Zeitschrift für angewandte Mathematik und Physik
Journal Prestige (SJR): 0.828 Citation Impact (citeScore): 2 Number of Followers: 2 Hybrid journal (It can contain Open Access articles) ISSN (Print) 00442275  ISSN (Online) 14209039 Published by SpringerVerlag [2468 journals] 
 On a mathematical model for cancer invasion with repellent pHtaxis and
nonlocal intraspecific interaction
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Abstract: Abstract Starting from a mesoscopic description of cell migration and intraspecific interactions, we obtain by upscaling an effective reaction–diffusion–taxis equation for the cell population density involving spatial nonlocalities in the source term and biasing its motility and growth behavior according to environmental acidity. We prove global existence, uniqueness, and boundedness of a nonnegative solution to a simplified version of the coupled system describing cell and acidity dynamics. A 1D study of pattern formation is performed. Numerical simulations illustrate the qualitative behavior of solutions.
PubDate: 20240214

 On some direct and inverse problems for an integrodifferential equation

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Abstract: Abstract The direct and two inverse problems defined for an integrodifferential equation on a bounded domain have been considered. The spectral problem of the integrodifferential equation constitutes the Legendre differential equation in space variable. Finding a spacedependent source term whenever the data at some time, say T, as overspecified condition, constitutes the Ist inverse problem. The 2nd inverse problem consists of recovering a timedependent coefficient in the source term from an integral type overspecified condition. The Fourier approach is used to have the analytical series solution of the problems. The existence and uniqueness results for the direct and inverse problems under certain regularity conditions on the data are presented.
PubDate: 20240214

 Dynamical behavior of solutions of a reaction–diffusion–advection
model with a free boundary
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Abstract: Abstract This paper is devoted to study the population dynamics of a single species in a onedimensional environment which is modeled by a reaction–diffusion–advection equation with free boundary condition. We find three critical values \(c_0\) , 2 and \(\beta ^*\) for the advection coefficient \(\beta \) with \(\beta ^*>2>c_0>0\) , which play key roles in the dynamics, and prove that a spreadingvanishing dichotomy result holds when \(2<\beta \leqslant c_0\) ; a small spreadingvanishing dichotomy result holds when \(c_0<\beta <2\) ; a virtual spreadingtransitionvanishing trichotomy result holds when \(2\leqslant \beta <\beta ^*\) ; only vanishing happens when \(\beta \geqslant \beta ^*\) ; a virtual vanishingtransitionvanishing trichotomy result holds when \(\beta \leqslant 2\) . When spreading or small spreading or virtual spreading happens for a solution, we make use of the traveling semiwave solutions to give a estimate for the asymptotic spreading speed and asymptotic profile of the right front.
PubDate: 20240214

 Nonlocal residual symmetries, Nth Bäcklund transformations and exact

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Abstract: Abstract The nonlocal residual symmetries of a generalized Broer–Kaup–Kupershmidt system are constructed using the truncated Painlevé expansion. By considering appropriate potential variables, these nonlocal symmetries are localized into Lie point symmetries by prolonging the generalized Broer–Kaup–Kupershmidt system to an enlarged system. Moreover, multiple residual symmetries of the studied system are derived, their localization is performed, and the Nth Bäcklund transformation theorem is presented. Furthermore, some exact interaction solutions between solitons and various kinds of nonlinear waves are formulated in view of the new consistent tanh expansion method.
PubDate: 20240212

 Properties of a class of quasiperiodic Schrödinger operators

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Abstract: Abstract In this paper, a class of models with deep physical meaning is studied through duality, and positive Lyapunov exponents and some spectral properties are obtained under certain conditions.
PubDate: 20240212

 Wellposedness, asymptotic stability and blowup results for a nonlocal
singular viscoelastic problem with logarithmic nonlinearity
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Abstract: Abstract Considered herein is the wellposedness, asymptotic stability and blowup of the initialboundary value problem for nonlocal singular viscoelastic wave equation with logarithmic nonlinearity \(u_{tt}\frac{1}{x}(x u_{x})_x\frac{1}{x}(x u_{xt})_x+\int \limits _{0}^{t}m(t\lambda )\frac{1}{x}(x u_{x}(x, \lambda ))_x \hbox {d}\lambda = u ^{r2}u\ln u \) subject to a nonlocal boundary condition. Through the effective combining of Galerkin approximation method, modified potential well theory, perturbed energy method, convexity theory and differentialintegral inequality techniques, we firstly demonstrate the global existence and uniqueness of weak solutions in certain weighted Sobolev spaces; Secondly, we establish the explicit polynomial and exponential energy decay estimates under some suitable conditions; Finally, we investigate the finite time blowup criterion and then derive its upper and lower bounds of blowup time. The above conclusions extend and improve some results in the literatures.
PubDate: 20240210

