundary conditions and Hardy-Leray type inequalities https://zbmath.org/1485.35085 2022-06-24T15:10:38.853281Z "Goldstein, Gisèle Ruiz" https://zbmath.org/authors/?q=ai:ruiz-goldstein.gisele "Goldstein, Jerome A." https://zbmath.org/authors/?q=ai:goldstein.jerome-a "Kömbe, Ismail" https://zbmath.org/authors/?q=ai:kombe.ismail "Tellioğlu Balekoğlu, Reyhan" https://zbmath.org/authors/?q=ai:tellioglu-balekoglu.reyhan Summary: We are primarily concerned with the absence of positive solutions of the following problem, $\begin{cases} \frac{\partial u}{\partial t}=\Delta(u^m)+V(x)u^m+\lambda u^q & \text{ in }\Omega\times (0, T),\\ u(x,0)=u_0(x)\geq 0 & \text{ in }\Omega,\\ \frac{\partial u}{\partial\nu}=\beta(x)u & \text{ on }\partial\Omega\times (0,T), \end{cases}$ where $$0<m<1$$, $$V\in L_{\mathrm{loc}}^1(\Omega)$$, $$\beta\in L_{\mathrm{loc}}^1(\partial\Omega)$$, $$\lambda\in\mathbb{R}$$, $$q>0$$, $$\Omega\subset\mathbb{R}^N$$ is a bounded open subset of $$\mathbb{R}^N$$ with smooth boundary $$\partial\Omega$$, and $$\frac{\partial u}{\partial\nu}$$ is the outer normal derivative of $$u$$ on $$\partial\Omega$$. Moreover, we also present some new sharp Hardy and Leray type inequalities with remainder terms that provide us concrete potentials to use in the partial differential equation of our interest. For the entire collection see [Zbl 07455846]. Retraction note: A new application of boundary integral behaviors of harmonic functions to the least harmonic majorant'' https://zbmath.org/1485.35086 2022-06-24T15:10:38.853281Z "Han, Minghua" https://zbmath.org/authors/?q=ai:han.minghua "Sun, Jianguo" https://zbmath.org/authors/?q=ai:sun.jianguo.2|sun.jianguo.1|sun.jianguo "Xue, Gaoying" https://zbmath.org/authors/?q=ai:xue.gaoying From the text: The Editors-in-Chief have retracted the article [ibid. 2017, Paper No. 67, 9 p. (2017; Zbl 1379.35041)] by \textit{M. Han} et al. because of overlap with articles from different authors that were simultaneously under consideration with another journal. Additionally, this article shows evidence of both authorship and peer review manipulation. The authors have not responded to any correspondence regarding this retraction. Estimates on fundamental solutions of parabolic magnetic Schrödinger operators and uniform parabolic equations with nonnegative potentials and their applications https://zbmath.org/1485.35087 2022-06-24T15:10:38.853281Z "Tang, Lin" https://zbmath.org/authors/?q=ai:tang.lin "Zhao, Yuan" https://zbmath.org/authors/?q=ai:zhao.yuan Summary: We study the fundamental solutions of parabolic magnetic Schrödinger operators and uniform parabolic operators with nonnegative potentials in the reverse Hölder class. The main aim of the paper is to give pointwise estimates of the heat kernel of the operators above, which improve and generalize the main results by \textit{K. Kurata} [J. Lond. Math. Soc., II. Ser. 62, No. 3, 885--903 (2000; Zbl 1013.35020)]. On a comparison theorem for parabolic equations with nonlinear boundary conditions https://zbmath.org/1485.35088 2022-06-24T15:10:38.853281Z "Kita, Kosuke" https://zbmath.org/authors/?q=ai:kita.kosuke "Ôtani, Mitsuharu" https://zbmath.org/authors/?q=ai:otani.mitsuharu Summary: In this article, a new type of comparison theorem for some second-order nonlinear parabolic systems with nonlinear boundary conditions is given, which can cover classical linear boundary conditions, such as the homogeneous Dirichlet or Neumann boundary condition. The advantage of our comparison theorem over the classical ones lies in the fact that it enables us to compare two solutions satisfying different types of boundary conditions. As an application of our comparison theorem, we can give some new results on the existence of blow-up solutions of some parabolic equations and systems with nonlinear boundary conditions. Comparison theorems for elliptic inequalities with lower-order derivatives that take into account the geometry of the domain https://zbmath.org/1485.35089 2022-06-24T15:10:38.853281Z "Kon'kov, A. A." https://zbmath.org/authors/?q=ai:konkov.andrej-a Summary: Comparison theorems are obtained with the help of which the spherical maximum of solutions of quasilinear elliptic inequalities containing lower-order derivatives is estimated in terms of solutions of the Cauchy problem for an ordinary differential equation with a right-hand side depending on the geometry of the domain. A Liouville-type theorem for fractional elliptic equation with exponential nonlinearity https://zbmath.org/1485.35090 2022-06-24T15:10:38.853281Z "Duong, Anh Tuan" https://zbmath.org/authors/?q=ai:duong.anh-tuan "Nguyen, Van Hoang" https://zbmath.org/authors/?q=ai:nguyen-van-hoang.1 Summary: In this paper, we are concerned with stable solutions to the fractional elliptic equation $(-\Delta )^s u=e^u \text{ in } \mathbb{R}^N,$ where $$(-\Delta )^s$$ is the fractional Laplacian with $$0<s<1$$. By developing a new technique with non-compactly supported test functions, we establish the nonexistence of stable solutions provided that $$N<10s$$. This result is optimal when $$s\uparrow 1$$. On the other hand, we believe that our technique can be used to study stable solutions of elliptic equations/systems involving the fractional Laplacian. Sharp regularity estimates for solutions of the continuity equation drifted by Sobolev vector fields https://zbmath.org/1485.35091 2022-06-24T15:10:38.853281Z "Bruè, Elia" https://zbmath.org/authors/?q=ai:brue.elia "Nguyen, Quoc-Hung" https://zbmath.org/authors/?q=ai:nguyen.quoc-hung|nguyen.quoc-hung.1 Summary: The aim of this note is to prove sharp regularity estimates for solutions of the continuity equation, associated to $$\boldsymbol{W}^{1,p}$$ vector fields for $$p>1$$. The regularity is of logarithmic order'' and is measured by means of suitable seminorms. Fractional differentiability for a class of double phase problems with measure data https://zbmath.org/1485.35092 2022-06-24T15:10:38.853281Z "Byun, Sun-Sig" https://zbmath.org/authors/?q=ai:byun.sun-sig "Shin, Pilsoo" https://zbmath.org/authors/?q=ai:shin.pilsoo "Song, Kyeong" https://zbmath.org/authors/?q=ai:song.kyeong Summary: This paper is concerned with a borderline case of double phase problems with a finite Radon measure on the right-hand side. We obtain sharp fractional regularity estimates for such non-uniformly elliptic problems. Regularity of solutions to a Vekua-type equation on compact Lie groups https://zbmath.org/1485.35093 2022-06-24T15:10:38.853281Z "de Moraes, Wagner Augusto Almeida" https://zbmath.org/authors/?q=ai:de-moraes.wagner-augusto-almeida Summary: We present sufficient conditions to have global hypoellipticity for a class of Vekua-type operators defined on a compact Lie group. When the group has the property that every non-trivial representation is not self-dual, we show that these sufficient conditions are also necessary. We also present results about the global solvability for this class of operators. Beale-Kato-Majda regularity criterion of smooth solutions for the Hall-MHD equations with zero viscosity https://zbmath.org/1485.35094 2022-06-24T15:10:38.853281Z "Gala, Sadek" https://zbmath.org/authors/?q=ai:gala.sadek "Galakhov, Eugeny" https://zbmath.org/authors/?q=ai:galakhov.eugeny "Ragusa, Maria Alessandra" https://zbmath.org/authors/?q=ai:ragusa.maria-alessandra "Salieva, Olga" https://zbmath.org/authors/?q=ai:salieva.olga-alekseevna Summary: In this paper, we investigate the Cauchy problem for the 3D incompressible Hall-MHD equations with zero viscosity. We prove the Beale-Kato-Majda regularity criterion of smooth solutions in terms of the velocity field and magnetic field in the homogeneous Besov spaces $${\dot{B}}_{\infty ,\infty }^0$$. Then we give a criterion on extension beyond $$T$$ of our local solution. Our result may be also regarded as an extension of the corresponding result of \textit{Y.-Z. Wang} and \textit{W. Zuo} [Commun. Pure Appl. Anal. 13, No. 3, 1327--1336 (2014; Zbl 1292.76012)]. Regularity of solutions to Kolmogorov equation with Gilbarg-Serrin matrix https://zbmath.org/1485.35095 2022-06-24T15:10:38.853281Z "Kinzebulatov, D." https://zbmath.org/authors/?q=ai:kinzebulatov.damir "Semënov, Yu. A." https://zbmath.org/authors/?q=ai:semenov.yuliy-a Summary: In $${\mathbb{R}}^d$$, $$d \ge 3$$, consider the divergence and the non-divergence form operators \begin{aligned} & -\Delta - \nabla \cdot (a-I) \cdot \nabla + b \cdot \nabla, \\ & - \Delta - (a-I) \cdot \nabla^2 + b \cdot \nabla, \end{aligned} where the second-order perturbations are given by the matrix $a-I=c|x|^{-2}x \otimes x, \quad c>-1.$ The vector field $$b:{\mathbb{R}}^d \rightarrow{\mathbb{R}}^d$$ is form-bounded with form-bound $$\delta >0$$. (This includes vector fields with entries in $$L^d$$, as well as vector fields having critical-order singularities.) We characterize quantitative dependence on $$c$$ and $$\delta$$ of the $$L^q \rightarrow W^{1,qd/(d-2)}$$ regularity of solutions of the corresponding elliptic and parabolic equations in $$L^q$$, $$q \ge 2 \vee ( d-2)$$. Gaussian bounds of fundamental matrix and maximal $$L^1$$ regularity for Lamé system with rough coefficients https://zbmath.org/1485.35096 2022-06-24T15:10:38.853281Z "Xu, Huan" https://zbmath.org/authors/?q=ai:xu.huan Summary: The purpose of this paper is twofold. First, we use a classical method to establish Gaussian bounds of the fundamental matrix of a generalized parabolic Lamé system with only bounded and measurable coefficients. Second, we derive a maximal $$L^1$$ regularity result for the abstract Cauchy problem associated with a composite operator. In a concrete example, we also obtain maximal $$L^1$$ regularity for the Lamé system, from which it follows that the Lipschitz seminorm of the solutions to the Lamé system is globally $$L^1$$-in-time integrable. As an application, we use a Lagrangian approach to prove a global-in-time well-posedness result for a viscous pressureless flow in a perturbation framework, but with possibly discontinuous densities. The method established in this paper might be a useful tool for studying many issues arising from viscous fluids with truly variable densities. A global regularity result for the 2D generalized magneto-micropolar equations https://zbmath.org/1485.35097 2022-06-24T15:10:38.853281Z "Zhang, Hui" https://zbmath.org/authors/?q=ai:zhang.hui|zhang.hui.11|zhang.hui.3|zhang.hui.7|zhang.hui.6|zhang.hui.8|zhang.hui.9|zhang.hui.2|zhang.hui.5|zhang.hui.4|zhang.hui.10|zhang.hui.1 Summary: In this paper, we proved the global (in time) regularity for smooth solution to the 2D generalized magneto-micropolar equations with zero viscosity. When there is no kinematic viscosity in the momentum equation, it is difficult to examine the bounds on the any derivatives of the velocity $$\|J^\varepsilon u\|_{L^2}$$. In order to overcome the main obstacle, we find a new unknown quantity which is by combining the vorticity and the microrotation angular velocity; the structure of the system including the combined quantity obeys a Beale-Kato-Majda criterion. Moreover, the maximal regularity of parabolic equations together with the classic commutator estimates allows us to derive the $$H^s$$ estimates for solutions of the system. A generalized (2 + 1)-dimensional Calogaro-Bogoyavlenskii-Schiff equation: symbolic computation, symmetry reductions, exact solutions, conservation laws https://zbmath.org/1485.35099 2022-06-24T15:10:38.853281Z "Moroke, M. C." https://zbmath.org/authors/?q=ai:moroke.m-c "Muatjetjeja, B." https://zbmath.org/authors/?q=ai:muatjetjeja.ben "Adem, A. R." https://zbmath.org/authors/?q=ai:adem.abdullahi-rashid Summary: Lie symmetry analysis is performed on a generalized (2 + 1)-dimensional Calogaro-Bogoyavlenskii-Schiff equation, which arises in the analysis of various problems in theoretical physics and many scientific applications. It should be pointed out that this equation was first derived by Bogoyavlenskii and Schiff in a unique ways, namely Bogoyavlenskii used the modified Lax formalism, whereas Schiff employed the same equation by reducing the self-dual Yang Mills equation. New analysis is shown that basically entails via symbolic computation to extract new symmetry reductions, new exact solutions and finally new conservation laws. Propagation dynamics for an age-structured population model in time-space periodic habitat https://zbmath.org/1485.35109 2022-06-24T15:10:38.853281Z "Pan, Yingli" https://zbmath.org/authors/?q=ai:pan.yingli Summary: How do environmental heterogeneity influence propagation dynamics of the age-structured invasive species? We investigate this problem by considering a yearly generation invasive species in time-space periodic habitat. Starting from an age-structured population growth law, we formulate a reaction-diffusion model with time-space periodic dispersal, mortality and recruitment. Thanks to the fundamental solution for linear part of the model, we reduce to study the dynamics of a time-space periodic semiflow which is defined by the solution map. By the recent developed dynamical theory in [\textit{J. Fang} et al., J. Funct. Anal. 272, No. 10, 4222--4262 (2017; Zbl 1398.35116)], we obtained the spreading speed and its coincidence with the minimal wave speed of time-space periodic traveling waves, as well as the variational characterization of spreading speed in terms of a principal eigenvalue problem. Such results are also proved back to the reaction-diffusion model. Spreading speeds and traveling waves for a time periodic DS-I-A epidemic model https://zbmath.org/1485.35110 2022-06-24T15:10:38.853281Z "Yang, Xiying" https://zbmath.org/authors/?q=ai:yang.xiying "Lin, Guo" https://zbmath.org/authors/?q=ai:lin.guo Summary: This paper is devoted to studying the speed of asymptotic spreading and minimal wave speed of traveling wave solutions for a time periodic and diffusive DS-I-A epidemic model, which describes the propagation threshold of disease spreading. The main feature of this model is the possible deficiency of the classical comparison principle such that many known results do not directly work. The speed of asymptotic spreading is estimated by constructing auxiliary equations and applying the classical theory of asymptotic spreading for Fisher type equation. The minimal wave speed is established by proving the existence and nonexistence of the nonconstant traveling wave solutions. Moreover, some numerical examples are presented to model the propagation dynamics of this system. Local existence and uniqueness of strong solutions to the two dimensional compressible primitive equations with density-dependent viscosity https://zbmath.org/1485.35120 2022-06-24T15:10:38.853281Z "Wang, Fengchao" https://zbmath.org/authors/?q=ai:wang.fengchao The author considers the two dimensional compressible primitive equations of an hydrostatic atmosphere model. Spatial dimensions are one horizontal and the vertical. The problem is in the rectangle (from the ground to a given height and on a segment in the horizontal dimension). Initial density and velocity profiles are exponentially decaying with height. Viscosities