A Note on a Class of Strongly Coupled Semilinear Reaction-diffusion Equations

A class of strongly coupled semilinear reaction-diffusion equations is discussed. The FitzHugh-Nagumo equation in [1] is only an example. The results that strongly coupled term generates an analytic semigroup extended the results in [4]. The results of the existence and uniqueness of solution and the stability of zero solution improved that in [1]. This paper also discusses the existence of periodic solution.     

关于半线性椭圆型方程爆破解的存在性

文中得到半线性椭圆型方程的爆破问题解的存在性,其中或者是Rn中的有界区域,C3,C4,C5,C6是正常数,并且C5,C3（0,1）．     

Global Existence and Nonexistence for a Strongly Coupled Parabolic System with Nonlinear Boundary Conditions

This paper deals with the strongly coupled parabolic system ut = v^m△u, vt = u^n△v, （x, t） ∈Ω × （0,T） subject to nonlinear boundary conditions 偏du/偏dη = u^αv^p, 偏du/偏dη= u^qv^β, （x, t） ∈ 偏dΩ × （0, T）, where Ω 包含 RN is a bounded domain, m, n are positive constants and α,β, p, q are nonnegative constants. Global existence and nonexistence of the positive solution of the above problem are studied and a new criterion is established. It is proved that the positive solution of the above problem exists globally if and only if α 〈 1,β 〈 1 and （m ＋p）（n ＋ q） ≤ （1 - α）（1 -β）.     

A Priori Estimates and Existence of Positive Solutions for a Quasilinear Elliptic Equation

On the basis of some new Liouville theorems, under suitableconditions, a priori estimates are obtained of positive solutionsof the problem where R^N (N 2) is a bounded smooth domain, p>1 and isa parameter, , q are given constants such that p–1<<p*–1, <q, p*=Np/(N–p) if N > p and p*= when N p, and a(x) is a continuous nonnegative function. Makinguse of the Leray–Schauder degree of a compact mappingand a priori estimates, the paper finds that the problem abovepossesses at least one positive solution. It also discussesthe corresponding perturbed problem, where a(x) is replacedby a(x)+, >0. The results are strikingly different from thoseobtained for the case =p–1.     

半线性椭圆型问题爆炸解的存在性与渐近行为

设Ω是RN（N≥3）中的C2有界区域,f是单调非减的非负连续可微函数满足f＇（a）∫a∞1/f（s）ds≤C0,　a>0.应用一种新型的非线性变换w（x）=∫u（x）∞　ds/f（s）将爆炸解问题△u=k（x）f（u）,u>0,x∈Ω,u|　Ω=∞转化成等价的带奇异项的Dirichlet问题,不仅得到了爆炸解问题解的最小爆炸速度,而且揭示了两类典型非线性爆炸解问题基本上是相同的.应用摄动方法,上下解方法得到了爆炸解的存在性.特别允许非线性项的系数不仅在Ω的内部子区域恒为零而且在Ω上可适当无界.随后再应用摄动方法,将所得结果推广到无界区域,得到了整体爆炸解的存在性以及在无穷远附近的最小爆炸速度（有关文献参见[1-33]）.     

一类半线性椭圆型方程组的正解

本文利用移动球面法证明了一类半线性椭圆型方程组正解的存在性与不存在性.     

Existence Results for a Class of Semilinear Elliptic Systems

In this paper, we study the existence of nontrivial solutions for the problem
{-△u=f（x,u,v）＋h1（x）in Ω
-△v=g（x,u,v）＋h2（x）inΩ
u=v=0 onδΩ
where Ω is bounded domain in R^N and h1,h2 ∈ L^2 （Ω）. The existence result is obtained by using the Leray-Schauder degree under the following condition on the nonlinearities f and g：
{lim s,｜t｜→＋∞f（x,s,t）/s=lim ｜s｜,t→＋∞g（x,s,t）/t=λ＋1 uniformly on Ω,
lim -s,｜t｜→＋∞f（x,s,t）/s=lim ｜s｜,-t→＋∞g（x,s,t）/t=λ-,uniformly on Ω,
where λ＋,λ-∈（0）∪σ（-△）,σ（-△）denote the spectrum of -△. The cases （i） where λ＋ = λ_ and （ii） where λ＋≠λ_ such that the closed interval with endpoints λ＋,λ_ contains at most one simple eigenvatue of -△ are considered.     