 Global wellposedness for 2D nonhomogeneous asymmetric fluids with
magnetic field and densitydependent viscosity
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Abstract: Abstract We study an initialboundary value problem of twodimensional nonhomogeneous asymmetric fluids with magnetic field and densitydependent viscosity \(\mu (\rho )\) . Applying Desjardins’ interpolation inequality and delicate energy estimates, we show the globalintime existence of a unique strong solution when \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\) is properly small. Moreover, we prove that the velocity, the microrotational velocity, and the magnetic field converge exponentially to zero in \(H^2\) as time goes to infinity.
PubDate: 20240210

 Normalized bound states for the Choquard equations in exterior domains

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Abstract: Abstract In this paper, we investigate the following nonlinear Choquard equation with prescribed \(L^2\) norm constraint $$\begin{aligned} \left\{ \begin{array}{ll} \Delta u=\lambda u+( x ^{1} * u ^2)u &{}\text{ in }\ {\Omega }, \\ u=0&{}\text{ on }\ {\partial \Omega }, \\ \int \limits _\Omega u ^2{\textrm{d}}x=a^2,\\ \end{array} \right. \end{aligned}$$ where \(a>0\) , \(\lambda \in \mathbb R\) appears as an unknown Lagrange multiplier and \(\Omega \subset \mathbb R^3\) is an exterior domain with smooth boundary \(\partial \Omega \ne \emptyset \) such that \(\mathbb R^3\backslash \Omega \) is bounded. By using the splitting lemma for the unconstrained problem in exterior domains, we prove the compactness of Palais–Smale sequences corresponding to the above problem at higher energy levels. Then combining the barycentric function and Brouwer degree theory, we establish the existence of positive normalized bound states for any \(a>0\) provided that \(\mathbb R^3\backslash \Omega \) is contained in a small ball and explain that the restriction on domain \(\Omega \) can be equivalently transferred to a. In addition, under the radial setting of domain \(\Omega \) , we use genus theory to obtain the existence and multiplicity of radial normalized solutions for any \(a>0\) . Finally, we point out that the main results can be extended to a general mass subcritical Choquard equation.
PubDate: 20240210

 Timeperiodic traveling wave solutions of a reaction–diffusion Zika
epidemic model with seasonality
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Abstract: Abstract In this paper, the full information about the existence and nonexistence of a timeperiodic traveling wave solution of a reaction–diffusion Zika epidemic model with seasonality, which is nonmonotonic, is investigated. More precisely, if the basic reproduction number, denoted by \(R_{0}\) , is larger than one, there exists a minimal wave speed \(c^* > 0\) satisfying for each \(c > c^*\) , the system admits a nontrivial timeperiodic traveling wave solution with wave speed c, and for \(c<c^*\) , there exist no nontrivial timeperiodic traveling waves such that if \(R_0 \leqslant 1\) , the system admits no nontrivial timeperiodic traveling waves.
PubDate: 20240210

 Stability and crossdiffusiondriven instability for a watervegetation
model with the infiltration feedback effect
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Abstract: Abstract This paper is devoted to a mathematical model with diffusion and crossdiffusion to describe the interaction between vegetation and soil water. First, the existence of Hopf bifurcation and crossdiffusiondriven Turing instability are discussed. Then, based on the nonlinear analysis, we obtain the exact parameters range for stationary patterns and show the dynamical behavior near Turing bifurcation point. It is found that the model has the properties of gap, strip and spot patterns. Moreover, the small wateruptake ability of vegetation roots promotes the growth of vegetation and the transitions of vegetation pattern. But with the continuous increase of the wateruptake ability of vegetation roots, the local vegetation biomass density increases and the isolation between vegetation patches also increases, which may induce the emergence of desertification. In addition, our results reveal that the water consumption rate induces the transitions of vegetation pattern and prohibits the increase of vegetation biomass density.
PubDate: 20240210

