Uniform Boundedness for Neutral Equations with Unbouned Delay

We consider a system of neutral equations with unbounded delay, and derive conditions on Liapunov functionals to ensure that the solutions are uniformly bounded and uniformly ultimately bounded.     

Interior estimates for some semilinear elliptic problem with critical nonlinearity

We study compactness properties for solutions of a semilinear elliptic equation with critical nonlinearity. For high dimensions, we are able to show that any solutions sequence with uniformly bounded energy is uniformly bounded in the interior of the domain. In particular, singularly perturbed Neumann equations admit pointwise concentration phenomena only at the boundary.     

一类四阶非自治微分方程解的有界和一致有界性

获得了一个四阶非线性微分方程的所有解是有界和一致有界的充分条件。     

Uniform a priori estimates for positive solutions of the Lane–Emden equation in the plane

We prove that positive solutions of the Lane–Emden equation in a two-dimensional smooth bounded domain are uniformly bounded for all large exponents.     

非线性守恒律高阶谱粘性法的收敛性

讨论守恒型方程周期边界问题的高阶谱粘性方法逼近解的收敛性.在逼近解一致有界的假设下,通过建立其高阶导数的上界估计,证明了高阶谱粘性方法逼近解具有同二阶谱粘性方法逼近解相类似的高频衰减性质.以此为基础,用补偿列紧法证明了高阶谱粘性方法逼近解收敛于守恒型方程的物理解.     

Stability and boundedness of solutions of neutral functional differential equations with finite delay

Sufficient and necessary criteria are established for the uniform stability and uniformly asymptotic stability of solutions of neutral functional differential equations (NFDEs) with finite delay by using the Liapunov functional approach. We also prove that the uniformly asymptotic stability of solutions implies the existence of bounded solution.     

具无限时滞中立型泛函微分方程解的稳定性与有界性

以相空间为基础，研究了具有无限时滞中立型泛函微分方程解的稳定性和有界性，建立了方程解为一致稳定，一致渐近稳定的充要性判据；证明了当方程右端泛函满足Lipschitz条件时，解的一致渐近稳定性蕴涵了有界解的存在性，推广了文献［4-6］中已有的相关结果.     

On a class of second order variational problems with constraints

We study the structure of optimal solutions for a class of constrained, second order variational problems on bounded intervals. We show that, for intervals of length greater than some positive constant, the optimal solutions are bounded inC ¹ by a bound independent of the length of the interval. Furthermore, for sufficiently large intervals, the ‘mass’ and ‘energy’ of optimal solutions are almost uniformly distributed.     

Weak solutions and supersolutions in $ L^1 $ for reaction-diffusion systems

We prove here that limits of nonnegative solutions to reaction-diffusion systems whose nonlinearities are bounded in always converge to supersolutions of the system. The motivation comes from the question of global existence in time of solutions for the wide class of systems preserving positivity and for which the total mass of the solution is uniformly bounded. We prove that, for a large subclass of these systems, weak solutions exist globally. RID="h1" ID="h1"A Philippe, mon ma?tre et ami     

Large time behavior of Euler-Poisson system for semiconductor

In this note,we present a framework for the large time behavior of general uniformly bounded weak entropy solutions to the Cauchy problem of Euler-Poisson system of semiconductor devices.It is shown that the solutions converges to the stationary solutions exponentially in time.No smallness and regularity conditions are assumed.     

The global existence of solutions for two classes of chemotaxis models

This paper is concerned with the global existence and uniform boundedness of solutions for two classes of chemotaxis models in two or three dimensional spaces. Firstly, by using detailed energy estimates, special interpolation relation and uniform Gronwall inequality, we prove the global existence of uniformly bounded solutions for a class of chemotaztic systems with linear chemotactic-sensitivity terms and logistic reaction terms. Secondly, by applying detailed analytic semigroup estimates and special iteration techniques, we obtain the global existence of uniformly bounded solutions for a class of chemotactic systems with nonlinear chemotacticsensitivity terms, which extends the global existence results of [6] to other general cases.     

