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Communications on Pure & Applied Analysis

May 2020 , Volume 19 , Issue 5

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Strong convergence of trajectory attractors for reaction–diffusion systems with random rapidly oscillating terms
Kuanysh A. Bekmaganbetov, Gregory A. Chechkin and Vladimir V. Chepyzhov
2020, 19(5): 2419-2443 doi: 10.3934/cpaa.2020106 +[Abstract](265) +[HTML](80) +[PDF](563.07KB)
Abstract:

We consider reaction–diffusion systems with random terms that oscillate rapidly in space variables. Under the assumption that the random functions are ergodic and statistically homogeneous we prove that the random trajectory attractors of these systems tend to the deterministic trajectory attractors of the averaged reaction-diffusion system whose terms are the average of the corresponding terms of the original system. Special attention is given to the case when the convergence of random trajectory attractors holds in the strong topology.

Uniform a priori estimates for elliptic problems with impedance boundary conditions
Théophile Chaumont-Frelet, Serge Nicaise and Jérôme Tomezyk
2020, 19(5): 2445-2471 doi: 10.3934/cpaa.2020107 +[Abstract](184) +[HTML](92) +[PDF](523.11KB)
Abstract:

We derive stability estimates in \begin{document}$ H^2 $\end{document} for elliptic problems with impedance boundary conditions that are uniform with respect to the impedance coefficient. Such estimates are of importance to establish sharp error estimates for finite element discretizations of contact impedance and high-frequency Helm-holtz problems. Though stability in \begin{document}$ H^2 $\end{document} is easily obtained by employing a ``bootstrap'' argument and well-established result for the corresponding Neumann problem, this strategy leads to a stability constant that increases with the impedance coefficient. Here, we propose alternative proofs to derive sharp and uniform stability constants for domains that are convex or smooth.

Global dynamics for a class of reaction-diffusion equations with distributed delay and neumann condition
Tarik Mohammed Touaoula
2020, 19(5): 2473-2490 doi: 10.3934/cpaa.2020108 +[Abstract](178) +[HTML](73) +[PDF](451.65KB)
Abstract:

In this paper, we investigate a class of non-monotone reaction-diffusion equations with distributed delay and a homogenous Neumann boundary condition. The main concern is the global attractivity of the unique positive steady state. To achieve this, we use an argument based on sub and super-solutions combined with the fluctuation method. We also give a condition under which the exponential stability of the positive steady state is reached. As particular examples, we apply our results to the diffusive Nicholson blowfly equation and the diffusive Mackey-Glass equation with distributed delay. We obtain some new results on exponential stability of the positive steady state for these models.

Global existence and non-existence analyses to a nonlinear Klein-Gordon system with damping terms under positive initial energy
Guangyu Xu, Chunlai Mu and Dan Li
2020, 19(5): 2491-2512 doi: 10.3934/cpaa.2020109 +[Abstract](223) +[HTML](138) +[PDF](475.73KB)
Abstract:

This paper considers the Cauchy problem for a nonlinear Klein-Gordon system with damping terms. In the existing works, the solution with low and critical initial energy was studied. We extend the previous results on following three aspects. Firstly, we consider the vacuum isolating phenomenon of solution under initial energy \begin{document}$ E(0)\leq0 $\end{document}. We find that the corresponding vacuum region is an ball and it expands to whole phase space as \begin{document}$ E(0) $\end{document} decays to \begin{document}$ -\infty $\end{document}. Secondly, we discuss the asymptotic behavior of blow-up solution and prove that the solution grows exponentially. The growth speed is estimated especially. Finally, the solution with arbitrary positive initial energy is studied. In this case, the initial conditions such that the solution exists globally and blows up in finite time are given, respectively.