 Boundedness and largetime behavior in a chemotaxis system with
signaldependent motility arising from tumor invasion
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Abstract: Abstract In this paper, we study the following chemotaxis system with signaldependent motility arising from tumor invasion $$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta (\varphi (v)u)+\rho u \mu u^l,&\qquad \quad x\in \Omega ,\,t>0,\\&v_t=\Delta v+ wz,&\qquad \quad x\in \Omega ,\,t>0,\\&w_t=wz,&\qquad \quad x\in \Omega ,\,t>0,\\&z_t=\Delta zz+u,&\qquad \quad x\in \Omega ,\,t>0 \end{aligned} \right. \end{aligned}$$ under homogeneous Neumann boundary conditions in a smooth bounded domain \( \Omega \subset {\mathbb {R}}^n(n\ge 1)\) , where the parameters \( \rho \ge 0,\mu \ge 0\) and \( l>1\) are constants, the motility function \(\varphi (v)\) satisfies \( \varphi (v)\in C^3([0,+\infty )), \varphi _1\le \varphi (v)\le \varphi _2\) and \( \varphi '(v) \le \varphi _3\) with \(\varphi _1,\varphi _2,\varphi _3>0.\) The purpose of this paper is to prove that the existence of global bounded solution for \(1\le n\le 3\) . For \(n\ge 4\) , we prove that the nonnegative classical solution (u, v, w, z) is globally bounded if \(l>\frac{n}{2}\) . In addition, we also show that all the global bounded solution will converge to the nontrivial constant steady state exponentially.
PubDate: 20240130

 Existence and asymptotic behavior of positive solutions to some
logarithmic Schrödinger–Poisson system
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Abstract: Abstract In this paper, we consider the following logarithmic Schrödinger–Poisson system $$\begin{aligned} \left\{ \begin{aligned}& \Delta u + V(x) u + \lambda K(x)\phi u = f(u) + u \log u^2,&x \in {\mathbb {R}}^{3},\\& \Delta \phi  \varepsilon ^4 \Delta _4 \phi = \lambda K(x) u^2,&x \in {\mathbb {R}}^{3},\\ \end{aligned} \right. \end{aligned}$$ which has increasingly received interest due to the indefiniteness of the energy functional and fourthorder term in Poisson equation. By using variational method, we prove the existence and multiplicity of positive solutions. Finally, we obtain the asymptotic behavior of positive solutions as \(\varepsilon \rightarrow 0^+\) and \(\lambda \rightarrow 0^+\) , respectively.
PubDate: 20240130

 Dynamical analysis of a spatial memory prey–predator system with
gestation delay and strong Allee effect
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Abstract: Abstract In this paper, we formulate a diffusive prey–predator system with strong Allee effect in prey, spatial memory, and gestation delay of predator. Turing instability and Hopf bifurcation are investigated extensively. The system more easily creates Turing patterns, when the predator owns sluggish memoryinduced diffusion. If the predator owns rapid memoryinduced diffusion, then the system may present abundant dynamics. Predator with long memory displays a spatial nonhomogeneous periodic spread once the system only contains spatial memory delay. Populations are in a homogeneous or nonhomogeneous periodic spread with gestation period. However, under the combined influences of memory delay and gestation delay, the stability switches appear. Strong Allee effect has obvious influences on the system. The positive equilibrium is linearly stable if there is no strong Allee effect; once strong Allee effect is reasonable, it is unstable. But if the Allee effect is huge enough, then two populations can die out. Finally, we investigate the results by numerical simulations, which demonstrate that spatial memory, gestation delay, and strong Allee effect are conductive to the dynamics of the system.
PubDate: 20240129

 Normalized solutions to planar Schrödinger equation with exponential
critical nonlinearity
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Abstract: Abstract This paper is concerned with the following planar Schrödinger equation $$\begin{aligned} \left\{ \begin{aligned}&\Delta u+\lambda u = f(u),&x \in {\mathbb {R}}^{2},\\&\mathop \int \limits _{{\mathbb {R}}^2}u^2dx=c,&\lambda \in {\mathbb {R}}^+. \end{aligned}\right. \end{aligned}$$ where \(f \in {\mathcal {C}}({\mathbb {R}},{\mathbb {R}})\) is of critical exponential growth. We obtain the existence of ground state normalized solutions \((u,\lambda )\) under general assumptions, and here \(\lambda \) stands for a Lagrange multiplier. Our theorems extend the results of Alves, Ji and Miyagaki (Calc Var 61:18, 2022) and Chang, Liu and Yan (J Geom Anal 33:83, 2023), where f satisfies a strong global assumption. In particular, some new estimates and approaches are introduced to overcome the lack of compactness resulting from the critical growth of f(u).
PubDate: 20240128