are some special functions of coordinates and density. For this problem, the local strong existence, uniqueness, and regularity result is proved. The initial data are supposed to obey regularity conditions, but may be large. Reviewer: Ilya A. Chernov (Petrozavodsk) Cauchy problem for the equation of longitudinal vibrations of a thick rod with allowance for transverse inertia https://zbmath.org/1485.35122 2022-06-24T15:10:38.853281Z "Umarov, Kh. G." https://zbmath.org/authors/?q=ai:umarov.khasan-galsanovich Summary: For a nonlinear differential equation of Sobolev type describing the longitudinal vibrations of a thick rod with allowance for its transverse inertia, we study the solvability of the Cauchy problem in the half-plane $$(x,t)\in \mathbb{R}^1\times [0,+\infty )$$ in the class of functions that, for each fixed value of the time variable $$t\geq 0$$, are continuous on the entire real line and have finite limits at infinity. Both sufficient conditions for the existence of a global solution of the Cauchy problem and sufficient conditions for its blowup on a finite time interval are found. Multiple end solutions to the Allen-Cahn equation in $$\mathbb{R}^2$$ https://zbmath.org/1485.35128 2022-06-24T15:10:38.853281Z "Kowalczyk, Michał" https://zbmath.org/authors/?q=ai:kowalczyk.michal "Liu, Yong" https://zbmath.org/authors/?q=ai:liu.yong.1 "Pacard, Frank" https://zbmath.org/authors/?q=ai:pacard.frank Summary: An entire solution of the Allen-Cahn equation $$\Delta u=f(u)$$, where $$f$$ is an odd function and has exactly three zeros at $$\pm 1$$ and 0, e.g. $$f(u)=u(u^2-1)$$, is called a $$2k$$ end solution if its nodal set is asymptotic to $$2k$$ half lines, and if along each of these half lines the function $$u$$ looks (up to a multiplication by $$-1$$) like the one dimensional, odd, heteroclinic solution $$H$$, of $$H^{\prime \prime}=f(H)$$. In this paper we present some recent advances in the theory of the multiple end solutions. We begin with the description of the moduli space of such solutions. Next we move on to study a special class of this solutions with just four ends. A special example is the saddle solutions $$U$$ whose nodal lines are precisely the straight lines $$y=\pm x$$. We describe completely connected components of the moduli space of four end solutions. Finally we establish a uniqueness result which gives a complete classification of these solutions. It says that all four end solutions are continuous deformations of the saddle solution. An integral equation method for the Helmholtz problem in the presence of small anisotropic inclusions https://zbmath.org/1485.35130 2022-06-24T15:10:38.853281Z "Lihiou, Houssem" https://zbmath.org/authors/?q=ai:lihiou.houssem "Khelifi, Abdessatar" https://zbmath.org/authors/?q=ai:khelifi.abdessatar Summary: We consider the Helmholtz problem with source term in an anisotropic domain of $$\mathbb{R}^3$$. The aim of this paper is to investigate the interplay between the geometry and analysis of elliptic equations under small perturbation of domain. The solving of this problem, anisotropic as well as isotropic case, is based on integral equations. We exhibit the Lippmann-Schwinger integral equation in the presence of finite number of anisotropic inclusions of small diameter. We derive some results for convergence estimates. An intermediate local-nonlocal eigenvalue elliptic problem https://zbmath.org/1485.35154 2022-06-24T15:10:38.853281Z "Delgado, Manuel" https://zbmath.org/authors/?q=ai:delgado.manuel.1 "Santos Júnior, Joao R." https://zbmath.org/authors/?q=ai:santos.joao-r-jun "Suárez, Antonio" https://zbmath.org/authors/?q=ai:suarez.antonio A posteriori verification of the positivity of solutions to elliptic boundary value problems https://zbmath.org/1485.35165 2022-06-24T15:10:38.853281Z "Tanaka, Kazuaki" https://zbmath.org/authors/?q=ai:tanaka.kazuaki "Asai, Taisei" https://zbmath.org/authors/?q=ai:asai.taisei Summary: The purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions of elliptic boundary value problems by assuming neither $$H^2$$-regularity nor $$L^{\infty }$$-error estimation, but only $$H^1_0$$-error estimation. In [the first author, J. Comput. Appl. Math. 370, Article ID 112647, 10 p. (2020; Zbl 1437.35397)], we proposed two approaches to verify the positivity of solutions of several semilinear elliptic boundary value problems. However, some cases require $$L^{\infty }$$-error estimation and, therefore, narrow applicability. In this paper, we extend one of the approaches and combine it with a priori error bounds for Laplacian eigenvalues to obtain a unified method that has wide application. We describe how to evaluate some constants required to verify the positivity of desired solutions. We apply our method to several problems, including those to which the previous method is not applicable. Retraction note: Boundary value behaviors for solutions of the equilibrium equations with angular velocity'' https://zbmath.org/1485.35172 2022-06-24T15:10:38.853281Z "Wang, Jiaofeng" https://zbmath.org/authors/?q=ai:wang.jiaofeng "Pu, Jun" https://zbmath.org/authors/?q=ai:pu.jun "Huang, Bin" https://zbmath.org/authors/?q=ai:huang.bin "Shi, Guojian" https://zbmath.org/authors/?q=ai:shi.guojian From the text: The Editors-in-Chief have retracted the article [ibid. 2015, Paper No. 230, 8 p. (2015; Zbl 1382.35239)] by \textit{J. Wang} et al. because it significantly overlaps with an article from other authors that was simultaneously under consideration at another journal [\textit{J. Shi} and \textit{Y. Liao}, J. Inequal. Appl. 2015, Paper No. 363, 8 p. (2015; Zbl 1350.35084)]. Additionally, the article also shows evidence of authorship and peer review manipulation. The authors have not responded to any correspondence regarding this retraction. Singular quasilinear elliptic systems with gradient dependence https://zbmath.org/1485.35183 2022-06-24T15:10:38.853281Z "Dellouche, Halima" https://zbmath.org/authors/?q=ai:dellouche.halima "Moussaoui, Abdelkrim" https://zbmath.org/authors/?q=ai:moussaoui.abdelkrim Summary: In this paper, we prove existence and regularity of positive solutions for singular quasilinear elliptic systems involving gradient terms. Our approach is based on comparison properties, a priori estimates and Schauder's fixed point theorem. Solutions to coupled critical elliptic systems involving attractive Hardy-type terms https://zbmath.org/1485.35184 2022-06-24T15:10:38.853281Z "Deng, Qian" https://zbmath.org/authors/?q=ai:deng.qian "Kang, Dongsheng" https://zbmath.org/authors/?q=ai:kang.dongsheng "Wu, Huimin" https://zbmath.org/authors/?q=ai:wu.huimin Summary: In this paper, a system of elliptic equations is studied, which involves coupled critical nonlinearities and attractive Hardy terms. By the local compactness analysis of $$(PS)_c$$ sequence and estimates on extremal functions corresponding to the best Sobolev constant, the existence of Mountain-Pass solutions is proved. Secondly, the asymptotic properties at the origin of solutions to the system are proved by the analytic technics and variational arguments. Gradient asymptotics of solutions to the Lamé systems in the presence of two nearly touching $$C^{1, \gamma }$$-inclusions https://zbmath.org/1485.35185 2022-06-24T15:10:38.853281Z "Hao, Xia" https://zbmath.org/authors/?q=ai:hao.xia "Zhao, Zhiwen" https://zbmath.org/authors/?q=ai:zhao.zhiwen Summary: In this paper, we establish the asymptotic expressions for the gradient of a solution to the Lamé systems with partially infinity coefficients as two rigid $$C^{1 , \gamma}$$-inclusions are very close but not touching. The novelty of these asymptotics lies in showing the optimality of the gradient blow-up rate in dimensions greater than two. Distributional solutions of anisotropic nonlinear elliptic systems with variable exponents: existence and regularity https://zbmath.org/1485.35186 2022-06-24T15:10:38.853281Z "Mokhtar, Naceri" https://zbmath.org/authors/?q=ai:mokhtar.naceri "Benboubker, Mohamed Badr" https://zbmath.org/authors/?q=ai:benboubker.mohamed-badr Summary: In this paper, we prove existence and maximal regularity for distributional solutions of anisotropic nonlinear elliptic systems with variable exponents where the right-hand side $$f$$ is in $$\big (\mathring{W}^{1,\overrightarrow{p}(\cdot)}(\Omega; {\mathbb{R}}^m)\big)^\ast$$ which is the dual space of the anisotropic Sobolev space $$\mathring{W}^{1,\overrightarrow{p}(\cdot)}(\Omega; {\mathbb{R}}^m)$$, and later the case where $$f$$ is in $$L^{q(x)}(\Omega; {\mathbb{R}}^m)$$, $$q(\cdot):{\overline{\Omega}}\rightarrow (1,+\infty)$$. The functional setting involves anisotropic Sobolev spaces with variable exponents as well as weak Lebesgue (Marcinkiewicz) spaces with variable exponents. Existence of least energy positive solutions to critical Schrödinger systems in $$\mathbb{R}^3$$ https://zbmath.org/1485.35187 2022-06-24T15:10:38.853281Z "You, Song" https://zbmath.org/authors/?q=ai:you.song "Zou, Wenming" https://zbmath.org/authors/?q=ai:zou.wenming Summary: In this paper, we consider the following coupled Schrödinger system with critical exponent: $\begin{cases} - \Delta u + \lambda_1 u = \mu_1 | u |^4 u + \beta | v |^3 | u | u \text{ in } \Omega, \\ - \Delta v + \lambda_2 v = \mu_2 | v |^4 v + \beta | u |^3 | v | v \text{ in } \Omega, \\ u = v = 0 \text{ on } \partial \Omega. \end{cases}$ Here, $$\Omega \subset \mathbb{R}^3$$ is a smooth bounded domain, $$- \lambda_1 ( \Omega ) < \lambda_1$$ , $$\lambda_2 < - \lambda^\ast ( \Omega )$$, $$\mu_1 , \mu_2 > 0$$, and $$\beta > 0$$, where $$\lambda_1 ( \Omega )$$ is the first eigenvalue of $$- \Delta$$ with Dirichlet boundary condition and $$\lambda^\ast ( \Omega ) \in ( 0 , \lambda_1 ( \Omega ) )$$. By a variational method, we establish the existence of least energy positive solutions for positive small $$\beta$$. Global regularity results for non-homogeneous growth fractional problems https://zbmath.org/1485.35192 2022-06-24T15:10:38.853281Z "Giacomoni, Jacques" https://zbmath.org/authors/?q=ai:giacomoni.jacques "Kumar, Deepak" https://zbmath.org/authors/?q=ai:kumar.deepak "Sreenadh, Konijeti" https://zbmath.org/authors/?q=ai:sreenadh.konijeti Summary: The main goal of this article is to show the global Hölder regularity of weak solutions to a class of problems involving the fractional $$(p, q)$$-Laplacian, denoted by $$(-\Delta)^{s_1}_p +(-\Delta)^{s_2}_q$$, for $$1<p,q<\infty$$ and $$s_1,s_2\in (0,1)$$. We use a suitable Caccioppoli inequality and a local boundedness result in order to prove the weak Harnack inequality. Consequently, by employing a suitable iteration process, we establish the interior Hölder continuity result for local weak solutions. The global Hölder regularity result we prove expands and improves the regularity results of \textit{J. Giacomoni, D. Kumar} and \textit{K. Sreenadh} [Interior and boundary regularity results for strongly nonhomogeneous $$p$$, $$q$$-fractional problems'', Adv. Calc. Var. (to appear, \url{arXiv: 2102.06080}] to the subquadratic case (that is, $$q<2)$$ and to the more general right-hand side, which requires a different and new approach. Moreover, we establish a non-local Harnack-type inequality for weak solutions and a strong maximum principle for weak super-solutions, which are of independent interest. On the number of concentrating solutions of a fractional Schrödinger-Poisson system with doubly critical growth https://zbmath.org/1485.35194 2022-06-24T15:10:38.853281Z "Qu, Siqi" https://zbmath.org/authors/?q=ai:qu.siqi "He, Xiaoming" https://zbmath.org/authors/?q=ai:he.xiaoming.1|he.xiaoming Summary: In this paper we study the existence, multiplicity and concentration of positive solutions for the fractional Schrödinger-Poisson system with doubly critical growth $\begin{cases} \varepsilon^{2s}(-\Delta)^s u+V( x)u= f(u)+\phi |u|^{2^*_s-3}u+|u|^{2^*_s-2}u,&x \in\mathbb{R}^3,\\ \varepsilon^{2s}(-\Delta)^s\phi =|u|^{2^*_s-1},& x \in\mathbb{R}^3, \end{cases}$ where $$s \in (\frac{3}{4},1)$$, $$\varepsilon$$ is a positive parameter, $$2^*_s = \frac{6}{3-2s}$$ is the fractional critical Sobolev exponent, $$(-\Delta)^s$$ is the fractional Laplacian operator, and $$f$$ is a continuous nonlinearity with subcritical growth. With the help of Nehari manifold and Ljusternik-Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum value for small values of the parameter $$\varepsilon$$. Lipschitz estimates for partial trace operators with extremal Hessian eigenvalues https://zbmath.org/1485.35195 2022-06-24T15:10:38.853281Z "Vitolo, Antonio" https://zbmath.org/authors/?q=ai:vitolo.antonio Summary: We consider the Dirichlet problem for partial trace operators which include the smallest and the largest eigenvalue of the Hessian matrix. It is related to two-player zero-sum differential games. No Lipschitz regularity result is known for the solutions, to our knowledge. If some eigenvalue is missing, such operators are nonlinear, degenerate, non-uniformly elliptic, neither convex nor concave. Here we prove an interior Lipschitz estimate under a non-standard assumption: that the solution exists in a larger, unbounded domain, and vanishes at infinity. In other words, we need a condition coming from far away. We also provide existence results showing that this condition is satisfied for a large class of solutions. On the occasion, we also extend a few qualitative properties of solutions, known for uniformly elliptic operators, to partial trace operators. Stable weak solutions to weighted Kirchhoff equations of Lane-Emden type https://zbmath.org/1485.35196 2022-06-24T15:10:38.853281Z "Wei, Yunfeng" https://zbmath.org/authors/?q=ai:wei.yunfeng "Yang, Hongwei" https://zbmath.org/authors/?q=ai:yang.hongwei "Yu, Hongwang" https://zbmath.org/authors/?q=ai:yu.hongwang Summary: This paper is concerned with the Liouville type theorem for stable weak solutions to the following weighted Kirchhoff equations: \begin{aligned}& -M \biggl(\int_{\mathbb{R}^N}\xi(z) \vert \nabla_Gu \vert^2\,dz \biggr){ \operatorname{div}}_G \bigl(\xi(z) \nabla_Gu \bigr) \\ &\quad=\eta(z) \vert u \vert^{p-1}u,\quad z=(x,y) \in \mathbb{R}^N = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}, \end{aligned} where $$M(t)=a+bt^k$$, $$t\geq0$$, with $$a,b,k\geq 0$$, $$a+b>0$$, $$k=0$$ if and only if $$b=0$$. Let $$N=N_1+N_2\geq 2$$, $$p>1+2k$$ and $$\xi(z),\eta(z)\in L^1_{\mathrm{loc}}(\mathbb{R}^N)\setminus\{ 0\}$$ be nonnegative functions such that $$\xi(z)\leq C\|z\|_G^{\theta}$$ and $$\eta(z)\geq C'\|z\|_G^d$$ for large $$\|z\|_G$$ with $$d>\theta-2$$. Here $$\alpha\geq0$$ and $$\|z\|_G=(|x|^{2(1+\alpha)}+|y|^2)^{\frac{1}{2(1+\alpha)}}. \operatorname{div}_G$$ (resp., $$\nabla_G)$$ is Grushin divergence (resp., Grushin gradient). Under some appropriate assumptions on $$k$$, $$\theta$$, $$d$$, and $$N_{\alpha}=N_1+(1+\alpha)N_2$$, the nonexistence of stable weak solutions to the problem is obtained. A distinguished feature of this paper is that the Kirchhoff function $$M$$ could be zero, which implies that the above problem is degenerate. Dirichlet problem for complex Hessian equations on $$k$$-pseudoconvex domains in $$\mathbb{C}^n$$ https://zbmath.org/1485.35198 2022-06-24T15:10:38.853281Z "Zhang, Qiqi" https://zbmath.org/authors/?q=ai:zhang.qiqi The interior gradient estimate for a class of mixed Hessian curvature equations https://zbmath.org/1485.35199 2022-06-24T15:10:38.853281Z "Zhou, Jundong" https://zbmath.org/authors/?q=ai:zhou.jundong Summary: In this paper, we are concerned with a class of mixed Hessian curvature equations with non-degeneration. By using the maximum principle and constructing an auxiliary function, we obtain the interior gradient estimate of $$(k-1)$$-admissible solutions. Asymptotic profile of ground states for the Schrödinger-Poisson-Slater equation https://zbmath.org/1485.35204 2022-06-24T15:10:38.853281Z "Liu, Zeng" https://zbmath.org/authors/?q=ai:liu.zeng "Moroz, Vitaly" https://zbmath.org/authors/?q=ai:moroz.vitaly Summary: We study the Schrödinger-Poisson-Slater equation $- \Delta u + u + \lambda ( I_2 \ast | u |^2 ) u = | u |^{p - 2} u \text{ in } \mathbb{R}^3,$ where $$p \in ( 3 , 6 )$$ and $$\lambda > 0$$. By using direct variational analysis based on the comparison of the ground state energy levels, we obtain a characterization of the limit profile of the positive ground states for $$\lambda \to \infty$$. Multiple entire solutions of fractional Laplacian Schrödinger equations https://zbmath.org/1485.35206 2022-06-24T15:10:38.853281Z "Wang, Jian" https://zbmath.org/authors/?q=ai:wang.jian.5|wang.jian.1|wang.jian.3|wang.jian.4|wang.jian.2|wang.jian.7|wang.jian.9 "Du, Zhuoran" https://zbmath.org/authors/?q=ai:du.zhuoran Summary: We consider the semi-linear fractional Schrödinger equation $$\begin{cases} (-\Delta)^su+V(x)u=f(x, u),&x\in\mathbb R^N,\\ u\in H^s(\mathbb R^N), \end{cases}$$ where both $V(x)$ and $f(x,u)$ are periodic in $x$, 0 belongs to a spectral gap of the operator $(-\Delta)^s+V$ and $f(x,u)$ is subcritical in $u$. We obtain the existence of nontrivial solutions by using a generalized linking theorem, and based on this existence we further establish infinitely many geometrically distinct solutions. We weaken the super-quadratic condition of $f$, which is usually assumed even in the standard Laplacian case so as to obtain the existence of solutions. On a critical exponential $$p \& N$$ equation type: existence and concentration of changing solutions https://zbmath.org/1485.35212 2022-06-24T15:10:38.853281Z "Costa, Gustavo S." https://zbmath.org/authors/?q=ai:costa.gustavo-s "Figueiredo, Giovany M." https://zbmath.org/authors/?q=ai:figueiredo.giovany-malcher Summary: In this paper we study a class of quasilinear equation with exponential critical growth. More precisely, we show existence of a family of nodal solutions, i.e, sign-changing solutions for the problem $\begin{cases} -\operatorname{div} \big(a (\epsilon^p |\nabla u|^p) \, \epsilon^p |\nabla u|^{p-2} \nabla u \big) \, +\, V(z) b(|u|^p) |u|^{p-2} u = f(u) \; \quad \text{in } \mathbb{R}^N, \\ u \in W^{1,p}(\mathbb{R}^N) \cap W^{1,N}(\mathbb{R}^N). \end{cases} \tag{$$P_\epsilon$$}$ Such nodal solutions concentrate on the minimum points set of the potential $$V$$, changes sign exactly once in $${\mathbb{R}}^N$$ and have exponential decay at infinity. Here we use variational methods and \textit{M. A. del Pino} and \textit{P. L. Felmer}'s technique [Calc. Var. Partial Differ. Equ. 4, No. 2, 121--137 (1996; Zbl 0844.35032)] in order to overcome the lack of compactness. A quasilinear transmission problem with application to Maxwell equations with a divergence-free $$\mathcal{D}$$-field https://zbmath.org/1485.35213 2022-06-24T15:10:38.853281Z "Dohnal, Tomáš" https://zbmath.org/authors/?q=ai:dohnal.tomas "Romani, Giulio" https://zbmath.org/authors/?q=ai:romani.giulio "Tietz, Daniel P." https://zbmath.org/authors/?q=ai:tietz.daniel-p Summary: Maxwell equations in the absence of free charges require initial data with a divergence-free displacement field $$\mathcal{D}$$. In materials in which the dependence $$\mathcal{D} = \mathcal{D}(\mathcal{E})$$ is nonlinear the quasilinear problem $${\nabla} \cdot \mathcal{D}(\mathcal{E}) = 0$$ is hence to be solved. In many applications, e.g. in the modelling of wave packets, an approximative asymptotic ansatz of the electric field $$\mathcal{E}$$ is used, which satisfies this divergence condition at $$t = 0$$ only up to a small residual. We search then for a small correction of the ansatz to enforce $${\nabla} \cdot \mathcal{D}(\mathcal{E}) = 0$$ at $$t = 0$$ and choose this correction in the form of a gradient field. In the usual case of a power type nonlinearity in $$\mathcal{D}(\mathcal{E})$$ this leads to the sum of the Laplace and $$p$$-Laplace operators. We also allow for the medium to consist of two different materials so that a transmission problem across an interface is produced. We prove the existence of the correction term for a general class of nonlinearities and provide regularity estimates for its derivatives, independent of the $$L^2$$-norm of the original ansatz. In this way, when applied to the wave packet setting, the correction term is indeed asymptotically smaller than the original ansatz. We also provide numerical experiments to support our analysis. Existence of multiple positive solutions for a $$p$$-Kirchhoff problem with singular nonlinearity https://zbmath.org/1485.35217 2022-06-24T15:10:38.853281Z "Li, Yuanxiao" https://zbmath.org/authors/?q=ai:li.yuanxiao Summary: In this paper, we study the following $$p$$-Kirchhoff-type elliptic problem: $\begin{cases} -M(\int_\Omega |\nabla u|^p\mathrm{d}x)\Delta_pu=\lambda \frac{f(x)}{u^{\gamma}}+g(x)\frac{u^{q-1}}{|x|^s}& \text{in }\Omega,\\ u=0& \text{on } \partial\Omega, \end{cases}$ where $$\lambda$$ and $$\gamma$$ are positive constants. Under some appropriate conditions on the singularity and the coefficients, the existence and multiplicity of positive solutions to this problem are obtained. Our approaches are based on the variational method and perturbation method. Liouville type theorem for stable solutions to weighted quasilinear problems in $$\mathbb{R}^N$$ https://zbmath.org/1485.35220 2022-06-24T15:10:38.853281Z "Sun, Fanrong" https://zbmath.org/authors/?q=ai:sun.fanrong Summary: In this paper, we prove the Liouville type theorem for stable $$W_{\text{loc}}^{1, p}$$ solutions of the weighted quasilinear problem $-\operatorname{div}\left(w_1(x)\left (s^2+|\nabla u|^2 \right)^{\frac{{p - 2}}{2}}\nabla u\right )=w_2(x)f(u)\quad \text{ in } \mathbb{R}^N,$ where $$s \geq 0$$ is a real number, $$f(u)$$ is either $$e^u$$ or $$-e^{\frac{1}{u}}$$ and $$w_1 (x),w_2 (x) \in L_{\text{loc}}^1 \left (\mathbb{R}^N \right )$$ be nonnegative functions so that $$w_1 (x) \leq C_1|x|^m$$ and $$w_2 (x) \geq C_2|x|^n$$ when $$|x|$$ is big enough. Here we need $$n>m$$. Existence of a positive solution for a class of Choquard equation with upper critical exponent https://zbmath.org/1485.35239 2022-06-24T15:10:38.853281Z "Pan, Hui-Lan" https://zbmath.org/authors/?q=ai:pan.hui-lan "Liu, Jiu" https://zbmath.org/authors/?q=ai:liu.jiu "Tang, Chun-Lei" https://zbmath.org/authors/?q=ai:tang.chunlei|tang.chun-lei Summary: In this paper, we investigate the existence of nontrivial solution for the following class of Choquard equation $-\Delta u+u=(I_\alpha \ast |u|^p)|u|^{p-2} u+\lambda|u|^{q-2}u \text{ in } \mathbb{R}^N, \tag{1}$ where $$N\in \mathbb{N}$$, $$N\geq 3$$, $$\alpha \in (0,N)$$, $$I_\alpha$$ is a Riesz potential, $$\lambda >0$$ is a parameter, $$p=\frac{N+\alpha }{N-2}$$ is the upper Hardy-Littlewood-Sobolev critical exponent and $$q\in (2,\frac{2N}{N-2}).$$ We prove that there exists $$\lambda_0>0$$ such that for $$\lambda \geq \lambda_0$$, problem (1) possesses one positive radial solution. Monotonicity and symmetry of positive solutions to degenerate quasilinear elliptic systems in half-spaces and strips https://zbmath.org/1485.35247 2022-06-24T15:10:38.853281Z "Le, Phuong" https://zbmath.org/authors/?q=ai:le.phuong-m|le.phuong-quynh "Vo, Hoang-Hung" https://zbmath.org/authors/?q=ai:vo.hoang-hung Summary: By means of the method of moving planes, we study the monotonicity of positive solutions to degenerate quasilinear elliptic systems in half-spaces. We also prove the symmetry of positive solutions to the systems in strips by using similar arguments. Our work extends the main results obtained in [\textit{A. Farina} et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 17, No. 4, 1207--1229 (2017; Zbl 1391.35164); \textit{L. Montoro} et al., Adv. Differ. Equ. 20, No. 7-8, 717--740 (2015; Zbl 1325.35066)] to the system, in which substantial differences with the single cases are presented. Positive solutions for double phase problems with combined nonlinearities https://zbmath.org/1485.35249 2022-06-24T15:10:38.853281Z "Liu, Zhenhai" https://zbmath.org/authors/?q=ai:liu.zhenhai "Papageorgiou, Nikolaos S." https://zbmath.org/authors/?q=ai:papageorgiou.nikolaos-s Let $$\Omega \subseteq \mathbb{R}^N$$ be a bounded domain with a Lipschitz boundary $$\partial \Omega$$. The authors consider a weighted Dirichlet concave-convex problem of the form $\begin{cases} -\Delta_{p}^{a}u(z)- \Delta_{q}u(z)=E(z) u(z)^{\tau-1}+\lambda u(z)^{r-1} \mbox{ in } \Omega,\\ u|_{\partial \Omega}=0, \quad 1<\tau <q<p<r, \quad \lambda >0, \quad u \geq 0, \end{cases}\tag{$$P_\lambda$$}$ where $$\Delta_p^a u= \operatorname{div}(a(z)|D u|^{p-2}D u)$$ for all $$u \in W^{1,p}_0(\Omega)$$ is the weighted $$p$$-Laplace operator, with $$a \in C^{0,1}(\overline{\Omega})$$ and $$a(z)\geq 0$$ for all $$z \in \overline{\Omega}$$, $$a \not \equiv 0$$. For $$a=1$$, we retrieve the classical $$p$$-Laplace operator. In the right-hand side, the first (concave) term is the $$(p-1)$$-sublinear function $$x \to E(z)x^{\tau-1}$$, $$x \geq 0$$, with $$E \in L^\infty(\Omega)\setminus\{0\}$$, $$E(z) \geq 0$$ for a.a. $$z \in \Omega$$ and the second (convex) term is the parametric $$(p-1)$$-superlinear function $$x \to \lambda x^{r-1}$$, $$x \geq 0$$. Using the Nehari method, the authors prove that for all small values of $$\lambda >0$$, the problem $$(P_\lambda)$$ admits at least two positive bounded solutions. Reviewer: Calogero Vetro (Palermo) Note on an elementary inequality and its application to the regularity of $$p$$-harmonic functions https://zbmath.org/1485.35250 2022-06-24T15:10:38.853281Z "Sarsa, Saara" https://zbmath.org/authors/?q=ai:sarsa.saara The authors study the Sobolev regularity of $$p$$-harmonic functions and show that $$|Du|^{\frac{p-2+s}{2}} Du$$ belongs to the Sobolev space $$W^{1,2}_{\mathrm{loc}},$$ for $$s> -1-\frac{p-1}{n-1}$$ and any $$p$$-harmonic function $$u.$$ The bound of $$s$$ satisfying the inequality $s > 2-\min\left\{p +\frac{n}{n-1}, 3 +\frac{p-1}{n-1}\right\},$ from the $$W^{1,2}_{\mathrm{loc}}$$-regularity of $$V_s(Du)$$ obtained by \textit{H. Dong} et al. [Adv. Math. 370, Article ID 107212, 39 p. (2020; Zbl 1442.35195)], has been improved to $$s> -1-\frac{p-1}{n-1}$$. The proof is based on an elementary inequality. Reviewer: Xiaoming He (Beijing) Global regularity for a class of Monge-Ampère type equations https://zbmath.org/1485.35254 2022-06-24T15:10:38.853281Z "Li, Mengni" https://zbmath.org/authors/?q=ai:li.mengni "Li, You" https://zbmath.org/authors/?q=ai:li.you Summary: In this paper we are concerned with the global regularity of solutions to the Dirichlet problem for a class of Monge-Ampère type equations. By employing the concept of $$(a, \eta)$$ type domain, we emphasize that the boundary regularity depends on the convexity of the domain in nature. The key idea of our proof is to provide more effective global Hölder estimates of convex solutions to the problem based on carefully choosing auxiliary functions and constructing sub-solutions. On a generalized diffusion problem: a complex network approach https://zbmath.org/1485.35258 2022-06-24T15:10:38.853281Z "Cantin, Guillaume" https://zbmath.org/authors/?q=ai:cantin.guillaume "Thorel, Alexandre" https://zbmath.org/authors/?q=ai:thorel.alexandre Summary: In this paper, we propose a new approach for studying a generalized diffusion problem, using complex networks of reaction-diffusion equations. We model the biharmonic operator by a network, based on a finite graph, in which the couplings between nodes are linear. To this end, we study the generalized diffusion problem, establishing results of existence, uniqueness and maximal regularity of the solution \textit{via} operator sums theory and analytic semigroups techniques. We then solve the complex network problem and present sufficient conditions for the solutions of both problems to converge to each other. Finally, we analyze their asymptotic behavior by establishing the existence of a family of exponential attractors. Pullback attractors of nonautonomous discrete $$p$$-Laplacian complex Ginzburg-Landau equations with fast-varying delays https://zbmath.org/1485.35268 2022-06-24T15:10:38.853281Z "Pu, Xiaoqin" https://zbmath.org/authors/?q=ai:pu.xiaoqin "Wang, Xuemin" https://zbmath.org/authors/?q=ai:wang.xuemin "Li, Dingshi" https://zbmath.org/authors/?q=ai:li.dingshi Summary: In this paper, we consider a class of nonautonomous discrete $$p$$-Laplacian complex Ginzburg-Landau equations with time-varying delays. We prove the existence and uniqueness of pullback attractor for these equations. The existing results of studying attractors for time-varying delay equations require that the derivative of the delay term should be less than 1 (called slow-varying delay). By using differential inequality technique, our results remove the constraints on the delay derivative. So, we can deal with the equations with fast-varying delays (without any constraints on the delay derivative). Entire solutions for a delayed lattice competitive system https://zbmath.org/1485.35269 2022-06-24T15:10:38.853281Z "Yan, Rui" https://zbmath.org/authors/?q=ai:yan.rui "Wang, Yang" https://zbmath.org/authors/?q=ai:wang.yang "Yao, Meiping" https://zbmath.org/authors/?q=ai:yao.meiping Summary: In this paper, we investigate the existence of entire solutions for a delayed lattice competitive system. Here the entire solutions are the solutions that exist for all $$(n,t)\in \mathbb{Z}\times \mathbb{R}$$. In order to prove the existence, we firstly embed the delayed lattice system into the corresponding larger system, of which the traveling front solutions are identical to those of the delayed lattice system. Then based on the comparison theorem and the sup-sub solutions method, we construct entire solutions which behave as two opposite traveling front solutions moving towards each other from both sides of $$x$$-axis and then annihilating. Moreover, our conclusions extend the invading way, which the superior species invade the inferior ones from both sides of $$x$$-axis and then the inferior ones extinct, into the lattice and delay case. A matrix Harnack estimate for a Kolmogorov type equation https://zbmath.org/1485.35276 2022-06-24T15:10:38.853281Z "Jiang, Feida" https://zbmath.org/authors/?q=ai:jiang.feida "Shen, Xinyi" https://zbmath.org/authors/?q=ai:shen.xinyi The authors study the properties of the solutions $$f$$ of equations of Kolmogorov type that are