Communications on Pure & Applied Analysis
May 2020 , Volume 19 , Issue 5
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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.
We derive stability estimates in
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.
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
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.
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.
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
We consider the nonlinear problem of inhomogeneous Allen-Cahn equation
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
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
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.
We provide three different characterizations of the space
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.
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
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.
We prove the interior
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 [
In this paper, we consider the following elliptic problem
and its perturbation problem, where
In this paper we consider the following semi-linear elliptic problem
Regarding the case
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
In this paper we study the existence of ground state solution and concentration of maxima for a class of strongly indefinite problem like
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.
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
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