 Stability and decay estimates of the 2D incompressible magnetomicropolar
fluid system with partial viscosity on a flat strip
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Abstract: Abstract In this paper, the authors establish the stability and the explicit decay estimates of the 2D incompressible magnetomicropolar fluid equations without magnetic diffusion and zero spin viscosity on a flat strip \(\Omega :={\mathbb {T}}\times [0,1]\) under the assumption that Navier type condition being imposed. The results are obtained heavily based on some timeweight energy estimates and a bootstrap argument.
PubDate: 20240128

 Green’s functions for an anisotropic elastic matrix containing an
elliptical incompressible liquid inclusion
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Abstract: Abstract We use the Stroh sextic formalism for anisotropic elasticity and Muskhelishvili’s complex variable formulation for isotropic elasticity to derive a fullfield closedform solution to the generalized plane strain problem of an elliptical incompressible liquid inclusion embedded in an infinite anisotropic elastic matrix subjected to a line force and a line dislocation. An explicit expression for the internal uniform hydrostatic tension within the liquid inclusion is obtained. Furthermore, in the case when the line force and line dislocation approach the elliptical liquid–solid interface, we develop a realform solution for the internal uniform hydrostatic tension in terms of the Barnett–Lothe tensors for the matrix.
PubDate: 20240128

 Mathematical derivation of a Reynolds equation for magnetomicropolar
fluid flows through a thin domain
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Abstract: Abstract In this paper, we study the asymptotic behavior of the stationary 3D magnetomicropolar fluid flow through a thin domain, whose thickness is given by a parameter \(0<\varepsilon \ll 1\) . Assuming that the magnetic Reynolds number is written in terms of the thickness \(\varepsilon \) , we prove that there exists a critical magnetic Reynolds number, namely \(Re_m^c=\varepsilon ^{2}\) , such that for every magnetic Reynolds number \(Re_m\) with order smaller or equal than \(Re_m^c\) , the magnetomicropolar fluid flow in the thin domain can be modeled asymptotically when \(\varepsilon \) tends to zero by a 2D Reynoldslike model, whose expression is also given.
PubDate: 20240128

 Bifurcation analysis of a delayed diffusive predator–prey model with
spatial memory and toxins
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Abstract: Abstract In this paper, we propose a diffusive predator–prey model with two delays, i.e., a spatial memory delay and a toxic effect delay. Initially, we analyze the global existence of the solution of the system. We then analyze the equilibria’s local stability without delays and investigate the Hopf bifurcation induced by one delay. Subsequently, we establish an analytical framework for constructing the stability switching curve in the delay space. Finally, we present numerical simulations to validate the theoretical results and verify the emergence of various spatial patterns in the system.
PubDate: 20240128

 Nonlinear perturbations of a periodic Kirchhoff–Boussinesqtype problems
in $$\mathbb {R}^{N}$$
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Abstract: Abstract This paper is concerned with the existence of a ground state solution for the following class of elliptic Kirchhoff–Boussinesqtype problems given by $$\begin{aligned} \Delta ^{2} u \pm \Delta _{p} u +V(x)u= f(u) +\gamma u ^{2_{**}2}u \ \ \text{ in } \ \ \mathbb {R}^{N}, \end{aligned}$$ where \(2< p< 2^{*}= \frac{2N}{N2}\) for \( N\ge 3\) and \(2_{**}= \infty \) for \(N=3\) , \(N=4\) , \(2_{**}= \frac{2N}{N4}\) for \(N\ge 5\) . Here V and f are continuous functions with V being either periodic or asymptote to infinity a periodic function. The function f(u) has subcritical growth and behaves like \( u ^{q2}u\) with \(p<q< 2_{**}\) . We show existence of a ground state solution using variational methods considering the subcritical case, i.e, \(\gamma =0\) and the critical case, i.e, \(\gamma =1\) .
PubDate: 20240127

 Wellposedness of the governing equations for a quasilinear viscoelastic
model with pressuredependent moduli in which both stress and strain
appear linearly
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Abstract: Abstract The response of a body described by a quasilinear viscoelastic constitutive relation, whose material moduli depend on the mechanical pressure (that is onethird the trace of stress) is studied. The constitutive relation stems from a class of implicit relations between the histories of the stress and the relative deformation gradient. Apriori thresholding is enforced through the pressure that ensures that the displacement gradient remains small. The resulting mixed variational problem consists of an evolutionary equation with the Volterra convolution operator; this equation is studied for wellposedness within the theory of maximal monotone graphs. For isotropic extension or compression, a semianalytic solution of the quasilinear viscoelastic problem is constructed under stress control. The equations are studied numerically with respect to monotone loading both with and without thresholding. In the example, the thresholding procedure ensures that the solution does not blowup in finite time.
PubDate: 20240116