generalizations of the ultra-parabolic equation $$f_t=f_{xx}-xf_y$$ to more dimensions. More precisely: $f_t=\sum_{i=1}^n\alpha_if_{x_ix_i}+\sum_{i=1}^k\beta_ix_i\left(\sum_{j=1}^{m_i} f_{x_{i_j}}\right) \quad \text{on} \; \mathbb R^N\times (0,T),$ where $$\alpha_i>0$$ and $$\beta_i\neq 0$$; $$1 \le k \le n$$, $$\sum_{i=1}^km_i=N-n$$ and $$i_j=n+m_1+\dots +m_{i-1}+j$$. They obtain a matrix Harnack inequality via nontrivial technical calculations. From that, by tracing or integrating on suitable paths, they deduce a number of scalar Harnack inequalities. For the entire collection see [Zbl 1455.58001]. Reviewer: Antonio Vitolo (Fisciano) Staffans-Weiss perturbations for maximal $$L^p$$-regularity in Banach spaces https://zbmath.org/1485.35277 2022-06-24T15:10:38.853281Z "Amansag, Ahmed" https://zbmath.org/authors/?q=ai:amansag.ahmed "Bounit, Hamid" https://zbmath.org/authors/?q=ai:bounit.hamid "Driouich, Abderrahim" https://zbmath.org/authors/?q=ai:driouich.abderrahim "Hadd, Said" https://zbmath.org/authors/?q=ai:hadd.said Summary: In this paper, we show that the concept of maximal $$L^p$$-regularity is stable under a large class of unbounded perturbations, namely Staffans-Weiss perturbations. To that purpose, we first prove that the analyticity of semigroups is preserved under this class of perturbations, which is a necessary condition for the maximal regularity. In UMD spaces, $${\mathcal{R}}$$-boundedness is exploited to give conditions guaranteeing the maximal regularity. For Banach spaces, a condition is imposed to prove maximal regularity. Moreover, we apply the obtained results to perturbed boundary value problems. Decay estimate and non-extinction of solutions of $$p$$-Laplacian nonlocal heat equations https://zbmath.org/1485.35280 2022-06-24T15:10:38.853281Z "Toualbia, Sarra" https://zbmath.org/authors/?q=ai:toualbia.sarra "Zaraï, Abderrahmane" https://zbmath.org/authors/?q=ai:zarai.abderrahmane "Boulaaras, Salah" https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud Summary: The main goal of this work is to study the initial boundary value problem of a nonlocal heat equations with logarithmic nonlinearity in a bounded domain. By using the logarithmic Sobolev inequality and potential wells method, we obtain the decay, blow-up and non-extinction of solutions under some conditions, and the results extend the results of a recent paper [\textit{L. Yan} and \textit{Z. Yang}, Bound. Value Probl. 2018, Paper No. 121, 11 p. (2018; Zbl 07509577)]. Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source https://zbmath.org/1485.35281 2022-06-24T15:10:38.853281Z "Park, Sun-Hye" https://zbmath.org/authors/?q=ai:park.sunhye Summary: In this paper, we study the wave equation with frictional damping, time delay in the velocity, and logarithmic source of the form $u_{tt}(x,t) - \Delta u (x,t) + \alpha u_t (x,t) + \beta u_t (x, t- \tau) = u(x,t) \ln \bigl\vert u(x,t) \bigr\vert^{\gamma}.$ There is much literature on wave equations with a polynomial nonlinear source, but not much on the equations with logarithmic source. We show the local and global existence of solutions using Faedo-Galerkin's method and the logarithmic Sobolev inequality. And then we investigate the decay rates and infinite time blow-up for the solutions through the potential well and perturbed energy methods. Local existence and lower bound of blow-up time to a Cauchy problem of a coupled nonlinear wave equations https://zbmath.org/1485.35283 2022-06-24T15:10:38.853281Z "Kafini, Mohammad" https://zbmath.org/authors/?q=ai:kafini.mohammad-mustafa "Al-Omari, Shadi" https://zbmath.org/authors/?q=ai:al-omari.shadi Summary: In this paper, we consider a Cauchy problem of a coupled linearly-damped wave equations with nonlinear sources. In the whole space, we establish the local existence and show that there are solutions with negative initial energy that blow up in a finite time. Moreover, under some conditions on the initial data, we estimate a lower bound of that time. Blow up of solutions of two singular nonlinear viscoelastic equations with general source and localized frictional damping terms https://zbmath.org/1485.35284 2022-06-24T15:10:38.853281Z "Boulaaras, Salah" https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud "Choucha, Abdelbaki" https://zbmath.org/authors/?q=ai:choucha.abdelbaki "Ouchenane, Djamel" https://zbmath.org/authors/?q=ai:ouchenane.djamel "Cherif, Bahri" https://zbmath.org/authors/?q=ai:cherif.bahri-belkacem Summary: This work studies the blow-up result of the solution of a coupled nonlocal singular viscoelastic equation with general source and localized frictional damping terms under some suitable conditions. This work is a natural continuation of the previous recent articles by the first author et al. [Appl. Anal. 99, 1--25 (2020; Zbl 1485.35298); Math. Methods Appl. Sci. 43, No. 10, 6140--6164 (2020; Zbl 1452.35094); Topol. Methods Nonlinear Anal. 56, No. 1, 283--312 (2020; Zbl 1465.35061)]. Growth of solutions with $$L^{2(p+2)}$$-norm for a coupled nonlinear viscoelastic Kirchhoff equation with degenerate damping terms https://zbmath.org/1485.35286 2022-06-24T15:10:38.853281Z "Choucha, Abdelbaki" https://zbmath.org/authors/?q=ai:choucha.abdelbaki "Hidan, Muajebah" https://zbmath.org/authors/?q=ai:hidan.muajebah "Cherif, Bahri" https://zbmath.org/authors/?q=ai:cherif.bahri-belkacem "Idris, Sahar Ahmed" https://zbmath.org/authors/?q=ai:idris.sahar-ahmed Summary: In this work, we consider a coupled nonlinear viscoelastic Kirchhoff equations with degenerate damping, dispersion and source terms. Under suitable hypothesis, we will prove that when the initial data are large enough (in the energy point of view), the energy grows exponentially and thus so the $$L^{2(p+2)}$$-norm. Scattering to a stationary solution for the superquintic radial wave equation outside an obstacle https://zbmath.org/1485.35295 2022-06-24T15:10:38.853281Z "Duyckaerts, Thomas" https://zbmath.org/authors/?q=ai:duyckaerts.thomas "Yang, Jianwei Urbain" https://zbmath.org/authors/?q=ai:yang.jianwei-urbain Summary: We consider the focusing wave equation outside a ball of $$\mathbb{R}^3$$, with Dirichlet boundary condition and a superquintic power nonlinearity. We classify all radial stationary solutions, and prove that all radial global solutions are asymptotically the sum of a stationary solution and a radiation term. Parabolic approximation of quasilinear wave equations with applications in nonlinear acoustics https://zbmath.org/1485.35296 2022-06-24T15:10:38.853281Z "Kaltenbacher, Barbara" https://zbmath.org/authors/?q=ai:kaltenbacher.barbara "Nikolić, Vanja" https://zbmath.org/authors/?q=ai:nikolic.vanja Global existence and decay for a system of two singular one-dimensional nonlinear viscoelastic equations with general source terms https://zbmath.org/1485.35298 2022-06-24T15:10:38.853281Z "Boulaaras, Salah" https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud "Guefaifia, Rafik" https://zbmath.org/authors/?q=ai:guefaifia.rafik "Mezouar, Nadia" https://zbmath.org/authors/?q=ai:mezouar.nadia Summary: The paper studies a system of two singular one-dimensional nonlinear equations that arise in generalized viscoelasticity with long-term memory, with general source terms and nonlocal boundary condition. We prove the existence of a global solution to the problem using the potential-well theory. Furthermore, we construct a Lyapunov functional and use it together with the perturbed energy method to prove a general decay result. Two inequalities for the first Robin eigenvalue of the Finsler Laplacian https://zbmath.org/1485.35303 2022-06-24T15:10:38.853281Z "di Blasio, Giuseppina" https://zbmath.org/authors/?q=ai:di-blasio.giuseppina "Gavitone, Nunzia" https://zbmath.org/authors/?q=ai:gavitone.nunzia Summary: Let $$\Omega\subset\mathbb{R}^n,n\ge 2,$$ be a bounded, connected, open set with Lipschitz boundary. Let $$F$$ be a suitable norm in $$\mathbb{R}^n$$ and let $$\Delta_Fu=\mathrm{div}\left(F_{\xi}(\nabla u)F(\nabla u)\right)$$ be the so-called Finsler Laplacian, with $$u\in H^1(\Omega)$$. In this paper, we prove two inequalities for $$\lambda_F(\beta,\Omega)$$, the first eigenvalue of $$\Delta_F$$ with Robin boundary conditions involving a positive function $$\beta (x)$$. As a consequence of our result, we obtain the asymptotic behavior of $$\lambda_F(\beta,\Omega)$$ when $$\beta$$ is a positive constant which goes to zero. Nonstandard Liouville tori and caustics in asymptotics in the form of Airy and Bessel functions for 2D standing coastal waves https://zbmath.org/1485.35307 2022-06-24T15:10:38.853281Z "Anikin, A. Yu." https://zbmath.org/authors/?q=ai:anikin.a-yu "Dobrokhotov, S. Yu." https://zbmath.org/authors/?q=ai:dobrokhotov.sergei-yu "Nazaikinskii, V. E." https://zbmath.org/authors/?q=ai:nazaikinskii.vladimir-e "Tsvetkova, A. V." https://zbmath.org/authors/?q=ai:tsvetkova.anna-v Summary: The spectral problem $$-\langle \nabla ,D(x)\nabla \psi \rangle = \lambda \psi$$ in a bounded two-dimensional domain $$\Omega$$ is considered, where $$D(x)$$ is a smooth function positive inside the domain and zero on the boundary whose gradient is different from zero on the boundary. This problem arises in the study of long waves trapped by the shore and by bottom irregularities. For its asymptotic solutions as $$\lambda \rightarrow \infty$$, explicit formulas are given when $$D(x)$$ has a special form that guarantees the complete integrability of the Hamiltonian system corresponding to the Hamiltonian $$H(x,p)=D(x)p^2$$. Since the problem is degenerate, the relevant Liouville tori are not in the standard phase space $$T^*\Omega$$, but in the extended'' phase space $$\boldsymbol{\Phi }\supset T^*\Omega$$, while their restrictions to $$T^*\Omega$$ are not compact and go to infinity'' with respect to momenta near the boundary of $$\Omega$$. As a result, nonstandard caustics emerge, formed by the boundary or its part, near which asymptotic eigenfunctions are expressed in terms of a Bessel function of composite argument. Standard caustics (within the domain) may also appear, which yield Airy functions in the asymptotics. Localization of eigenfunctions in a narrow Kirchhoff plate https://zbmath.org/1485.35309 2022-06-24T15:10:38.853281Z "Bakharev, F. L." https://zbmath.org/authors/?q=ai:bakharev.fedor-l "Matveenko, S. G." https://zbmath.org/authors/?q=ai:matveenko.sergei-g Summary: The asymptotics of eigenvalues and eigenfunctions of the Dirichlet problem for the biharmonic operator in a narrow two-dimensional domain (a thin Kirchhoff plate with rigidly clamped edges) as its width tends to zero is studied. The effect of localization of eigenfunctions is described, which consists in their exponential decay when removing away from the most wide plate region. Asymptotics of the spectrum of the mixed boundary value problem for the Laplace operator in a thin spindle-shaped domain https://zbmath.org/1485.35310 2022-06-24T15:10:38.853281Z "Nazarov, S. A." https://zbmath.org/authors/?q=ai:nazarov.sergei-aleksandrovich "Taskinen, J." https://zbmath.org/authors/?q=ai:taskinen.jari Summary: The asymptotics is examined for solutions to the spectral problem for the Laplace operator in a $$d$$-dimensional thin, of diameter $$O(h)$$, spindle-shaped domain $$\Omega^h$$ with the Dirichlet condition on small, of size $$h\ll 1$$, terminal zones $$\Gamma^h_\pm$$ and the Neumann condition on the remaining part of the boundary $$\partial \Omega^h$$. In the limit as $$h\rightarrow +0$$, an ordinary differential equation on the axis $$(-1,1)\ni z$$ of the spindle arises with a coefficient degenerating at the points $$z=\pm 1$$ and moreover, without any boundary condition because the requirement on the boundedness of eigenfunctions makes the limit spectral problem well-posed. Error estimates are derived for the one-dimensional model but in the case of $$d=3$$ it is necessary to construct boundary layers near the sets $$\Gamma^h_\pm$$ and in the case of $$d=2$$ it is necessary to deal with selfadjoint extensions of the differential operator. The extension parameters depend linearly on $$\ln h$$ so that its eigenvalues are analytic functions in the variable $$1/|\ln h|$$. As a result, in all dimensions the one-dimensional model gets the power-law accuracy $$O(h^{\delta_d})$$ with an exponent $$\delta_d>0$$. First (the smallest) eigenvalues, positive in $$\Omega^h$$ and null in $$(-1,1)$$, require individual treatment. Also, infinite asymptotic series are discussed, as well as the static problem (without the spectral parameter) and related shapes of thin domains. Semigroup theory for the Stokes operator with Navier boundary condition on $$L^p$$ spaces https://zbmath.org/1485.35315 2022-06-24T15:10:38.853281Z "Amrouche, Chérif" https://zbmath.org/authors/?q=ai:amrouche.cherif "Escobedo, Miguel" https://zbmath.org/authors/?q=ai:escobedo.miguel "Ghosh, Amrita" https://zbmath.org/authors/?q=ai:ghosh.amrita The authors consider the incompressible Navier-Stokes equations in a bounded domain with $$C^{1,1}$$ boundary, completed with slip boundary condition. They study the general semigroup theory for the Stokes operator with Navier boundary condition with (possibly) minimal regularity on $$\alpha$$ (the inverse of the slip length) which gives existence, uniqueness, and regularity of both strong and weak solutions. It is also shown that for $$\alpha$$ large, the weak and strong solutions of both the linear and nonlinear systems are bounded uniformly with respect to $$\alpha$$. For the entire collection see [Zbl 1467.76003]. Reviewer: Fatma Gamze Duzgun (Ankara) Navier-Stokes/Allen-Cahn system with generalized Navier boundary condition https://zbmath.org/1485.35318 2022-06-24T15:10:38.853281Z "Chen, Ya-zhou" https://zbmath.org/authors/?q=ai:chen.yazhou "He, Qiao-lin" https://zbmath.org/authors/?q=ai:he.qiaolin "Huang, Bin" https://zbmath.org/authors/?q=ai:huang.bin "Shi, Xiao-ding" https://zbmath.org/authors/?q=ai:shi.xiaoding A three-dimensional, two-phase compressible fluid flow with a generalized Navier boundary condition is studied using the principle of minimum energy dissipation. The existence of local-in-time strong solutions is proved, and the moving contact lines on the solid boundary are studied. Reviewer: Piotr Biler (Wrocław) Convergence of the pressure in the homogenization of the Stokes equations in randomly perforated domains https://zbmath.org/1485.35319 2022-06-24T15:10:38.853281Z "Giunti, Arianna" https://zbmath.org/authors/?q=ai:giunti.arianna "Höfer, Richard M." https://zbmath.org/authors/?q=ai:hofer.richard-m The authors prove a homogenization result, especially focusing on the pressure, for the steady incompressible Stokes equations $$-\Delta u_{\varepsilon }+\nabla p_{\varepsilon }=f$$, $$\nabla \cdot u_{\varepsilon }=0$$, posed in a domain $$D^{\varepsilon }$$, which is obtained by removing from a bounded set $$D\subset \mathbb{R}^{d}$$, $$d>2$$ a random number of small balls having random centers and radii: $$D^{\varepsilon }=D\setminus H^{\varepsilon }$$, with $$H^{\varepsilon }=\cup _{z_{i}\in \Phi \cap \frac{1}{ \varepsilon }D}B_{\varepsilon \frac{d}{d-2}\rho _{i}}(\varepsilon z_{i})$$, where $$\Phi$$ is a Poisson point process on $$\mathbb{R}^{d}$$ with homogeneous intensity rate $$\lambda >0$$, and the radii $$\{\rho _{i}\}_{z_{i}\in \Phi }\sqsubseteq \mathbb{R}_{+}$$ are identically and independently distributed unbounded random variables which satisfy $$\left\langle \rho ^{(d-2)+\beta }\right\rangle <+\infty$$ for some $$\beta >0$$, $$\left\langle \cdot \right\rangle$$ being the expectation under the probability measure on the radii $$\rho _{i}$$. The homogeneous Dirichlet boundary condition $$u_{\varepsilon }=0$$ is imposed in $$\partial D^{\varepsilon }$$. The source term $$f$$ is supposed to belong to $$H^{-1}(D; \mathbb{R}^{d})$$ and the above problem has a solution $$(u_{\varepsilon },p_{\varepsilon })$$ which belongs to $$H_{0}^{1}(D^{\varepsilon };\mathbb{R} ^{d})\times L_{0}^{2}(D^{\varepsilon };\mathbb{R})$$. The authors introduce the homogenized problem (Brinkman's equation): $$-\Delta u_{h}+\mu u_{h}+\nabla p_{h}=f$$, $$\nabla \cdot u_{h}=0$$, posed in $$D$$, where $$\mu =C_{d}\lambda \left\langle \rho ^{d-2}\right\rangle I$$, $$C_{d}>0$$ being a constant which only depends on the dimension $$d$$. The main result of the paper proves that for $$P$$-almost every $$\omega \in \Omega$$ there exists a family of sets $$E^{\varepsilon }\sqsubseteq \mathbb{R}^{d}$$ and a sequence $$r_{\varepsilon }\rightarrow 0$$ such that $$H^{\varepsilon }\sqsubseteq E^{\varepsilon }$$ and for $$\varepsilon \downarrow 0^{+}$$ $$Cap(E^{\varepsilon }\setminus H^{\varepsilon })\rightarrow 0$$, where $$Cap$$ denotes the harmonic capacity in $$\mathbb{R}^{d}$$. Further, the modification $$\widetilde{p} _{\varepsilon }$$ of the pressure defined as $$\widetilde{p}_{\varepsilon }=p_{\varepsilon }-\frac{1}{\left\vert D_{r_{\varepsilon }}\setminus E^{\varepsilon }\right\vert }\int_{D_{r_{\varepsilon }}\setminus E^{\varepsilon }}p_{\varepsilon }$$ in $$D_{r_{\varepsilon }}\setminus E^{\varepsilon }$$ and $$\widetilde{p}_{\varepsilon }=p_{\varepsilon }$$ in $$(D\setminus D_{r_{\varepsilon }})\cup E^{\varepsilon }$$, satisfies for all $$q<\frac{d}{d-1}$$ $$\widetilde{p}_{\varepsilon }\rightarrow p_{h}$$ in $$L_{0}^{q}(D;\mathbb{R})$$. Here $$D_{r}=\{x\in D:dist(x,\partial D)>r\}$$. The key tool for the proof is an estimate on the Bogovski operator in $$D\setminus E^{\varepsilon }$$. The authors also build appropriate test functions. Reviewer: Alain Brillard (Riedisheim) On Liouville-type theorems for the stationary MHD and the Hall-MHD systems in $$\mathbb{R}^3$$ https://zbmath.org/1485.35321 2022-06-24T15:10:38.853281Z "Chae, Dongho" https://zbmath.org/authors/?q=ai:chae.dongho "Kim, Junha" https://zbmath.org/authors/?q=ai:kim.junha "Wolf, Jörg" https://zbmath.org/authors/?q=ai:wolf.jorg This is an extension of results in [\textit{D. Chae} and \textit{J. Wolf}, Calc. Var. Partial Differ. Equ. 58, No. 3, Paper No. 111, 11 p. (2019; Zbl 1433.35218)] to the case of three-dimensional magnetohydrodynamics equations as well as a generalization of [\textit{D. Chae} and \textit{J. Wolf}, J. Differ. Equations 295, 233--248 (2021; Zbl 1470.35255)]. Under some growth conditions for the mean oscillations of potential functions, the corresponding stationary solutions are shown to be trivial. Reviewer: Piotr Biler (Wrocław) On global well-posedness to 3D Navier-Stokes-Landau-Lifshitz equations https://zbmath.org/1485.35322 2022-06-24T15:10:38.853281Z "Duan, Ning" https://zbmath.org/authors/?q=ai:duan.ning "Zhao, Xiaopeng" https://zbmath.org/authors/?q=ai:zhao.xiaopeng Summary: In this paper, we prove the global well-posedness of solutions for the Cauchy problem of three-dimensional incompressible Navier-Stokes-Landau-Lifshitz equations under the condition that $$\|u_0\|_{H^{\frac12}}+\|\nabla d_0\|_{H^{\frac12+\varepsilon}} (\varepsilon > 0)$$ is sufficiently small. This result can be seen as an improvement of the previous paper [\textit{R. Wei} et al., AMM, Appl. Math. Mech., Engl. Ed. 39, No. 10, 1499--1528 (2018; Zbl 1402.35230)]. Homogenization of a lubrication problem in oscillating domain by two-scale convergence method https://zbmath.org/1485.35324 2022-06-24T15:10:38.853281Z "Koroleva, Y. O." https://zbmath.org/authors/?q=ai:koroleva.yu-o "Korolev, A. V." https://zbmath.org/authors/?q=ai:korolev.alexey-v|korolev.aleksei-v Summary: In present paper we do a homogenization with respect to a small parameter of a boundary-value problem describing fluid flow between two moving in space and time rough surfaces. The two-scale convergence method was used to justify the behavior of the flow in the limit. A remark on the non-uniqueness in $$L^{\infty}$$ of the solutions to the two-dimensional Stokes problem in exterior domains https://zbmath.org/1485.35326 2022-06-24T15:10:38.853281Z "Maremonti, Paolo" https://zbmath.org/authors/?q=ai:maremonti.paolo The author deals with the two-dimensional Stokes initial boundary value problem in exterior domains, and the case of initial data $$u_{0}\in L^{\infty}\left( \Omega\right) ,$$ divergence free in weak sense. \ The author also investigates the property $$\left\Vert u\left( t\right) \right\Vert _{\infty}\leq c\left\Vert u_{0}\right\Vert _{\infty}$$ for all $$t>0$$ and $$c$$ independent of $$u$$, and non-uniqueness of the solutions. Reviewer: Fatma Hıra (Atakum) Statistical solutions for a nonautonomous modified Swift-Hohenberg equation https://zbmath.org/1485.35327 2022-06-24T15:10:38.853281Z "Wang, Jintao" https://zbmath.org/authors/?q=ai:wang.jintao "Zhang, Xiaoqian" https://zbmath.org/authors/?q=ai:zhang.xiaoqian "Zhao, Caidi" https://zbmath.org/authors/?q=ai:zhao.caidi A Swift-Hohenberg fourth order evolution problem for $$u_t+\Delta^2u+s\Delta u+\alpha u+b|\nabla u|^2+u^3=g(t,x)$$ is studied in bounded domains of $$\mathbb R^n$$, $$n\le 3$$. Equations of this type are used e.g. in hydrodynamics and some pattern formation models. Under some supplementary assumptions such as $$|b|<4$$, the existence of a unique pullback attractor, and then a family of invariant probability measures, and finally statistical solutions is proved. Reviewer: Piotr Biler (Wrocław) Exact dynamical behavior for a dual Kaup-Boussinesq system by symmetry reduction and coupled trial equations method https://zbmath.org/1485.35334 2022-06-24T15:10:38.853281Z "Li, Wen-He" https://zbmath.org/authors/?q=ai:li.wenhe "Wang, Yong" https://zbmath.org/authors/?q=ai:wang.yong.10|wang.yong.3|wang.yong.7|wang.yong.9|wang.yong.5|wang.yong.8|wang.yong.2|wang.yong|wang.yong.6|wang.yong.1|wang.yong.11 Summary: We propose a coupled trial equation method for a coupled differential equations system. Furthermore, according to the invariant property under the translation, we give the symmetry reduction of a dual Kaup-Boussinesq system, and then we use the proposed trial equation method to construct its exact solutions which describe its dynamical behavior. In particular, we get a cosine function solution with a constant propagation velocity, which shows an important periodic behavior of the system. Modelling and nonlinear boundary stabilization of the modified generalized Korteweg-de Vries-Burgers equation https://zbmath.org/1485.35340 2022-06-24T15:10:38.853281Z "Smaoui, N." https://zbmath.org/authors/?q=ai:smaoui.nejib "Chentouf, B." https://zbmath.org/authors/?q=ai:chentouf.boumediene "Alalabi, A." https://zbmath.org/authors/?q=ai:alalabi.ala Summary: In this paper, we study the modelling and nonlinear boundary stabilization problem of the modified generalized Korteweg-de Vries-Burgers equation (MGKdVB) when the spatial domain is $$[0,1]$$. First, the MGKdVB equation is derived using the long-wave approximation and perturbation method. Then, two nonlinear boundary controllers are proposed for this equation and the $$L^2$$-global exponential stability of the solution is shown. Numerical simulations are given to illustrate the efficiency of the developed control schemes. Orbital stability and asymptotic stability of mKdV breather-type soliton solutions https://zbmath.org/1485.35342 2022-06-24T15:10:38.853281Z "Wang, Juan" https://zbmath.org/authors/?q=ai:wang.juan "Tian, Lixin" https://zbmath.org/authors/?q=ai:tian.lixin "Zhang, Yingnan" https://zbmath.org/authors/?q=ai:zhang.yingnan.1 Summary: In this paper, we study the stability of modified Korteweg-de Vries equation breather. By using variable separation method, we obtain the exact breather-type soliton solutions. Moreover, this kind of solutions are globally stable in $$H^2$$ topology, and we describe a simple, mathematical proof of the orbital stability and asymptotic stability of breather-type soliton solutions under a class of small perturbation. Dynamic behaviors for a perturbed nonlinear Schrödinger equation with the power-law nonlinearity in a non-Kerr medium https://zbmath.org/1485.35343 2022-06-24T15:10:38.853281Z "Chai, Jun" https://zbmath.org/authors/?q=ai:chai.jun "Tian, Bo" https://zbmath.org/authors/?q=ai:tian.bo "Zhen, Hui-Ling" https://zbmath.org/authors/?q=ai:zhen.hui-ling "Sun, Wen-Rong" https://zbmath.org/authors/?q=ai:sun.wen-rong "Liu, De-Yin" https://zbmath.org/authors/?q=ai:liu.de-yin Summary: Effects of quantic nonlinearity on the propagation of the ultrashort optical pulses in a non-Kerr medium, like an optical fiber, can be described by a perturbed nonlinear Schrödinger equation with the power law nonlinearity, which is studied in this paper from a planar-dynamic-system view point. We obtain the equivalent two-dimensional planar dynamic system of such an equation, for which, according to the bifurcation theory and qualitative theory, phase portraits are given. Through the analysis of those phase portraits, we present the relations among the Hamiltonian, orbits of the dynamic system and types of the analytic solutions. Analytic expressions of the periodic-wave solutions, kink- and bell-shaped solitary-wave solutions are derived, and we find that the periodic-wave solutions can be reduced to the kink- and bell-shaped solitary-wave solutions. Low and high energy solutions of oscillatory non-autonomous Schrödinger equations with magnetic field https://zbmath.org/1485.35350 2022-06-24T15:10:38.853281Z "Zhang, Youpei" https://zbmath.org/authors/?q=ai:zhang.youpei "Tang, Xianhua" https://zbmath.org/authors/?q=ai:tang.xian-hua This article is concerned with the study of the equation $-\Delta_A u=\lambda \beta(x)|u|^q u+f(u)u\quad\mbox{ in } \Omega,$ where $$\Omega$$ is a smooth and bounded domain and $$\Delta_A$$ is the magnetic Laplace operator, $$q\geq 0$$, $$\beta\in L^\infty(\Omega)$$ and $$f:[0,\infty)\to \mathbb{R}$$ is an oscillating function either near zero or near infinity. Using a variational approach, the authors establish various properties of solutions and their asymptotics. Reviewer: Marius Ghergu (Dublin) Collective dynamics in science and society https://zbmath.org/1485.35357 2022-06-24T15:10:38.853281Z "Bellomo, N." https://zbmath.org/authors/?q=ai:bellomo.nicola "Brezzi, F." https://zbmath.org/authors/?q=ai:brezzi.franco This is an editorial statement by the two authors. It presents articles published in a special issue devoted to active particle methods applied to modeling, qualitative analysis, and simulation of the collective dynamics of large systems of interacting living entities in science and society. Reviewer: Adrian Muntean (Karlstad) Two flow approaches to the Loewner-Nirenberg problem on manifolds https://zbmath.org/1485.35362 2022-06-24T15:10:38.853281Z "Li, Gang" https://zbmath.org/authors/?q=ai:li.gang.11|li.gang.1|li.gang.6|li.gang.4|li.gang.8|li.gang.10|li.gang.2|li.gang.9 The flow approach is applied to study the generalized Loewner-Nirenberg problem on a Riemannian manifold $$(M,g)$$ with boundary: \begin{align*} \begin{cases} \frac{4(n-1)}{n-2}\Delta u - R_g u - n(n-1) u^{\frac{n+2}{n-2}} = 0, & \text{in}\ \mathring{M}, \\ u(p) \to \infty, & \text{as}\ p\to \partial M. \end{cases} \end{align*} The first main result is Theorem 1.1. The author stated that the Cauchy-Dirichlet problem \begin{align*} \begin{cases} u_t = \frac{4(n-1)}{n-2}\Delta u - R_g u - n(n-1) u^{\frac{n+2}{n-2}} & \text{in}\ \mathring{M}, \\ u(p,0) u_0(p), & p\to \partial M, \\ u(q,t)=\phi(q,t), & q\in \partial M \end{cases} \end{align*} has a solution $$u(t,p)$$ under certain mild technical assumptions. Moreover, as $$t\to\infty$$, the solution $$u(t,p)$$ converges to a limit $$u_{\infty}$$ which solves the above Loewner-Nirenberg problem. In the second main result, i.e., Theorem 1.2, the author used limiting solutions of the Cauchy-Dirichlet problem regarding the Yamabe flow to solve the Loewner-Nirenberg problem. Reviewer: Ruobing Zhang (Stony Brook) Sublinear equations driven by Hörmander operators https://zbmath.org/1485.35365 2022-06-24T15:10:38.853281Z "Biagi, Stefano" https://zbmath.org/authors/?q=ai:biagi.stefano "Pinamonti, Andrea" https://zbmath.org/authors/?q=ai:pinamonti.andrea "Vecchi, Eugenio" https://zbmath.org/authors/?q=ai:vecchi.eugenio Summary: We characterize the existence of a unique positive weak solution for a Dirichlet boundary value problem driven by a linear second-order differential operator modeled on Hörmander vector fields, where the right hand side has sublinear growth. A global existence result for a semilinear wave equation with lower order terms on compact Lie groups https://zbmath.org/1485.35368 2022-06-24T15:10:38.853281Z "Palmieri, Alessandro" https://zbmath.org/authors/?q=ai:palmieri.alessandro Summary: In this paper, we study the semilinear wave equation with lower order terms (damping and mass) and with power type nonlinearity $$|u|^p$$ on compact Lie groups. We will prove the global in time existence of small data solutions in the evolution energy space without requiring any lower bounds for $$p>1$$. In our approach, we employ some results from Fourier analysis on compact Lie groups. Local and blowing-up solutions for an integro-differential diffusion equation and system https://zbmath.org/1485.35374 2022-06-24T15:10:38.853281Z "Borikhanov, Meiirkhan" https://zbmath.org/authors/?q=ai:borikhanov.meiirkhan-b "Torebek, Berikbol T." https://zbmath.org/authors/?q=ai:torebek.berikbol-tillabayuly Summary: In the present paper, the semilinear integro-differential diffusion equation and system with singular in time sources are considered. An analog of Duhamel's principle for the linear integro-differential diffusion equation is proved. Using Duhamel's principle, a representation of the solution and the well-posedness of the initial problem for the linear integro-differential diffusion equation are established. The results on the existence of local integral solutions and the nonexistence of global solutions to the semilinear integro-differential diffusion equation and system are presented. Symmetry and monotonicity of positive solutions for a system involving fractional p\&q-Laplacian in $$\mathbb{R}^n$$ https://zbmath.org/1485.35377 2022-06-24T15:10:38.853281Z "Cao, Linfen" https://zbmath.org/authors/?q=ai:cao.linfen "Fan, Linlin" https://zbmath.org/authors/?q=ai:fan.linlin Summary: In this paper, we study a nonlinear system involving the fractional p\&q-Laplacian in $$\mathbb{R}^n$$ $\begin{cases} (-\Delta)_p^{s_1}u(x)+(-\Delta)_q^{s_2}u(x)=f(u(x), v(x)), & x\in\mathbb{R}^n,\\ (-\Delta)_p^{s_1}v(x)+(-\Delta)_q^{s_2}v(x)=g(u(x), v(x)), & x\in{\mathbb{R}}^n,\\ u, v>0, & x\in\mathbb{R}^n. \end{cases}$ where $$0<s_1$$, $$s_2<1$$, $$p, q>2$$. By using the direct method of moving planes, we prove that the positive solution $$(u, v)$$ of system above must be radially symmetric and monotone decreasing in the whole space. Conservation laws, analytical solutions and stability analysis for the time-fractional Schamel-Zakharov-Kuznetsov-Burgers equation https://zbmath.org/1485.35380 2022-06-24T15:10:38.853281Z "EL-Kalaawy, O. H." https://zbmath.org/authors/?q=ai:el-kalaawy.omar-h "Moawad, S. M." https://zbmath.org/authors/?q=ai:moawad.salah-m "Tharwat, M. M." https://zbmath.org/authors/?q=ai:tharwat.mohamed-mahmoud|tharwat.mohammed-m "Al-Denari, Rasha B." https://zbmath.org/authors/?q=ai:al-denari.rasha-b Summary: In this paper, we consider the $$(3+1)$$-dimensional time-fractional Schamel-Zakharov-Kuznetsov-Burgers (SZKB) equation. With the help of the Riemann-Liouville derivatives, the Lie point symmetries of the $$(3+1)$$-dimensional time-fractional SZKB equation are derived. By applying the Lie point symmetry method as well as Erdélyi-Kober fractional operator, we get the similarity reductions of the time-fractional SZKB equation. Conservation laws of the time-fractional SZKB are constructed. Moreover, we obtain its power series solutions with the convergence analysis. In addition, the analytical solution is obtained by modified trial equation method. Finally, stability is analyzed graphically in different planes. Lie symmetry reductions and conservation laws for fractional order coupled KdV system https://zbmath.org/1485.35391 2022-06-24T15:10:38.853281Z "Jafari, Hossein" https://zbmath.org/authors/?q=ai:jafari.hossein "Sun, Hong Guang" https://zbmath.org/authors/?q=ai:sun.hongguang "Azadi, Marzieh" https://zbmath.org/authors/?q=ai:azadi.marzieh Summary: Lie symmetry analysis is achieved on a new system of coupled KdV equations with fractional order, which arise in the analysis of several problems in theoretical physics and numerous scientific phenomena. We determine the reduced fractional ODE system corresponding to the governing factional PDE system. In addition, we develop the conservation laws for the system of fractional order coupled KdV equations. On fractional logarithmic Schrödinger equations https://zbmath.org/1485.35396 2022-06-24T15:10:38.853281Z "Li, Qi" https://zbmath.org/authors/?q=ai:li.qi|li.qi.1 "Peng, Shuangjie" https://zbmath.org/authors/?q=ai:peng.shuangjie "Shuai, Wei" https://zbmath.org/authors/?q=ai:shuai.wei Summary: We study the following fractional logarithmic Schrödinger equation: $(-\Delta )^s u+V(x)u=u\log{u}^2, \quad x\in\mathbb{R}^N,$ where $$N\ge 1$$, $$(-\Delta )^s$$ denotes the fractional Laplace operator, $$0< s< 1$$ and $$V(x)\in\mathcal{C}(\mathbb{R}^N)$$. Under different assumptions on the potential $$V(x)$$, we prove the existence of positive ground state solution and least energy sign-changing solution for the equation. It is known that the corresponding variational functional is not well defined in $${H}^s(\mathbb{R}^N)$$, and inspired by \textit{T. Cazenave} [Nonlinear Anal., Theory Methods Appl. 7, 1127--1140 (1983; Zbl 0529.35068)], we first prove that the variational functional is well defined in a subspace of $${H}^s(\mathbb{R}^N)$$. Then, by using minimization method and Lions' concentration-compactness principle, we prove that the existence results. A fractional Laplacian problem with critical nonlinearity https://zbmath.org/1485.35399 2022-06-24T15:10:38.853281Z "Long, Xiuhong" https://zbmath.org/authors/?q=ai:long.xiuhong "Wang, Jixiu" https://zbmath.org/authors/?q=ai:wang.jixiu Summary: In this paper, using the mountain pass theorem we obtain a positive solution to the fractional Laplacian problem $$\begin{cases} (-\Delta)^su = g(x)(u - k)^{q-1}_+ + u^{\frac{n+2s}{n-2s}} &\text{in }\Omega,\\ u > 0 &\text{in }\Omega,\\ u = 0 &\text{on }\partial\Omega, \end{cases}$$ where $\Omega\subset \mathbb R^n$ is a bounded smooth domain, $0 < s < 1$, $2 \leq q < 2n/(n - 2s)$ and $k\in (0, \infty)$ is an arbitrary number. The function $g : \Omega\to \mathbb R$ is a nonnegative continuous function satisfying some integrability condition. Inverse problem for a Cahn-Hilliard type system modeling tumor growth https://zbmath.org/1485.35426 2022-06-24T15:10:38.853281Z "Sakthivel, K." https://zbmath.org/authors/?q=ai:sakthivel.kumarasamy "Arivazhagan, A." https://zbmath.org/authors/?q=ai:arivazhagan.anbu "Barani Balan, N." https://zbmath.org/authors/?q=ai:baranibalan.n|barani-balan.natesan Summary: In this paper, we address an inverse problem of reconstructing a space-dependent semilinear coefficient in the tumor growth model described by a system of semilinear partial differential equations (PDEs) with Dirichlet boundary condition using boundary-type measurement. We establish a new higher order weighted Carleman estimate for the given system with the help of Dirichlet boundary conditions. By deriving a suitable regularity of solutions for this nonlinear system of PDEs and the new Carleman estimate, we prove Lipschitz-type stability for the tumor growth model. Asymptotic stability for a free boundary tumor model with a periodic supply of external nutrients https://zbmath.org/1485.35434 2022-06-24T15:10:38.853281Z "Huang, Yaodan" https://zbmath.org/authors/?q=ai:huang.yaodan Summary: For tumor growth, the morphological instability provides a mechanism for invasion via tumor fingering and fragmentation. This work considers the asymptotic stability of a free boundary tumor model with a periodic supply of external nutrients. The model consists of two elliptic equations describing the concentration of nutrients and the distribution of the internal pressure in the tumor tissue, respectively. The effect of the parameter $$\mu$$ representing a measure of mitosis on the morphological stability are taken into account. It was recently established in [the author et al., Discrete Contin. Dyn. Syst. 39, No. 5, 2473--2510 (2019; Zbl 1414.35236)] that there exists a critical value $$\mu_\ast$$ such that the unique spherical periodic positive solution is linearly stable for $$0 \mu_\ast$$. In this paper, we further prove that the spherical periodic positive solution is asymptotically stable for $$0 < \mu < \mu_\ast$$ for the fully nonlinear problem. A free boundary problem of some modified Leslie-gower predator-prey model with nonlocal diffusion term https://zbmath.org/1485.35435 2022-06-24T15:10:38.853281Z "Niu, Shiwen" https://zbmath.org/authors/?q=ai:niu.shiwen "Cheng, Hongmei" https://zbmath.org/authors/?q=ai:cheng.hongmei "Yuan, Rong" https://zbmath.org/authors/?q=ai:yuan.rong Summary: This paper is mainly considered a Leslie-Gower predator-prey model with nonlocal diffusion term and a free boundary condition. The model describes the evolution of the two species when they initially occupy the bounded region $$[0,h_0]$$. We first show that the problem has a unique solution defined for all $$t>0$$. Then, we establish the long-time dynamical behavior, including Spreading-vanishing dichotomy and Spreading-vanishing criteria. The dynamics of partially degenerate nonlocal diffusion systems with free boundaries https://zbmath.org/1485.35436 2022-06-24T15:10:38.853281Z "Zhang, Heting" https://zbmath.org/authors/?q=ai:zhang.heting "Li, Lei" https://zbmath.org/authors/?q=ai:li.lei.4|li.lei.3|li.lei.6|li.lei.5|li.lei.1|li.lei|li.lei.7|li.lei.2 "Wang, Mingxin" https://zbmath.org/authors/?q=ai:wang.mingxin|wang.mingxin.1 Summary: We consider a class of partially degenerate nonlocal diffusion systems with free boundaries. Such problems can describe the evolution of one species with nonlocal diffusion and the other without diffusion or with much slower diffusion. The existence, uniqueness, and regularity of global solutions are first proven. The criteria of spreading and vanishing are also established for the Lotka-Volterra type competition and prey-predator growth terms. Moreover, we investigate long-time behaviors of the solution and propose estimates of spreading speeds when spreading occurs. A free boundary problem with subcritical exponents in Orlicz spaces https://zbmath.org/1485.35437 2022-06-24T15:10:38.853281Z "Zheng, Jun" https://zbmath.org/authors/?q=ai:zheng.jun "Tavares, Leandro S." https://zbmath.org/authors/?q=ai:tavares.leandro-s Summary: Given certain functions $$\varphi ,q,h$$ and a nonnegative constant $$\lambda$$, we establish regularities for the free boundary problem ${\mathcal{J}}(u)=\int_{\varOmega }(G(|\nabla u|)+qF(u^+)+hu+\lambda \chi_{\{u>0\}} )\text{d}x\rightarrow \text{min},$ over the set $$\{u\in W^{1,G}(\varOmega ): u-\varphi \in W^{1,G}_0(\varOmega )\}$$ in the setting of Orlicz spaces, where the functions $$G$$ and $$F$$ satisfy the structural conditions of Tolksdorf's type. Moreover, $$F$$ allows for subcritical exponents. The main results obtained in this paper include: \begin{itemize} \item[(i)] the local $$C^{1,\alpha}$$- and Log-Lipschitz continuities of minimizers in the subcritical case for $$\lambda =0$$ and $$\lambda \ge 0$$, respectively; \item[(ii)] the growth rates near the free boundary for non-negative minimizers in the subcritical case for $$\lambda \ge 0$$, which give the optimal growth rates of non-negative minimizers for the $$p$$-Laplacian problems; \item[(iii)] the local Lipschitz continuity of non-negative minimizers for $$\lambda >0$$ under the natural growth condition that $$F(t)\lesssim 1+G(t)$$ for $$t \ge 0$$. \end{itemize} All the results presented in this paper are new even for the free boundary problems of $$p$$-Laplacian. Regularity results for nonlinear Young equations and applications https://zbmath.org/1485.35439 2022-06-24T15:10:38.853281Z "Addona, Davide" https://zbmath.org/authors/?q=ai:addona.davide "Lorenzi, Luca" https://zbmath.org/authors/?q=ai:lorenzi.luca "Tessitore, Gianmario" https://zbmath.org/authors/?q=ai:tessitore.gianmario Summary: In this paper we provide sufficient conditions which ensure that the nonlinear equation $$\mathrm{d}y(t)=Ay(t)\mathrm{d}t+\sigma (y(t))\mathrm{d}x(t), t\in (0,T]$$, with $$y(0)=\psi$$ and $$A$$ being an unbounded operator, admits a unique mild solution such that $$y(t)\in D(A)$$ for any $$t\in (0,T]$$, and we compute the blow-up rate of the norm of $$y(t)$$ as $$t\rightarrow 0^+$$. We stress that the regularity of $$y$$ is independent of the smoothness of the initial datum $$\psi$$, which in general does not belong to $$D(A)$$. As a consequence we get an integral representation of the mild solution $$y$$ which allows us to prove a chain rule formula for smooth functions of $$y$$. Regularity results for full nonlinear mixed type integral-differential operators https://zbmath.org/1485.47073 2022-06-24T15:10:38.853281Z "Tang, Lin" https://zbmath.org/authors/?q=ai:tang.lin Summary: We introduce a class of full nonlinear integral-differential equations with nonsymmetric positive kernels and study the regularity of their solutions. More precisely, we establish a nonlocal version of A-B-P estimate, a Harnack inequality, Hölder regularity, and $$C^{1,\alpha}$$ of the solutions. Singular perturbation of an elastic energy with a singular weight https://zbmath.org/1485.49003 2022-06-24T15:10:38.853281Z "Misiats, Oleksandr" https://zbmath.org/authors/?q=ai:misiats.oleksandr "Topaloglu, Ihsan" https://zbmath.org/authors/?q=ai:topaloglu.ihsan "Vasiliu, Daniel" https://zbmath.org/authors/?q=ai:vasiliu.daniel Summary: We study the singular perturbation of an elastic energy with a singular weight. The minimization of this energy results in a multi-scale pattern formation. We derive an energy scaling law in terms of the perturbation parameter and prove that, although one cannot expect periodicity of minimizers, the energy of a minimizer is uniformly distributed across the sample. Finally, following the approach developed by \textit{G. Alberti} and \textit{S. Müller} [Commun. Pure Appl. Math. 54, No. 7, 761--825 (2001; Zbl 1021.49012)] we prove that a sequence of minimizers of the perturbed energies converges to a Young measure supported on piecewise-linear periodic functions of slope $$\pm 1$$ whose period depends on the location in the domain and the weights in the energy. A general convergence result for viscosity solutions of Hamilton-Jacobi equations and non-linear semigroups https://zbmath.org/1485.49021 2022-06-24T15:10:38.853281Z "Kraaij, Richard C." https://zbmath.org/authors/?q=ai:kraaij.richard-clemens The author investigates a stability result for viscosity solutions of Hamilton-Jacoby type equation of the form $f-\lambda Hf=h$ in the context of non-compact spaces. More precisely, the main result extends to that spaces the Feng-Kurtz version of the Barles-Perthame procedure, which turns out to be a well established procedure allowing to approximate and to pass to the limit in viscosity super- and sub-solutions without assuming a-priori estimates. The first part of the paper is devoted to an historical presentation of the context of viscosity solutions and the theory of semigroups. Then the author moves to the basic study of of the equation $$f-\lambda Hf=h$$ and related pseudo-resolvents. Finally, in the last sections, the main stability results are proved; as a consequence, a convergence result for nonlinear semigroups is obtained. Morover, at the very end some examples are discussed. Reviewer: Luca Lussardi (Torino) Maximal regularity for local minimizers of non-autonomous functionals https://zbmath.org/1485.49044 2022-06-24T15:10:38.853281Z "Hästö, Peter" https://zbmath.org/authors/?q=ai:hasto.peter-a "Ok, Jihoon" https://zbmath.org/authors/?q=ai:ok.jihoon Summary: We establish local $$C^{1,\alpha}$$-regularity for some $$\alpha\in(0,1)$$ and $$C^{\alpha}$$-regularity for any $$\alpha\in(0,1)$$ of local minimizers of the functional $v\mapsto\int_\Omega\varphi(x,|Dv|)dx,$ where $$\varphi$$ satisfies a $$(p,q)$$-growth condition. Establishing such a regularity theory with sharp, general conditions has been an open problem since the 1980s. In contrast to previous results, we formulate the continuity requirement on $$\varphi$$ in terms of a single condition for the map $$(x,t)\mapsto\varphi(x,t)$$, rather than separately in the $$x$$- and $$t$$-directions. Thus we can obtain regularity results for functionals without assuming that the gap $$q/p$$ between the upper and lower growth bounds is close to $$1$$. Moreover, for $$\varphi(x,t)$$ with particular structure, including $$p$$-, Orlicz-, $$p(x)$$- and double phase-growth, our single condition implies known, essentially optimal, regularity conditions. Hence, we handle regularity theory for the above functional in a universal way. Prescribing Morse scalar curvatures: critical points at infinity https://zbmath.org/1485.58014 2022-06-24T15:10:38.853281Z "Mayer, Martin" https://zbmath.org/authors/?q=ai:mayer.martin Summary: The problem of prescribing conformally the scalar curvature of a closed Riemannian manifold as a