A stochastic threshold for an epidemic model with isolation and a non linear incidence
Tomás Caraballo, Mohamed El Fatini, Idriss Sekkak, Regragui Taki and Aziz Laaribi
2020, 19(5): 2513-2531 doi: 10.3934/cpaa.2020110 +[Abstract](210) +[HTML](77) +[PDF](1125.72KB)
Abstract:

In this paper, we study a stochastic epidemic model with isolation and nonlinear incidence. In particular, we propose a stochastic threshold for the model without any sharp sufficient assumptions on model parameters as compared to existing works. Firstly, we establish the uniqueness of the global positive solution according to Lyapunov function method. Secondly, we prove stochastic permanence of the solutions. Then, we establish sufficient condition for the extinction. Thirdly, we investigate necessary and sufficient conditions for persistence in mean of the disease. Finally, we provide some numerical simulations to illustrate our theoretical results.

Paradoxical phenomena and chaotic dynamics in epidemic models subject to vaccination
Alfonso Ruiz Herrera
2020, 19(5): 2533-2548 doi: 10.3934/cpaa.2020111 +[Abstract](121) +[HTML](72) +[PDF](609.5KB)
Abstract:

An alternative to the constant vaccination strategy could be the administration of a large number of doses on "immunization days" with the aim of maintaining the basic reproduction number to be below one. This strategy, known as pulse vaccination, has been successfully applied for the control of many diseases especially in low-income countries. In this paper, we analytically prove (without being computer-aided) the existence of chaotic dynamics in the classical SIR model with pulse vaccination. To the best of our knowledge, this is the first time in which a theoretical proof of chaotic dynamics is given for an epidemic model subject to pulse vaccination. In a realistic public health context, our analysis suggests that the combination of an insufficient vaccination coverage and high birth rates could produce chaotic dynamics and an increment of the number of infectious individuals.

Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium maxwell distributions
Andrea Bondesan, Laurent Boudin, Marc Briant and Bérénice Grec
2020, 19(5): 2549-2573 doi: 10.3934/cpaa.2020112 +[Abstract](123) +[HTML](63) +[PDF](701.99KB)
Abstract:

We consider the Boltzmann operator for mixtures with cutoff Maxwellian, hard potential, or hard-sphere collision kernels. In a perturbative regime around the global Maxwellian equilibrium, the linearized Boltzmann multi-species operator \begin{document}$ \mathbf{L} $\end{document} is known to possess an explicit spectral gap \begin{document}$ \lambda_{ \mathbf{L}} $\end{document}, in the global equilibrium weighted \begin{document}$ L^2 $\end{document} space. We study a new operator \begin{document}$ \mathbf{ L^{\varepsilon}} $\end{document} obtained by linearizing the Boltzmann operator for mixtures around local Maxwellian distributions, where all the species evolve with different small macroscopic velocities of order \begin{document}$ \varepsilon $\end{document}, \begin{document}$ \varepsilon >0 $\end{document}. This is a non-equilibrium state for the mixture. We establish a quasi-stability property for the Dirichlet form of \begin{document}$ \mathbf{ L^{\varepsilon}} $\end{document} in the global equilibrium weighted \begin{document}$ L^2 $\end{document} space. More precisely, we consider the explicit upper bound that has been proved for the entropy production functional associated to \begin{document}$ \mathbf{L} $\end{document} and we show that the same estimate holds for the entropy production functional associated to \begin{document}$ \mathbf{ L^{\varepsilon}} $\end{document}, up to a correction of order \begin{document}$ \varepsilon $\end{document}.

Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation
Suting Wei and Jun Yang
2020, 19(5): 2575-2616 doi: 10.3934/cpaa.2020113 +[Abstract](117) +[HTML](70) +[PDF](675.56KB)
Abstract:

We consider the nonlinear problem of inhomogeneous Allen-Cahn equation

where \begin{document}$ \Omega $\end{document} is a bounded domain in \begin{document}$ \mathbb R^2 $\end{document} with smooth boundary, \begin{document}$ \epsilon $\end{document} is a small positive parameter, \begin{document}$ \nu $\end{document} denotes the unit outward normal of \begin{document}$ \partial \Omega $\end{document}, \begin{document}$ V $\end{document} is a positive smooth function on \begin{document}$ \bar\Omega $\end{document}. Let \begin{document}$ \Gamma\subset\Omega $\end{document} be a smooth curve dividing \begin{document}$ \Omega $\end{document} into two disjoint regions and intersecting orthogonally with \begin{document}$ \partial\Omega $\end{document} at exactly two points \begin{document}$ P_1 $\end{document} and \begin{document}$ P_2 $\end{document}. Moreover, by considering \begin{document}$ {\mathbb R}^2 $\end{document} as a Riemannian manifold with the metric \begin{document}$ g = V(y)\,({\mathrm d}{y}_1^2+{\mathrm d}{y}_2^2) $\end{document}, we assume that: the curve \begin{document}$ \Gamma $\end{document} is a non-degenerate geodesic in the Riemannian manifold \begin{document}$ ({\mathbb R}^2, g) $\end{document}, the Ricci curvature of the Riemannian manifold \begin{document}$ ({\mathbb R}^2, g) $\end{document} along the normal \begin{document}$ \mathbf{n} $\end{document} of \begin{document}$ \Gamma $\end{document} is positive at \begin{document}$ \Gamma $\end{document}, the generalized mean curvature of the submanifold \begin{document}$ \partial\Omega $\end{document} in \begin{document}$ ({\mathbb R}^2, g) $\end{document} vanishes at \begin{document}$ P_1 $\end{document} and \begin{document}$ P_2 $\end{document}. Then for any given integer \begin{document}$ N\geq 2 $\end{document}, we construct a solution exhibiting \begin{document}$ N $\end{document}-phase transition layers near \begin{document}$ \Gamma $\end{document} (the zero set of the solution has \begin{document}$ N $\end{document} components, which are curves connecting \begin{document}$ \partial\Omega $\end{document} and directed along the direction of \begin{document}$ \Gamma $\end{document}) with mutual distance \begin{document}$ O(\epsilon|\log \epsilon|) $\end{document}, provided that \begin{document}$ \epsilon $\end{document} stays away from a discrete set of values to avoid the resonance of the problem. Asymptotic locations of these layers are governed by a Toda system.

Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle
Jeongmin Han
2020, 19(5): 2617-2640 doi: 10.3934/cpaa.2020114 +[Abstract](100) +[HTML](53) +[PDF](435.15KB)
Abstract:

We prove in this article that functions satisfying a dynamic programming principle have a local interior Lipschitz type regularity. This DPP is partly motivated by the connection to the normalized parabolic \begin{document}$ p $\end{document}-Laplace operator.

Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution
Anupam Sen and T. Raja Sekhar
2020, 19(5): 2641-2653 doi: 10.3934/cpaa.2020115 +[Abstract](145) +[HTML](61) +[PDF](472.24KB)
Abstract:

We study the interactions between classical elementary waves and delta shock wave in quasilinear hyperbolic system of conservation laws. This governing system describes a thin film of a perfectly soluble anti-surfactant solution in the limit of large capillary and P\begin{document}$ \acute{e} $\end{document}clet numbers. This system is one of the example of non-strictly hyperbolic system whose Riemann solution consists of delta shock wave as well as classical elementary waves such as shock waves, rarefaction waves and contact discontinuities. The global structure of the perturbed Riemann solutions are constructed and analyzed case by case when delta shock wave is involved.

Existence of weak solution for mean curvature flow with transport term and forcing term
Keisuke Takasao
2020, 19(5): 2655-2677 doi: 10.3934/cpaa.2020116 +[Abstract](107) +[HTML](63) +[PDF](502.62KB)
Abstract:

We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove the global existence of the weak solutions for the mean curvature flow with the terms, by using the modified Allen-Cahn equation that holds useful properties such as the monotonicity formula.