GLOBAL ENTROPY SOLUTIONS TO AN INHOMOGENEOUS ISENTROPIC COMPRESSIBLE EULER SYSTEM

In this article, we develop a new technique to prove the global existene of entropy solutions to an inhomogeneous isentropic compressible Euler equations through the compensated compactness and vanishing viscosity method. In particular, the entropy solutions are uniformly bounded independent of time.     

Mathematical model for the control of a pest population with impulsive perturbations on diseased pest

In this paper, a stage-structured pest management SI model with impulsive perturbations on infected pest is introduced. Sufficient conditions of the global attractivity of pest-extinction periodic solution and permanence of the system are obtained. We also prove that all solutions of system are uniformly ultimately bounded.     

一类带非局部反应项的抛物方程组全局解的一致有界性

本文研究一类带非局部反应项的抛物方程组全局解的一致有界性,证明了该类方程组全局解关于时间和空间变量都是有界的.     

Global boundedness and stability of solutions for prey-taxis model with handling and searching predators

In this paper, we show the global boundedness and stability of solutions for prey-taxis model with handling and searching predators in a two-dimensional bounded domain with smooth boundary. First, entropy-like equations and boundedness criteria are derived, and it is proved that the system has a unique uniformly bounded global classical solution. In addition, we show that prey-only steady state is globally asymptotically stable if the predator is weak. The convergence rate of solutions to the steady states is derived in the paper.     

A priori estimate for non-uniform elliptic equations

A priori estimate for non-uniform elliptic equations with periodic boundary conditions is concerned. The domain considered consists of two sub-regions, a connected high permeability region and a disconnected matrix block region with low permeability. Let ? denote the size ratio of one matrix block to the whole domain. It is shown that in the connected high permeability sub-region, the Hölder and the Lipschitz estimates of the non-uniform elliptic solutions are bounded uniformly in ?. But Hölder gradient estimate and Lp estimate of the second order derivatives of the solutions in general are not bounded uniformly in ?.     

连串反应中控制火焰的耦合GKS-CGL方程组解的极限行为

本文考虑连串反应中控制火焰的耦合广义Kuramoto Sivashinsky-Ginzburg Landau(GKS-CGL)方程组的周期初值问题,主要研究其解在系数g→0和δ→0时的极限行为.首先,采用Galerkin方法,通过构造一系列精细的先验估计,得到GKS-CGL方程组周期初值问题整体光滑解的存在唯一性.其次,利用一致有界估计证得GKS-CGL方程组极限解收敛,并给出解的收敛率估计.     

On parabolic equations in one space dimension

Several negative results are presented concerning the solvability in Sobolev classes of the Cauchy problem for the inhomogeneous second-order uniformly parabolic equations without lower order terms in one space dimension. The main coefficient is assumed to be a bounded measurable function of (t, x) bounded away from 0. We also discuss upper and lower estimates of certain kind on the fundamental solutions of such equations.     

Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations

The asymptotic behavior of solutions of the three-dimensional Navier-Stokes equations is considered on bounded smooth domains with no-slip boundary conditions and on periodic domains. Asymptotic regularity conditions are presented to ensure that the convergence of a Leray-Hopf weak solution to its weak ω-limit set (weak in the sense of the weak topology of the space H of square-integrable divergence-free velocity fields with the appropriate boundary conditions) are achieved also in the strong topology. It is proved that the weak ω-limit set is strongly compact and strongly attracts the corresponding solution if and only if all the solutions in the weak ω-limit set are continuous in the strong topology of H. Corresponding results for the strong convergence towards the weak global attractor of Foias and Temam are also presented. In this case, it is proved that the weak global attractor is strongly compact and strongly attracts the weak solutions, uniformly with respect to uniformly bounded sets of weak solutions, if and only if all the global weak solutions in the weak global attractor are strongly continuous in H.     

A priori bounds on difference quotients of solutions to some linear uniformly elliptic difference equations

Second order difference quotients of solutions to a class of linear uniformly elliptic difference Dirichlet problems are bounded in terms of quantities which depend on the coefficients of the operator, the inhomogenous term, the boundary values and the domain-which we take to be a rectangle. The results we obtain have theoretical and practical applications.