given Morse function reduces to solving an elliptic partial differential equation with critical Sobolev exponent. Two ways of attacking this problem consist in subcritical approximations or negative pseudogradient flows. We show under a mild nondegeneracy assumption the equivalence of both approaches with respect to zero weak limits, in particular a one-to-one correspondence of zero weak limit finite energy subcritical blow-up solutions, zero weak limit critical points at infinity of negative type and sets of critical points with negative Laplacian of the function to be prescribed. Existence and uniqueness of invariant measures of 3D stochastic MHD-$$\alpha$$ model driven by degenerate noise https://zbmath.org/1485.60063 2022-06-24T15:10:38.853281Z "Zhang, Rangrang" https://zbmath.org/authors/?q=ai:zhang.rangrang Summary: In this paper, we establish the existence and uniqueness of invariant measures of the 3D stochastic magnetohydrodynamic-$$\alpha$$ model (MHD-$$\alpha)$$ driven by degenerate additive noise. We firstly study the Feller property of solutions and establish the existence of invariant measures by utilizing the classical Krylov-Bogoliubov theorem. Then, we prove the uniqueness of invariant measures for the corresponding transition semigroup by utilizing the notion of asymptotic strong Feller proposed by \textit{M. Hairer} and \textit{J. C. Mattingly} [Ann. Math. (2) 164, No. 3, 993--1032 (2006; Zbl 1130.37038)]. The proof not only requires the investigation of degenerate noise, but also the study of highly nonlinear, unbounded drifts. The anisotropic integrability logarithmic regularity criterion to the 3D micropolar fluid equations https://zbmath.org/1485.76024 2022-06-24T15:10:38.853281Z "Alghamdi, Ahmad Mohammad" https://zbmath.org/authors/?q=ai:alghamdi.ahmad-mohammad "Gala, Sadek" https://zbmath.org/authors/?q=ai:gala.sadek "Kim, Jae-Myoung" https://zbmath.org/authors/?q=ai:kim.jaemyoung "Ragusa, Maria Alessandra" https://zbmath.org/authors/?q=ai:ragusa.maria-alessandra Summary: The aim of this paper is to establish the regularity criterion of weak solutions to the 3D micropolar fluid equations by one directional derivative of the pressure in anisotropic Lebesgue spaces. We improve the regularity criterion for weak solutions previously given by \textit{Y. Jia} et al. [Abstr. Appl. Anal. 2012, Article ID 395420, 13 p. (2012; Zbl 1246.76140)]. Spatiotemporal optical similaritons in dual-core waveguide with an external source https://zbmath.org/1485.78005 2022-06-24T15:10:38.853281Z "Soloman Raju, Thokala" https://zbmath.org/authors/?q=ai:raju.thokala-soloman Summary: We explore analytically and numerically the existence of exact asymptotic spatiotemporal optical self-similar light bullets to the nonlinear Schrödinger equation with gain in the presence of an external source in $$(3+1)$$-dimensions. This model appertains to the description of self-similar wave propagation through asymmetric planar dual-core waveguide (DWG) amplifiers. The asymmetric DWG is composed of two adjoining, closely spaced, upper and lower waveguides, in which the lower one acts as a passive waveguide while the upper waveguide is an active one. Due to the linear coupling between them, we can control the dynamical behaviors of the wave propagating through the passive waveguide by controlling the wave in active waveguide. We explicate the mechanism to control the dynamical behaviors of these self-similar waves for two specific cases: (i) when the gain and width are hyperbolic functions and (ii) when the gain and width are periodic functions. Contextuality and dichotomizations of random variables https://zbmath.org/1485.81007 2022-06-24T15:10:38.853281Z "Kujala, Janne V." https://zbmath.org/authors/?q=ai:kujala.janne-v "Dzhafarov, Ehtibar N." https://zbmath.org/authors/?q=ai:dzhafarov.ehtibar-n Summary: The Contextuality-by-Default approach to determining and measuring the (non)contextuality of a system of random variables requires that every random variable in the system be represented by an equivalent set of dichotomous random variables. In this paper we present general principles that justify the use of dichotomizations and determine their choice. The main idea in choosing dichotomizations is that if the set of possible values of a random variable is endowed with a pre-topology (V-space), then the allowable dichotomizations split the space of possible values into two linked subsets (linkedness'' being a weak form of pre-topological connectedness). We primarily focus on two types of random variables most often encountered in practice: categorical and real-valued ones (including continuous random variables, greatly underrepresented in the contextuality literature). A categorical variable (one with a finite number of unordered values) is represented by all of its possible dichotomizations. If the values of a random variable are real numbers, then they are dichotomized by intervals above and below a variable cut point. Endpoint symmetries of helicity amplitudes https://zbmath.org/1485.81042 2022-06-24T15:10:38.853281Z "Zwicky, Roman" https://zbmath.org/authors/?q=ai:zwicky.roman Summary: We investigate helicity amplitudes (HAs) of $$A \to BC$$-type decays for arbitrary spin towards the kinematic endpoint. We show that HAs are proportional to product of Clebsch-Gordan coefficients (CGC) and the velocity to a non-negative power. The latter can be zero in which case the HA is non-vanishing at the endpoint. At the kinematic endpoint the explicit breaking of rotational symmetry, through the external momenta, is restored and the findings can be interpreted as a special case of the Wigner-Eckart theorem. Our findings are useful for i) checking theoretical computations and ii) the case where there is a sequence of decays, say $$B \to B_1 B_2$$ with the pair ($$B_1 B_2$$) not interacting (significantly) with the $$C$$-particle. Angular observables, which are ratios of HAs, are given by ratios of CGC at the endpoint. We briefly discuss power corrections in the velocity to the leading order. Gravitational waves from density perturbations in an early matter domination era https://zbmath.org/1485.83010 2022-06-24T15:10:38.853281Z "Dalianis, Ioannis" https://zbmath.org/authors/?q=ai:dalianis.ioannis "Kouvaris, Chris" https://zbmath.org/authors/?q=ai:kouvaris.chris Universal gravitational-wave signatures from heavy new physics in the electroweak sector https://zbmath.org/1485.83011 2022-06-24T15:10:38.853281Z "Eichhorn, Astrid" https://zbmath.org/authors/?q=ai:eichhorn.astrid "Lumma, Johannes" https://zbmath.org/authors/?q=ai:lumma.johannes "Pawlowski, Jan M." https://zbmath.org/authors/?q=ai:pawlowski.jan-m "Reichert, Manuel" https://zbmath.org/authors/?q=ai:reichert.manuel "Yamada, Masatoshi" https://zbmath.org/authors/?q=ai:yamada.masatoshi Linearized propagation equations for metric fluctuations in a general (non-vacuum) background geometry https://zbmath.org/1485.83012 2022-06-24T15:10:38.853281Z "Fanizza, G." https://zbmath.org/authors/?q=ai:fanizza.giovanna|fanizza.giuseppe "Gasperini, M." https://zbmath.org/authors/?q=ai:gasperini.maurizio "Pavone, E." https://zbmath.org/authors/?q=ai:pavone.e "Tedesco, L." https://zbmath.org/authors/?q=ai:tedesco.luigi|tedesco.leanna Reconstruction of primordial power spectrum of curvature perturbation from the merger rate of primordial black hole binaries https://zbmath.org/1485.83014 2022-06-24T15:10:38.853281Z "Kimura, Rampei" https://zbmath.org/authors/?q=ai:kimura.rampei "Suyama, Teruaki" https://zbmath.org/authors/?q=ai:suyama.teruaki "Yamaguchi, Masahide" https://zbmath.org/authors/?q=ai:yamaguchi.masahide "Zhang, Ying-li" https://zbmath.org/authors/?q=ai:zhang.ying-li|zhang.yingli Synthetic gravitational waves from a rolling axion monodromy https://zbmath.org/1485.83016 2022-06-24T15:10:38.853281Z "Özsoy, Ogan" https://zbmath.org/authors/?q=ai:ozsoy.ogan Anisotropic separate universe and Weinberg's adiabatic mode https://zbmath.org/1485.83017 2022-06-24T15:10:38.853281Z "Tanaka, Takahiro" https://zbmath.org/authors/?q=ai:tanaka.takahiro "Urakawa, Yuko" https://zbmath.org/authors/?q=ai:urakawa.yuko Gravitational-wave cosmological distances in scalar-tensor theories of gravity https://zbmath.org/1485.83018 2022-06-24T15:10:38.853281Z "Tasinato, Gianmassimo" https://zbmath.org/authors/?q=ai:tasinato.gianmassimo "Garoffolo, Alice" https://zbmath.org/authors/?q=ai:garoffolo.alice "Bertacca, Daniele" https://zbmath.org/authors/?q=ai:bertacca.daniele "Matarrese, Sabino" https://zbmath.org/authors/?q=ai:matarrese.sabino Fingerprints of the quantum space-time in time-dependent quantum mechanics: an emergent geometric phase https://zbmath.org/1485.83023 2022-06-24T15:10:38.853281Z "Chakraborty, Anwesha" https://zbmath.org/authors/?q=ai:chakraborty.anwesha "Nandi, Partha" https://zbmath.org/authors/?q=ai:nandi.partha "Chakraborty, Biswajit" https://zbmath.org/authors/?q=ai:chakraborty.biswajit Summary: We show the emergence of Berry phase in a forced harmonic oscillator system placed in the quantum space-time of Moyal type, where the time $$t$$' is also an operator. An effective commutative description of the system gives a time dependent generalised harmonic oscillator system with perturbation linear in position and momentum. The system is then diagonalised to get a generalised harmonic oscillator and then its adiabatic evolution over time-period $$\mathcal{T}$$ is studied in Heisenberg picture to compute the expression of geometric phase-shift. Gravitational waves from dark sectors, oscillating inflatons, and mass boosted dark matter https://zbmath.org/1485.83035 2022-06-24T15:10:38.853281Z "Bhoonah, Amit" https://zbmath.org/authors/?q=ai:bhoonah.amit "Bramante, Joseph" https://zbmath.org/authors/?q=ai:bramante.joseph "Nerval, Simran" https://zbmath.org/authors/?q=ai:nerval.simran "Song, Ningqiang" https://zbmath.org/authors/?q=ai:song.ningqiang The cosmological evolution of self-interacting dark matter https://zbmath.org/1485.83043 2022-06-24T15:10:38.853281Z "Egana-Ugrinovic, Daniel" https://zbmath.org/authors/?q=ai:egana-ugrinovic.daniel "Essig, Rouven" https://zbmath.org/authors/?q=ai:essig.rouven "Gift, Daniel" https://zbmath.org/authors/?q=ai:gift.daniel "LoVerde, Marilena" https://zbmath.org/authors/?q=ai:loverde.marilena Stealth dark energy in scordatura DHOST theory https://zbmath.org/1485.83046 2022-06-24T15:10:38.853281Z "Gorji, Mohammad Ali" https://zbmath.org/authors/?q=ai:gorji.mohammad-ali "Motohashi, Hayato" https://zbmath.org/authors/?q=ai:motohashi.hayato "Mukohyama, Shinji" https://zbmath.org/authors/?q=ai:mukohyama.shinji Asymptotic analysis of the Boltzmann equation for dark matter relic abundance https://zbmath.org/1485.83053 2022-06-24T15:10:38.853281Z "Morrison, Logan A." https://zbmath.org/authors/?q=ai:morrison.logan-a "Patel, Hiren H." https://zbmath.org/authors/?q=ai:patel.hiren-h "Ulbricht, Jaryd F." https://zbmath.org/authors/?q=ai:ulbricht.jaryd-f Reconstructing teleparallel gravity with cosmic structure growth and expansion rate data https://zbmath.org/1485.83058 2022-06-24T15:10:38.853281Z "Said, Jackson Levi" https://zbmath.org/authors/?q=ai:said.jackson-levi "Mifsud, Jurgen" https://zbmath.org/authors/?q=ai:mifsud.jurgen "Sultana, Joseph" https://zbmath.org/authors/?q=ai:sultana.joseph "Adami, Kristian Zarb" https://zbmath.org/authors/?q=ai:adami.kristian-zarb Complex scalar field reheating and primordial black hole production https://zbmath.org/1485.83067 2022-06-24T15:10:38.853281Z "Carrion, Karim" https://zbmath.org/authors/?q=ai:carrion.karim "Hidalgo, Juan Carlos" https://zbmath.org/authors/?q=ai:hidalgo.juan-carlos "Montiel, Ariadna" https://zbmath.org/authors/?q=ai:montiel.ariadna "Padilla, Luis E." https://zbmath.org/authors/?q=ai:padilla.luis-e Testing gravity of a disformal Kerr black hole in quadratic degenerate higher-order scalar-tensor theories by quasi-periodic oscillations https://zbmath.org/1485.83070 2022-06-24T15:10:38.853281Z "Chen, Songbai" https://zbmath.org/authors/?q=ai:chen.songbai "Wang, Zejun" https://zbmath.org/authors/?q=ai:wang.zejun "Jing, Jiliang" https://zbmath.org/authors/?q=ai:jing.jiliang Effects of the shape of curvature peaks on the size of primordial black holes https://zbmath.org/1485.83074 2022-06-24T15:10:38.853281Z "Escrivà, Albert" https://zbmath.org/authors/?q=ai:escriva.albert "Romano, Antonio Enea" https://zbmath.org/authors/?q=ai:romano.antonio-enea Static response and Love numbers of Schwarzschild black holes https://zbmath.org/1485.83076 2022-06-24T15:10:38.853281Z "Hui, Lam" https://zbmath.org/authors/?q=ai:hui.lam "Joyce, Austin" https://zbmath.org/authors/?q=ai:joyce.austin "Penco, Riccardo" https://zbmath.org/authors/?q=ai:penco.riccardo "Santoni, Luca" https://zbmath.org/authors/?q=ai:santoni.luca "Solomon, Adam R." https://zbmath.org/authors/?q=ai:solomon.adam-ross Gravitational waves from type II axion-like curvaton model and its implication for NANOGrav result https://zbmath.org/1485.83079 2022-06-24T15:10:38.853281Z "Kawasaki, Masahiro" https://zbmath.org/authors/?q=ai:kawasaki.masahiro "Nakatsuka, Hiromasa" https://zbmath.org/authors/?q=ai:nakatsuka.hiromasa Perturbation spectra and renormalization-group techniques in double-field inflation and quantum gravity cosmology https://zbmath.org/1485.83096 2022-06-24T15:10:38.853281Z "Anselmi, Damiano" https://zbmath.org/authors/?q=ai:anselmi.damiano Mode coupling on a geometrodynamical quantization of an inflationary universe https://zbmath.org/1485.83099 2022-06-24T15:10:38.853281Z "Brizuela, David" https://zbmath.org/authors/?q=ai:brizuela.david "de León, Irene" https://zbmath.org/authors/?q=ai:de-leon.irene Tachyonic preheating in Palatini $$R^2$$ inflation https://zbmath.org/1485.83103 2022-06-24T15:10:38.853281Z "Karam, Alexandros" https://zbmath.org/authors/?q=ai:karam.alexandros "Tomberg, Eemeli" https://zbmath.org/authors/?q=ai:tomberg.eemeli-s "Veermäe, Hardi" https://zbmath.org/authors/?q=ai:veermae.hardi Cosmological simulation in tides: power spectra, halo shape responses, and shape assembly bias https://zbmath.org/1485.83109 2022-06-24T15:10:38.853281Z "Akitsu, Kazuyuki" https://zbmath.org/authors/?q=ai:akitsu.kazuyuki "Li, Yin" https://zbmath.org/authors/?q=ai:li.yin "Okumura, Teppei" https://zbmath.org/authors/?q=ai:okumura.teppei Power spectrum in stochastic inflation https://zbmath.org/1485.83111 2022-06-24T15:10:38.853281Z "Ando, Kenta" https://zbmath.org/authors/?q=ai:ando.kenta "Vennin, Vincent" https://zbmath.org/authors/?q=ai:vennin.vincent Sourced fluctuations in generic slow contraction https://zbmath.org/1485.83113 2022-06-24T15:10:38.853281Z "Artymowski, Michał" https://zbmath.org/authors/?q=ai:artymowski.michal "Ben-Dayan, Ido" https://zbmath.org/authors/?q=ai:ben-dayan.ido "Thattarampilly, Udaykrishna" https://zbmath.org/authors/?q=ai:thattarampilly.udaykrishna Canonical description of cosmological backreaction https://zbmath.org/1485.83122 2022-06-24T15:10:38.853281Z "Bojowald, Martin" https://zbmath.org/authors/?q=ai:bojowald.martin "Ding, Ding" https://zbmath.org/authors/?q=ai:ding.ding Comparing multi-field primordial