一类拟线性椭圆型偏微分方程的先验界的估计

近几年对边值问题－ｄｉｖ（｜Ｄｕ｜^p-2Ｄｕ）＝λｆ（ｕ）｝在Ω上ｕ｜(?)Ω＝０正解方面已经得到了许多结果．这里λ＞０，Ω是有界区域和对ｓ≥０，ｆ（ｓ）≥０．在本文中在条件Ｎ≥ｐ＞１，Ω＝Ｂ_１＝｛ｘ∈Ｒ^N，｜ｘ｜＜１｝和ｆ∈Ｃ¹（０，∞）∩Ｃ⁰（［０，∞）），ｆ（０）＝０，研究了这类问题的正对称解的先验界估计．     

四阶半线性椭圆变分不等式正解的存在性

本文通过Leray—Schauder度，给出四阶半线性椭圆变分不等式正解的存在性结果．     

Everywhere Regularity of Solutions to a Class of Strongly Coupled Degenerate Parabolic Systems

A class of strongly coupled degenerate parabolic system is considered. Sufficient conditions will be given to show that bounded weak solutions are Hölder continuous everywhere. The general theory will be applied to a generalized porous media type Shigesada-Kawasaki-Teramoto model in population dynamics.     

Existence and Uniqueness of Smooth Solution for a Four-waves Coupled System

In this paper, we consider a four-waves coupled system which describes the interaction between particles. Based on the uniform bound and strong convergence property in lower order norm, local existence and uniqueness of smooth solution is established by a limiting argument. Moreover, we show the solution exists globally in two dimensional case under certain condition on the size for $L^2$ norm of the initial data.     

A Priori Estimates and Existence of Positive Solutions to Quasilinear Elliptic Equations in General Form

In this paper we prove the existence of a positive solution to the following superlinear elliptic Dirichlet problem, - Σ^n_{i,j=1}aij(x, u, Du)D_{ij}u = f(x, u, Du) in Ω, \quad u = 0 on ∂Ω where f satisfies certain growth conditions.     

Existence and Stability of Positive Solutions for an Elliptic Cooperative System

By discussing the properties of a linear cooperative system, the necessary and sufficient conditions for the existence of positive solutions of an elliptic cooperative system in terms of the principal eigenvalue of the associated linear system are established, and some local stability results for the positive solutions are also obtained.     

A Priori Feedback Estimates for Multiscale Reaction-Diffusion Systems

We study the approximation of a multiscale reaction–diffusion system posed on both macroscopic and microscopic space scales. The coupling between the scales is done through micro–macro flux conditions. Our target system has a typical structure for reaction–diffusion flow problems in media with distributed microstructures (also called, double porosity materials). Besides ensuring basic estimates for the convergence of two-scale semi-discrete Galerkin approximations, we provide a set of a priori feedback estimates and a local feedback error estimator that help in designing a distributed-high-errors strategy to allow for a computationally e?cient zooming in and out from microscopic structures. The error control on the feedback estimates relies on two-scale-energy, regularity, and interpolation estimates as well as on a fine bookeeping of the sources responsible with the propagation of the (multiscale) approximation errors. The working technique based on a priori feedback estimates is in principle applicable to a large class of systems of PDEs with dual structure admitting strong solutions.     

Existence and Nonexistence of Entire Positive Solutions of Semilinear Elliptic Systems

We show that entire positive solutions exist for the semilinear elliptic system Δu = p(x)v^α, Δv = q(x)u^β on R^N, N ≥ 3, for positive α and β, provided that the nonnegative functions p and q are continuous and satisfy appropriate decay conditions at infinity. We also show that entire solutions fail to exist if the functions p and q are of slow decay.