$ BV $ functions on open domains: the Wiener case and a Fomin differentiable case
Davide Addona, Giorgio Menegatti and Michele Miranda jr.
2020, 19(5): 2679-2711 doi: 10.3934/cpaa.2020117 +[Abstract](108) +[HTML](54) +[PDF](581.52KB)
Abstract:

We provide three different characterizations of the space \begin{document}$ BV(O, \gamma) $\end{document} of the functions of bounded variation with respect to a centred non-degenerate Gaussian measure \begin{document}$ \gamma $\end{document} on open domains \begin{document}$ O $\end{document} in Wiener spaces. Throughout these different characterizations we deduce a sufficient condition in order to belong to \begin{document}$ BV(O, \gamma) $\end{document} by means of the Ornstein-Uhlenbeck semigroup and we provide an explicit formula for one-dimensional sections of functions of bounded variation. Finally, we apply our techniques to Fomin differentiable probability measures \begin{document}$ \nu $\end{document} on a Hilbert space \begin{document}$ X $\end{document}, and we infer a characterization of the space \begin{document}$ BV(O, \nu) $\end{document} of the functions of bounded variation with respect to \begin{document}$ \nu $\end{document} on open domains \begin{document}$ O\subseteq X $\end{document}.

Low modes regularity criterion for a chemotaxis-Navier-Stokes system
Mimi Dai and Han Liu
2020, 19(5): 2713-2735 doi: 10.3934/cpaa.2020118 +[Abstract](103) +[HTML](55) +[PDF](434.75KB)
Abstract:

In this paper we study the regularity problem of a three dimensional chemotaxis-Navier-Stokes system. A new regularity criterion in terms of only low modes of the oxygen concentration and the fluid velocity is obtained via a wavenumber splitting approach. The result improves certain existing criteria in the literature.

Asymptotic analysis for 1D compressible Navier-Stokes-Vlasov equations
Haibo Cui, Wenjun Wang and Lei Yao
2020, 19(5): 2737-2750 doi: 10.3934/cpaa.2020119 +[Abstract](113) +[HTML](64) +[PDF](422.38KB)
Abstract:

In this paper, we study the asymptotic analysis of 1D compressible Navier-Stokes-Vlasov equations. By taking advantage of the one space dimension, we obtain the hydrodynamic limit for compressible Navier-Stokes-Vlasov equations with the pressure \begin{document}$ P(\rho) = A\rho^{\gamma} $\end{document} \begin{document}$ (\gamma>1) $\end{document}. The proof relies on weak convergence method.

Limiting dynamical behavior of random fractional FitzHugh-Nagumo systems driven by a Wong-Zakai approximation process
Dingshi Li, Xiaohu Wang and Junyilang Zhao
2020, 19(5): 2751-2776 doi: 10.3934/cpaa.2020120 +[Abstract](110) +[HTML](58) +[PDF](532.19KB)
Abstract:

In this paper, we study the long term behavior of non-autonomous fractional FitzHugh-Nagumo systems with random forcing given by an approximation of white noise, called Wong-Zakai approximation. We first prove the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximation fractional FitzHugh-Nagumo systems, and then establish the upper semicontinuity of attractors of system driven by a linear multiplicative Wong-Zakai approximations as random forcing approaches white noise in some sense.