feature models with the Planck data https://zbmath.org/1485.83124 2022-06-24T15:10:38.853281Z "Braglia, Matteo" https://zbmath.org/authors/?q=ai:braglia.matteo "Chen, Xingang" https://zbmath.org/authors/?q=ai:chen.xingang "Hazra, Dhiraj Kumar" https://zbmath.org/authors/?q=ai:hazra.dhiraj-kumar Entanglement entropy of cosmological perturbations for S-brane ekpyrosis https://zbmath.org/1485.83125 2022-06-24T15:10:38.853281Z "Brahma, Suddhasattwa" https://zbmath.org/authors/?q=ai:brahma.suddhasattwa "Brandenberger, Robert" https://zbmath.org/authors/?q=ai:brandenberger.robert-h "Wang, Ziwei" https://zbmath.org/authors/?q=ai:wang.ziwei Cosmological constraints with the effective fluid approach for modified gravity https://zbmath.org/1485.83127 2022-06-24T15:10:38.853281Z "Cardona, Wilmar" https://zbmath.org/authors/?q=ai:cardona.wilmar "Arjona, Rubén" https://zbmath.org/authors/?q=ai:arjona.ruben "Estrada, Alejandro" https://zbmath.org/authors/?q=ai:estrada.alejandro "Nesseris, Savvas" https://zbmath.org/authors/?q=ai:nesseris.savvas Beyond perturbation theory in inflation https://zbmath.org/1485.83128 2022-06-24T15:10:38.853281Z "Celoria, Marco" https://zbmath.org/authors/?q=ai:celoria.marco "Creminelli, Paolo" https://zbmath.org/authors/?q=ai:creminelli.paolo "Tambalo, Giovanni" https://zbmath.org/authors/?q=ai:tambalo.giovanni "Yingcharoenrat, Vicharit" https://zbmath.org/authors/?q=ai:yingcharoenrat.vicharit The cosmic neutrino background as a collection of fluids in large-scale structure simulations https://zbmath.org/1485.83130 2022-06-24T15:10:38.853281Z "Chen, Joe Zhiyu" https://zbmath.org/authors/?q=ai:chen.joe-zhiyu "Upadhye, Amol" https://zbmath.org/authors/?q=ai:upadhye.amol "Wong, Yvonne Y. Y." https://zbmath.org/authors/?q=ai:wong.yvonne-y-y Cosmology of strongly interacting fermions in the early universe https://zbmath.org/1485.83134 2022-06-24T15:10:38.853281Z "Domènech, Guillem" https://zbmath.org/authors/?q=ai:domenech.guillem "Sasaki, Misao" https://zbmath.org/authors/?q=ai:sasaki.misao Sourcing curvature modes with entropy perturbations in non-singular bouncing cosmologies https://zbmath.org/1485.83142 2022-06-24T15:10:38.853281Z "Ijjas, Anna" https://zbmath.org/authors/?q=ai:ijjas.anna "Kolevatov, Roman" https://zbmath.org/authors/?q=ai:kolevatov.roman Ultralocality and slow contraction https://zbmath.org/1485.83143 2022-06-24T15:10:38.853281Z "Ijjas, Anna" https://zbmath.org/authors/?q=ai:ijjas.anna "Sullivan, Andrew P." https://zbmath.org/authors/?q=ai:sullivan.andrew-p "Pretorius, Frans" https://zbmath.org/authors/?q=ai:pretorius.frans "Steinhardt, Paul J." https://zbmath.org/authors/?q=ai:steinhardt.paul-j "Cook, William G." https://zbmath.org/authors/?q=ai:cook.william-g Spectrum of cuscuton bounce https://zbmath.org/1485.83152 2022-06-24T15:10:38.853281Z "Kim, J. Leo" https://zbmath.org/authors/?q=ai:kim.j-leo "Geshnizjani, Ghazal" https://zbmath.org/authors/?q=ai:geshnizjani.ghazal A first comparison of kinetic field theory with Eulerian standard perturbation theory https://zbmath.org/1485.83154 2022-06-24T15:10:38.853281Z "Kozlikin, Elena" https://zbmath.org/authors/?q=ai:kozlikin.elena "Lilow, Robert" https://zbmath.org/authors/?q=ai:lilow.robert "Fabis, Felix" https://zbmath.org/authors/?q=ai:fabis.felix "Bartelmann, Matthias" https://zbmath.org/authors/?q=ai:bartelmann.matthias Non-minimally coupled curvaton https://zbmath.org/1485.83158 2022-06-24T15:10:38.853281Z "Liu, Lei-Hua" https://zbmath.org/authors/?q=ai:liu.lei-hua "Prokopec, Tomislav" https://zbmath.org/authors/?q=ai:prokopec.tomislav Mass varying neutrinos with different quintessence potentials https://zbmath.org/1485.83161 2022-06-24T15:10:38.853281Z "Mandal, Sayan" https://zbmath.org/authors/?q=ai:mandal.sayan "Chitov, Gennady Y." https://zbmath.org/authors/?q=ai:chitov.gennady-y "Avsajanishvili, Olga" https://zbmath.org/authors/?q=ai:avsajanishvili.olga "Singha, Bijit" https://zbmath.org/authors/?q=ai:singha.bijit "Kahniashvili, Tina" https://zbmath.org/authors/?q=ai:kahniashvili.tina Small-scale CMB anisotropies induced by the primordial magnetic fields https://zbmath.org/1485.83164 2022-06-24T15:10:38.853281Z "Minoda, Teppei" https://zbmath.org/authors/?q=ai:minoda.teppei "Ichiki, Kiyotomo" https://zbmath.org/authors/?q=ai:ichiki.kiyotomo "Tashiro, Hiroyuki" https://zbmath.org/authors/?q=ai:tashiro.hiroyuki The full Boltzmann hierarchy for dark matter-massive neutrino interactions https://zbmath.org/1485.83169 2022-06-24T15:10:38.853281Z "Mosbech, Markus R." https://zbmath.org/authors/?q=ai:mosbech.markus-r "Boehm, Celine" https://zbmath.org/authors/?q=ai:boehm.celine "Hannestad, Steen" https://zbmath.org/authors/?q=ai:hannestad.steen "Mena, Olga" https://zbmath.org/authors/?q=ai:mena.olga "Stadler, Julia" https://zbmath.org/authors/?q=ai:stadler.julia "Wong, Yvonne Y. Y." https://zbmath.org/authors/?q=ai:wong.yvonne-y-y Scalar-tensor mixing from icosahedral inflation https://zbmath.org/1485.83170 2022-06-24T15:10:38.853281Z "Nicolis, Alberto" https://zbmath.org/authors/?q=ai:nicolis.alberto "Sun, Guanhao" https://zbmath.org/authors/?q=ai:sun.guanhao Multifield inflation beyond $$N_{\mathrm{field}}=2$$: non-Gaussianities and single-field effective theory https://zbmath.org/1485.83172 2022-06-24T15:10:38.853281Z "Pinol, Lucas" https://zbmath.org/authors/?q=ai:pinol.lucas A manifestly covariant theory of multifield stochastic inflation in phase space: solving the discretisation ambiguity in stochastic inflation https://zbmath.org/1485.83173 2022-06-24T15:10:38.853281Z "Pinol, Lucas" https://zbmath.org/authors/?q=ai:pinol.lucas "Renaux-Petel, Sébastien" https://zbmath.org/authors/?q=ai:renaux-petel.sebastien "Tada, Yuichiro" https://zbmath.org/authors/?q=ai:tada.yuichiro The effective field theory and perturbative analysis for log-density fields https://zbmath.org/1485.83177 2022-06-24T15:10:38.853281Z "Rubira, Henrique" https://zbmath.org/authors/?q=ai:rubira.henrique "Voivodic, Rodrigo" https://zbmath.org/authors/?q=ai:voivodic.rodrigo Hearing Higgs with gravitational wave detectors https://zbmath.org/1485.83178 2022-06-24T15:10:38.853281Z "Salvio, Alberto" https://zbmath.org/authors/?q=ai:salvio.alberto An $$n$$-th order Lagrangian forward model for large-scale structure https://zbmath.org/1485.83179 2022-06-24T15:10:38.853281Z "Schmidt, Fabian" https://zbmath.org/authors/?q=ai:schmidt.fabian Modeling galaxies in redshift space at the field level https://zbmath.org/1485.83180 2022-06-24T15:10:38.853281Z "Schmittfull, Marcel" https://zbmath.org/authors/?q=ai:schmittfull.marcel "Simonović, Marko" https://zbmath.org/authors/?q=ai:simonovic.marko "Ivanov, Mikhail M." https://zbmath.org/authors/?q=ai:ivanov.mikhail-m "Philcox, Oliver H. E." https://zbmath.org/authors/?q=ai:philcox.oliver-h-e "Zaldarriaga, Matias" https://zbmath.org/authors/?q=ai:zaldarriaga.matias Optimal computation of anisotropic galaxy three point correlation function multipoles using 2DFFTLOG formalism https://zbmath.org/1485.83185 2022-06-24T15:10:38.853281Z "Umeh, Obinna" https://zbmath.org/authors/?q=ai:umeh.obinna Primordial non-Gaussianity from G-inflation https://zbmath.org/1485.83186 2022-06-24T15:10:38.853281Z "Zhang, Fengge" https://zbmath.org/authors/?q=ai:zhang.fengge "Gong, Yungui" https://zbmath.org/authors/?q=ai:gong.yungui "Lin, Jiong" https://zbmath.org/authors/?q=ai:lin.jiong "Lu, Yizhou" https://zbmath.org/authors/?q=ai:lu.yizhou "Yi, Zhu" https://zbmath.org/authors/?q=ai:yi.zhu Local primordial non-Gaussianity in the relativistic galaxy bispectrum https://zbmath.org/1485.85002 2022-06-24T15:10:38.853281Z "Maartens, Roy" https://zbmath.org/authors/?q=ai:maartens.roy "Jolicoeur, Sheean" https://zbmath.org/authors/?q=ai:jolicoeur.sheean "Umeh, Obinna" https://zbmath.org/authors/?q=ai:umeh.obinna "De Weerd, Eline M." https://zbmath.org/authors/?q=ai:de-weerd.eline-m "Clarkson, Chris" https://zbmath.org/authors/?q=ai:clarkson.chris-a Neutrino scattering off a black hole surrounded by a magnetized accretion disk https://zbmath.org/1485.85006 2022-06-24T15:10:38.853281Z "Dvornikov, Maxim" https://zbmath.org/authors/?q=ai:dvornikov.maxim The imitation game: Proca stars that can mimic the Schwarzschild shadow https://zbmath.org/1485.85008 2022-06-24T15:10:38.853281Z "Herdeiro, Carlos A. R." https://zbmath.org/authors/?q=ai:herdeiro.carlos-a-r "Pombo, Alexandre M." https://zbmath.org/authors/?q=ai:pombo.alexandre-m "Radu, Eugen" https://zbmath.org/authors/?q=ai:radu.eugen "Cunha, Pedro V. P." https://zbmath.org/authors/?q=ai:cunha.pedro-v-p "Sanchis-Gual, Nicolas" https://zbmath.org/authors/?q=ai:sanchis-gual.nicolas On the primordial information available to galaxy redshift surveys https://zbmath.org/1485.85012 2022-06-24T15:10:38.853281Z "McQuinn, Matthew" https://zbmath.org/authors/?q=ai:mcquinn.matthew Coherence of oscillations in matter and supernova neutrinos https://zbmath.org/1485.85015 2022-06-24T15:10:38.853281Z "Porto-Silva, Yago P." https://zbmath.org/authors/?q=ai:porto-silva.yago-p "Smirnov, Alexei Yu." https://zbmath.org/authors/?q=ai:smirnov.alexei-yu The Ly$$\alpha$$ forest flux correlation function: a perturbation theory perspective https://zbmath.org/1485.85025 2022-06-24T15:10:38.853281Z "Chen, Shi-Fan" https://zbmath.org/authors/?q=ai:chen.shi-fan "Vlah, Zvonimir" https://zbmath.org/authors/?q=ai:vlah.zvonimir "White, Martin" https://zbmath.org/authors/?q=ai:white.martin The nature of non-Gaussianity and statistical isotropy of the 408 MHz Haslam synchrotron map https://zbmath.org/1485.85028 2022-06-24T15:10:38.853281Z "Rahman, Fazlu" https://zbmath.org/authors/?q=ai:rahman.fazlu "Chingangbam, Pravabati" https://zbmath.org/authors/?q=ai:chingangbam.pravabati "Ghosh, Tuhin" https://zbmath.org/authors/?q=ai:ghosh.tuhin A note on advection-diffusion cholera model with bacterial hyperinfectivity https://zbmath.org/1485.92169 2022-06-24T15:10:38.853281Z "Wu, Xiaoqing" https://zbmath.org/authors/?q=ai:wu.xiaoqing "Shan, Yinghui" https://zbmath.org/authors/?q=ai:shan.yinghui "Gao, Jianguo" https://zbmath.org/authors/?q=ai:gao.jianguo In the recent paper by \textit{X. Wang} and \textit{F.-B. Wang} [J. Math. Anal. Appl. 480, No. 2, Article ID 123407, 29 p. (2019; Zbl 1423.92241)] a system of advection-diffusion equations was suggested to model the transmission of cholera. The authors complement these results proving two new theorems in the paper under review. Namely, Theorem 1.2 establishes local asymptotic stability and global attractivity of the cholera-free equilibrium $$E_{0}$$ for the case when the basic reproduction number $$\mathcal{R}_{0}=1.$$ For $$\mathcal{R}_{0}>1,$$ Theorem 1.3 furnishes sufficient conditions for global asymptotic stability of the positive equilibrium $$E^{\ast}.$$ Reviewer: Svitlana P. Rogovchenko (Kristiansand) The existence of localized vegetation patterns in a systematically reduced model for dryland vegetation https://zbmath.org/1485.92177 2022-06-24T15:10:38.853281Z "Jaïbi, Olfa" https://zbmath.org/authors/?q=ai:jaibi.olfa "Doelman, Arjen" https://zbmath.org/authors/?q=ai:doelman.arjen "Chirilus-Bruckner, Martina" https://zbmath.org/authors/?q=ai:chirilus-bruckner.martina "Meron, Ehud" https://zbmath.org/authors/?q=ai:meron.ehud Summary: In this paper we consider the 2-component reaction-diffusion model that was recently obtained by a systematic reduction of the 3-component Gilad et al. model for dryland ecosystem dynamics \textit{E. Gilad} et al. [Ecosystem engineers: from pattern formation to habitat creation'', Phys. Rev. Lett. 93, No. 9, Article ID 098105, 4 p. (2004; \url{doi:10.1103/PhysRevLett.93.098105})]. The nonlinear structure of this model is more involved than other more conceptual models, such as the extended \textit{C. A. Klausmeier} [Regular and irregular patterns in semiarid vegetation'', Science 284, No. 5421, 1826--1828 (1999; \url{doi:10.1126/science.284.5421.1826})] model, and the analysis a priori is more complicated. However, the present model has a strong advantage over these more conceptual models in that it can be more directly linked to ecological mechanisms and observations. Moreover, we find that the model exhibits a richness of analytically tractable patterns that exceeds that of Klausmeier-type models. Our study focuses on the 4-dimensional dynamical system associated with the reaction-diffusion model by considering traveling waves in 1 spatial dimension. We use the methods of geometric singular perturbation theory to establish the existence of a multitude of heteroclinic/homoclinic/periodic orbits that jump' between (normally hyperbolic) slow manifolds, representing various kinds of localized vegetation patterns. The basic 1-front invasion patterns and 2-front spot/gap patterns that form the starting point of our analysis have a direct ecological interpretation and appear naturally in simulations of the model. By exploiting the rich nonlinear structure of the model, we construct many multi-front patterns that are novel, both from the ecological and the mathematical point of view. In fact, we argue that these orbits/patterns are not specific for the model considered here, but will also occur in a much more general (singularly perturbed reaction-diffusion) setting. We conclude with a discussion of the ecological and mathematical implications of our findings. Boundary null controllability of degenerate heat equation as the limit of internal controllability https://zbmath.org/1485.93058 2022-06-24T15:10:38.853281Z "Araújo, B. S. V." https://zbmath.org/authors/?q=ai:araujo.b-s-v "Demarque, R." https://zbmath.org/authors/?q=ai:demarque.reginaldo "Viana, L." https://zbmath.org/authors/?q=ai:viana.laura|viana.leonardo-p|viana.luiz Summary: In this paper, we recover the boundary null controllability for the degenerate heat equation by analyzing the asymptotic behavior of an eligible family of state-control pairs $$((u_\varepsilon, h_\varepsilon))_{\varepsilon > 0}$$ solving corresponding singularly perturbed internal null controllability problems. As in other situations studied in the literature, our approach relies on Carleman estimates and meticulous weak convergence results. However, for the degenerate parabolic case, some specific trace operator inequalities must be obtained, in order to justify correctly the passage to the limit argument. Existence of global attractors for the coupled system of suspension bridge equations https://zbmath.org/1485.93481 2022-06-24T15:10:38.853281Z "Aliev, A. B." https://zbmath.org/authors/?q=ai:aliev.akbar-b "Farhadova, Y. M." https://zbmath.org/authors/?q=ai:farhadova.y-m Summary: In this paper we study the mathematical model of the bridge problem where the roadbed and the tensioning cable have a common point. The correctness of the considered problem is proved and in the linear case the exponential energy decay of the system is shown. In the case of non-focused non-linear source terms we show the existence of an absorbing set and the asymptotic compactness of the nonlinear semigroup generated by the corresponding dynamic system. By using these results we show that the same nonlinear semigroup has a global minimal attractor.