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Communications on Pure and Applied Analysis

May 2022 , Volume 21 , Issue 5

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Global Well-posedness and Optimal Decay Rate of the Quasi-static Incompressible Navier–Stokes–Fourier–Maxwell–Poisson System
Yuan Xu, Fujun Zhou and Weihua Gong
2022, 21(5): 1537-1565 doi: 10.3934/cpaa.2022028 +[Abstract](398) +[HTML](93) +[PDF](574.16KB)
Abstract:

This work aims to establish global classical solution and optimal \begin{document}$ L^p $\end{document} (\begin{document}$ p\ge 2 $\end{document}) time decay rate of the quasi-static incompressible Navier–Stokes–Fourier–Maxwell–Poisson system with small initial data in \begin{document}$ \mathbb{R}^3 $\end{document}. The optimal \begin{document}$ L^2 $\end{document} time decay rate for higher order spatial derivatives is also given. To deal with the difficulty induced by the degeneration of the coupled Maxwell equation, we adopt the vector-valued form of the electric field \begin{document}$ E $\end{document} to obtain the time decay rate.

Exact travelling solution for a reaction-diffusion system with a piecewise constant production supported by a codimension-1 subspace
Anton S. Zadorin
2022, 21(5): 1567-1580 doi: 10.3934/cpaa.2022030 +[Abstract](353) +[HTML](100) +[PDF](448.66KB)
Abstract:

A generalisation of reaction diffusion systems and their travelling solutions to cases when the productive part of the reaction happens only on a surface in space or on a line on plane but the degradation and the diffusion happen in bulk are important for modelling various biological processes. These include problems of invasive species propagation along boundaries of ecozones, problems of gene spread in such situations, morphogenesis in cavities, intracellular reaction etc. Piecewise linear approximations of reaction terms in reaction-diffusion systems often result in exact solutions of propagation front problems. This article presents an exact travelling solution for a reaction-diffusion system with a piecewise constant production restricted to a codimension-1 subset. The solution is monotone, propagates with the unique constant velocity, and connects the trivial solution to a nontrivial nonhomogeneous stationary solution of the problem. The properties of the solution closely parallel the properties of monotone travelling solutions in classical bistable reaction-diffusion systems.

Shock polars for non-polytropic compressible potential flow
Volker W. Elling
2022, 21(5): 1581-1594 doi: 10.3934/cpaa.2022032 +[Abstract](263) +[HTML](114) +[PDF](481.42KB)
Abstract:

We consider compressible potential flow for general equations of state. Assuming hyperbolicity and convex equation of state, we prove that shock polars have a unique critical point (in each half), as well as a unique sonic point, with critical and strong shocks always on the subsonic side. We also show existence of normal and oblique shocks, as well as monotonicity of density, enthalpy, pressure along each half-polar, with Mach number monotone only along the subsonic part.

Global behavior for the classical solution of compressible viscous micropolar fluid with cylinder symmetry
Lan Huang, Zhiying Sun, Xin-Guang Yang and Alain Miranville
2022, 21(5): 1595-1620 doi: 10.3934/cpaa.2022033 +[Abstract](269) +[HTML](98) +[PDF](472.38KB)
Abstract:

This paper is concerned with the global solutions of the 3D compressible micropolar fluid model in the domain to a subset of \begin{document}$ R^3 $\end{document} bounded with two coaxial cylinders that present the solid thermo-insulated walls, which is in a thermodynamical sense perfect and polytropic. Compared with the classical Navier-Stokes equations, the angular velocity \begin{document}$ w $\end{document} in this model brings benefit that is the damping term -\begin{document}$ uw $\end{document} can provide extra regularity of \begin{document}$ w $\end{document}. At the same time, the term \begin{document}$ uw^2 $\end{document} is bad, it increases the nonlinearity of our system. Moreover, the regularity and exponential stability in \begin{document}$ H^4 $\end{document} also are proved.