$ L^{p(\cdot)} $-regularity of Hessian for nondivergence parabolic and elliptic equations with measurable coefficients
Junjie Zhang, Shenzhou Zheng and Haiyan Yu
2020, 19(5): 2777-2796 doi: 10.3934/cpaa.2020121 +[Abstract](115) +[HTML](60) +[PDF](472.41KB)
Abstract:

We prove the interior \begin{document}$ L^{p(\cdot)} $\end{document}-estimates for the Hessian of strong solutions to nondivergence parabolic equations \begin{document}$ u_{t}(x,t)-a_{ij}(x,t)D_{ij}u(x,t) = f(x,t) $\end{document} and elliptic equations \begin{document}$ a_{ij}(x)D_{ij}u(x) = f(x) $\end{document}, respectively. Besides a natural assumption that \begin{document}$ p(\cdot) $\end{document} is \begin{document}$ \log $\end{document}-Hölder continuous, we also assume that the coefficients \begin{document}$ a_{ij}(x,t) $\end{document} and \begin{document}$ a_{ij}(x) $\end{document} are merely measurable in one of spatial variables and have small BMO semi-norms with respect to other variables.

Connecting orbits in Hilbert spaces and applications to P.D.E
Panayotis Smyrnelis
2020, 19(5): 2797-2818 doi: 10.3934/cpaa.2020122 +[Abstract](112) +[HTML](55) +[PDF](555.47KB)
Abstract:

We prove a general theorem on the existence of heteroclinic orbits in Hilbert spaces, and present a method to reduce the solutions of some P.D.E. problems to such orbits. In our first application, we give a new proof in a slightly more general setting of the heteroclinic double layers (initially constructed by Schatzman [20]), since this result is particularly relevant for phase transition systems. In our second application, we obtain a solution of a fouth order P.D.E. satisfying similar boundary conditions.

Ground state solution and multiple solutions to elliptic equations with exponential growth and singular term
Yanjun Liu and Chungen Liu
2020, 19(5): 2819-2838 doi: 10.3934/cpaa.2020123 +[Abstract](141) +[HTML](63) +[PDF](474.75KB)
Abstract:

In this paper, we consider the following elliptic problem

and its perturbation problem, where \begin{document}$ N\geq 2 $\end{document}, \begin{document}$ 0<\eta<N $\end{document}, \begin{document}$ V(x) \geq V_{0 }> 0 $\end{document} and \begin{document}$ f(x, t) $\end{document} has a critical exponential growth behavior. By using the variational technique and the indirection method, the existence of a positive ground state solution is proved. For the perturbation problem, the existence of two distinct nontrivial weak solutions is proved.

Nonexistence results on the space or the half space of $ -\Delta u+\lambda u = |u|^{p-1}u $ via the Morse index
Abdelbaki Selmi, Abdellaziz Harrabi and Cherif Zaidi
2020, 19(5): 2839-2852 doi: 10.3934/cpaa.2020124 +[Abstract](131) +[HTML](64) +[PDF](426.92KB)
Abstract:

In this paper we consider the following semi-linear elliptic problem

where \begin{document}$ \mathcal{O} = \mathbb{R}^N $\end{document}; or \begin{document}$ \mathcal{O} = \mathbb{R}^N_+ = \{x = (x',x_N),\, x'\in \mathbb{R}^{N-1},x_N>0\} $\end{document} with Dirichlet boundary conditions. Here \begin{document}$ N\geq2 $\end{document}, \begin{document}$ p>1 $\end{document} and \begin{document}$ \lambda $\end{document} is a positive real parameter. The main goal ofthis work is to analyze the influence of the linear term \begin{document}$ \lambda u $\end{document}, in order to classify regular stable solutions possibly unbounded and sign-changing. Our analysis reveals the nonexistence of nontrivial stable solutions (respectively solutions which are stable outside a compact set) for all \begin{document}$ p> 1 $\end{document} (respectively for all \begin{document}$ p\geq \frac{N+2}{N-2} $\end{document}, or \begin{document}$ 1<p<\frac{N+2}{N-2} $\end{document} and \begin{document}$ |u|^{p-1}<\frac{\lambda (p+1)}{2} $\end{document}). Inspired by [6,9,16,23], we establish a monotonicity formula to discuss the supercritical case.