Random attractors for non-autonomous stochastic Brinkman-Forchheimer equations on unbounded domains
Shu Wang, Mengmeng Si and Rong Yang
2022, 21(5): 1621-1636 doi: 10.3934/cpaa.2022034 +[Abstract](319) +[HTML](111) +[PDF](394.89KB)
Abstract:

In this paper, we study the asymptotic behavior of the non-autonomous stochastic 3D Brinkman-Forchheimer equations on unbounded domains. We first define a continuous non-autonomous cocycle for the stochastic equations, and then prove that the existence of tempered random attractors by Ball's idea of energy equations. Furthermore, we obtain that the tempered random attractors are periodic when the deterministic non-autonomous external term is periodic in time.

Monotonicity and nonexistence of positive solutions for pseudo-relativistic equation with indefinite nonlinearity
Yuxia Guo and Shaolong Peng
2022, 21(5): 1637-1648 doi: 10.3934/cpaa.2022037 +[Abstract](261) +[HTML](92) +[PDF](368.02KB)
Abstract:

In this paper, we consider the following general pseudo-relativistic Schrödinger equation with indefinite nonlinearities:

where \begin{document}$ s\in(0,1) $\end{document}, mass \begin{document}$ m>0 $\end{document} and \begin{document}$ a $\end{document} is a non-decreasing functions. We prove the nonexistence and the monotonicity of the positive bounded solution for the above equation via the direct method of moving planes.

Semi-classical states for fractional Schrödinger equations with magnetic fields and fast decaying potentials
Xiaoming An and Xian Yang
2022, 21(5): 1649-1672 doi: 10.3934/cpaa.2022038 +[Abstract](266) +[HTML](101) +[PDF](502.51KB)
Abstract:

This paper deals with the following fractional magnetic Schrödinger equations

where \begin{document}$ \varepsilon>0 $\end{document} is a parameter, \begin{document}$ s\in(0,1) $\end{document}, \begin{document}$ N\geq3 $\end{document}, \begin{document}$ 2+2s/(N-2s)<p<2_s^*: = 2N/(N-2s) $\end{document}, \begin{document}$ A\in C^{0,\alpha}({\mathbb R}^N,{\mathbb R}^N) $\end{document} with \begin{document}$ \alpha\in(0,1] $\end{document} is a magnetic field, \begin{document}$ V:{\mathbb R}^N\to{\mathbb R} $\end{document} is a nonnegative continuous potential. By variational methods and penalized idea, we show that the problem has a family of solutions concentrating at a local minimum of \begin{document}$ V $\end{document} as \begin{document}$ \varepsilon\to 0 $\end{document}. There is no restriction on the decay rates of \begin{document}$ V $\end{document}. Especially, \begin{document}$ V $\end{document} can be compactly supported. The appearance of \begin{document}$ A $\end{document} and the nonlocal of \begin{document}$ (-\Delta)^s $\end{document} makes the proof more difficult than that in [7], which considered the case \begin{document}$ A\equiv 0 $\end{document}.

Analysis of one-sided 1-D fractional diffusion operator
Yulong Li, Aleksey S. Telyakovskiy and Emine Çelik
2022, 21(5): 1673-1690 doi: 10.3934/cpaa.2022039 +[Abstract](378) +[HTML](100) +[PDF](466.17KB)
Abstract:

This work establishes the parallel between the properties of classic elliptic PDEs and the one-sided 1-D fractional diffusion equation, that includes the characterization of fractional Sobolev spaces in terms of fractional Riemann-Liouville (R-L) derivatives, variational formulation, maximum principle, Hopf's Lemma, spectral analysis, and theory on the principal eigenvalue and its characterization, etc. As an application, the developed results provide a novel perspective to study the distribution of complex roots of a class of Mittag-Leffler functions and, furthermore, prove the existence of real roots.

Two-sided estimates of total bandwidth for Schrödinger operators on periodic graphs
Evgeny Korotyaev and Natalia Saburova
2022, 21(5): 1691-1714 doi: 10.3934/cpaa.2022042 +[Abstract](219) +[HTML](88) +[PDF](605.69KB)
Abstract:

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We obtain two-sided estimates of the total bandwidth for the Schrödinger operators in terms of geometric parameters of the graph and the potentials. In particular, we show that these estimates are sharp. It means that these estimates become identities for specific graphs and potentials. The proof is based on the Floquet theory and trace formulas for fiber operators. The traces are expressed as finite Fourier series of the quasimomentum with coefficients depending on the potentials and cycles of the quotient graph from some specific cycle sets. In order to obtain our results we estimate these Fourier coefficients in terms of geometric parameters of the graph and the potentials.

Hölder-Logarithmic type approximation for nonlinear backward parabolic equations connected with a pseudo-differential operator
Dinh Nguyen Duy Hai
2022, 21(5): 1715-1734 doi: 10.3934/cpaa.2022043 +[Abstract](436) +[HTML](136) +[PDF](421.28KB)
Abstract:

In this paper, we deal with the backward problem for nonlinear parabolic equations involving a pseudo-differential operator in the \begin{document}$ n $\end{document}-dimensional space. We prove that the problem is ill-posed in the sense of Hadamard, i.e., the solution, if it exists, does not depend continuously on the data. To regularize the problem, we propose two modified versions of the so-called optimal filtering method of Seidman [T.I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33 (1996), 162–170]. According to different a priori assumptions on the regularity of the exact solution, we obtain some sharp optimal estimates of the Hölder-Logarithmic type in the Sobolev space \begin{document}$ H^q(\mathbb{R}^n) $\end{document}.

On an exponentially decaying diffusive chemotaxis system with indirect signals
Pan Zheng and Jie Xing
2022, 21(5): 1735-1753 doi: 10.3934/cpaa.2022044 +[Abstract](308) +[HTML](159) +[PDF](485.0KB)
Abstract:

This paper deals with an exponentially decaying diffusive chemotaxis system with indirect signal production or consumption

under homogeneous Neumann boundary conditions in a smoothly bounded domain \begin{document}$ \Omega\subset \mathbb{R}^{n} $\end{document}, \begin{document}$ n\geq2 $\end{document}, where the nonlinear diffusivity \begin{document}$ D $\end{document} and chemosensitivity \begin{document}$ S $\end{document} are supposed to satisfy

with the constants \begin{document}$ \beta^{-}\geq \beta^{+}>0 $\end{document}, \begin{document}$ K_{1},K_{2},K_{3}>0 $\end{document} and \begin{document}$ \alpha,\gamma\geq0 $\end{document}. When \begin{document}$ h(v,w) = -v+w $\end{document}, we study the global existence and boundedness of solutions for the above system provided that \begin{document}$ \alpha\in[0,\frac{2}{n}) $\end{document}, \begin{document}$ \beta^{-}\geq \beta^{+}>\frac{n}{2} $\end{document}, \begin{document}$ \gamma>1 $\end{document} and the initial mass of \begin{document}$ u_{0} $\end{document} is small enough. Moreover, it is proved that the global bounded solution \begin{document}$ (u,v,w) $\end{document} converges to \begin{document}$ (\overline{u_{0}},\overline{u_{0}},\overline{u_{0}}) $\end{document} in the \begin{document}$ L^{\infty} $\end{document}-norm as \begin{document}$ t\rightarrow \infty $\end{document}, where \begin{document}$ \overline{u_{0}} = \frac{1}{|\Omega|}\int_{\Omega}u_{0}(x)dx $\end{document}. When \begin{document}$ h(v,w) = -vw $\end{document}, it is shown that this system possesses a unique uniformly bounded classical solution if \begin{document}$ \alpha\geq0 $\end{document}, \begin{document}$ \gamma>0 $\end{document} and \begin{document}$ \beta^{-}\geq \beta^{+}>\frac{n}{2} $\end{document}. Furthermore, if \begin{document}$ n = 2 $\end{document}, \begin{document}$ \alpha\geq0 $\end{document}, \begin{document}$ \gamma\geq0 $\end{document}, and \begin{document}$ \beta^{-}\geq \beta^{+}>\varepsilon $\end{document} with some \begin{document}$ \varepsilon>0 $\end{document}, we only obtain the global existence of solutions for the above system.

Symmetry and monotonicity of positive solutions for a class of general pseudo-relativistic systems
Xueying Chen, Guanfeng Li and Sijia Bao
2022, 21(5): 1755-1772 doi: 10.3934/cpaa.2022045 +[Abstract](317) +[HTML](96) +[PDF](429.69KB)
Abstract:

In this paper, we focus on a class of general pseudo-relativistic systems

where \begin{document}$ m \in (0, +\infty) $\end{document} and \begin{document}$ s, t \in (0,1) $\end{document}. Before giving the main results, we first introduce a decay at infinity and a narrow region principle. Then we implement the direct method of moving planes to show the radial symmetry and monotonicity of positive solutions for the above system in both the unit ball and the whole space.