Regarding the case \begin{document}$ \mathcal{O} = \mathbb{R}^N $\end{document}, we obtain a complete classification which states that problem \begin{document}$ (P) $\end{document} has regular solutions which are stable outside a compact set if and only if \begin{document}$ p\in (1,\infty) $\end{document} and \begin{document}$ N = 2 $\end{document}; or \begin{document}$ p\in(1,\frac{N+2}{N-2}) $\end{document} and \begin{document}$ N\geq3. $\end{document}

Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay
Jingdong Wei, Jiangbo Zhou, Wenxia Chen, Zaili Zhen and Lixin Tian
2020, 19(5): 2853-2886 doi: 10.3934/cpaa.2020125 +[Abstract](146) +[HTML](112) +[PDF](561.82KB)
Abstract:

In this paper, we investigate the existence and nonexistence of traveling wave solutions in a nonlocal dispersal epidemic model with spatio-temporal delay. It is shown that this model admits a nontrivial positive traveling wave solution when the basic reproduction number \begin{document}$ R_0>1 $\end{document} and the wave speed \begin{document}$ c\geq c^* $\end{document} (\begin{document}$ c^* $\end{document} is the critical speed) and this model has no traveling wave solutions when \begin{document}$ R_0\leq1 $\end{document} or \begin{document}$ c<c^* $\end{document}. This indicates that \begin{document}$ c^* $\end{document} is the minimal wave speed.

Existence of ground state solution and concentration of maxima for a class of indefinite variational problems
Claudianor O. Alves and Geilson F. Germano
2020, 19(5): 2887-2906 doi: 10.3934/cpaa.2020126 +[Abstract](132) +[HTML](54) +[PDF](461.72KB)
Abstract:

In this paper we study the existence of ground state solution and concentration of maxima for a class of strongly indefinite problem like

where \begin{document}$ N \geq 1 $\end{document}, \begin{document}$ \epsilon $\end{document} is a positive parameter, \begin{document}$ f: \mathbb{R} \to \mathbb{R} $\end{document} is a continuous function with subcritical growth and \begin{document}$ V,A: \mathbb{R}^{N} \to \mathbb{R} $\end{document} are continuous functions verifying some technical conditions. Here \begin{document}$ V $\end{document} is a \begin{document}$ \mathbb{Z}^N $\end{document}-periodic function, \begin{document}$ 0 \not\in \sigma(-\Delta + V) $\end{document}, the spectrum of \begin{document}$ -\Delta +V $\end{document}, and

Weak-strong uniqueness of incompressible magneto-viscoelastic flows
Wenjing Zhao
2020, 19(5): 2907-2917 doi: 10.3934/cpaa.2020127 +[Abstract](121) +[HTML](56) +[PDF](347.72KB)
Abstract:

Our aim in this paper is to prove the weak-strong uniqueness property of solutions to a hydrodynamic system that models the dynamics of incompressible magneto-viscoelastic flows. The proof is based on the relative energy approach for the compressible Navier-Stokes system.

Bôcher-type results for the fourth and higher order equations on singular manifolds with conical metrics
Fangshu Wan
2020, 19(5): 2919-2948 doi: 10.3934/cpaa.2020128 +[Abstract](123) +[HTML](63) +[PDF](520.61KB)
Abstract:

We obtain the Bôcher-type theorems and present the sharp characterization of the asymptotic behavior at the isolated singularities of solutions of some fourth and higher order equations on singular manifolds with conical metrics. It is seen that the equations on singular manifolds with conical metrics are equivalent to weighted elliptic equations in \begin{document}$ B \backslash \{0\} $\end{document}, where \begin{document}$ B \subset \mathbb{R}^N $\end{document} is the unit ball. The weights can be singular at \begin{document}$ x = 0 $\end{document}. We present the sharp asymptotic behavior of nonnegative solutions of the weighted elliptic equations near \begin{document}$ x = 0 $\end{document} and the Liouville-type results for the degenerate elliptic equations in \begin{document}$ \mathbb{R}^N \backslash \{0\} $\end{document}.

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