Formation of singularities of solutions to the Cauchy problem for semilinear Moore-Gibson-Thompson equations
Sen Ming, Han Yang and Xiongmei Fan
2022, 21(5): 1773-1792 doi: 10.3934/cpaa.2022046 +[Abstract](240) +[HTML](82) +[PDF](476.76KB)
Abstract:

This paper is devoted to investigating formation of singularities for solutions to semilinear Moore-Gibson-Thompson equations with power type nonlinearity \begin{document}$ |u|^{p} $\end{document}, derivative type nonlinearity \begin{document}$ |u_{t}|^{p} $\end{document} and combined type nonlinearities \begin{document}$ |u_{t}|^{p}+|u|^{q} $\end{document} in the case of single equation, combined type nonlinearities \begin{document}$ |v_{t}|^{p_{1}}+|v|^{q_{1}} $\end{document}, \begin{document}$ |u_{t}|^{p_{2}}+|u|^{q_{2}} $\end{document}, combined and power type nonlinearities \begin{document}$ |v_{t}|^{p_{1}}+|v|^{q_{1}} $\end{document}, \begin{document}$ |u|^{q_{2}} $\end{document}, combined and derivative type nonlinearities \begin{document}$ |v_{t}|^{p_{1}}+|v|^{q_{1}} $\end{document}, \begin{document}$ |u_{t}|^{p_{2}} $\end{document} in the case of coupled system, respectively. More precisely, blow-up results of solutions to problems in the sub-critical and critical cases are derived by applying test function technique. Moreover, upper bound lifespan estimates of solutions to the coupled systems are investigated. The main new contribution is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent.

The number of limit cycles from the perturbation of a quadratic isochronous system with two switching lines
Ai Ke, Maoan Han and Wei Geng
2022, 21(5): 1793-1809 doi: 10.3934/cpaa.2022047 +[Abstract](317) +[HTML](122) +[PDF](633.04KB)
Abstract:

In this paper, we give an upper bound (for \begin{document}$ n\geq3 $\end{document}) and the least upper bound (for \begin{document}$ n = 1,2 $\end{document}) of the number of limit cycles bifurcated from period annuli of a quadratic isochronous system under the piecewise polynomial perturbations of degree \begin{document}$ n $\end{document}, respectively. The results improve the conclusions in [19].

Exponential attractors for two-dimensional nonlocal diffusion lattice systems with delay
Lin Yang, Yejuan Wang and Peter E. Kloeden
2022, 21(5): 1811-1831 doi: 10.3934/cpaa.2022048 +[Abstract](232) +[HTML](121) +[PDF](457.49KB)
Abstract:

In this paper, we study the long term dynamical behavior of a two-dimensional nonlocal diffusion lattice system with delay. First some sufficient conditions for the construction of an exponential attractor are presented for infinite dimensional autonomous dynamical systems with delay. Then, the existence of exponential attractors for the two-dimensional nonlocal diffusion delay lattice system is established by using the new method of tail-estimates of solutions and overcoming the difficulties caused by the nonlocal diffusion operator and the multi-dimensionality.

The number of limit cycles by perturbing a piecewise linear system with three zones
Xiaolei Zhang, Yanqin Xiong and Yi Zhang
2022, 21(5): 1833-1855 doi: 10.3934/cpaa.2022049 +[Abstract](322) +[HTML](94) +[PDF](696.42KB)
Abstract:

First, this paper provides a new proof for the expression of the generalized first order Melnikov function on piecewise smooth differential systems with multiply straight lines. Then, by using the Melnikov function, we consider the limit cycle bifurcation problem of a 3-piecewise near Hamiltonian system with two switching lines, obtaining \begin{document}$ 2n+3[\frac{n+1}{2}] $\end{document} limit cycles near the double generalized homoclinic loop.

2021 Impact Factor: 1.273
5 Year Impact Factor: 1.282
2021 CiteScore: 2.2

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