非线性反应扩散方程及其定常问题解的研究
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摘要
本文主要研究非线性反应扩散方程及其定常问题解的性质.非线性反应扩散方程来自于物理、化学和生物等科学领域,具有强烈的实际背景.因此,对于它们的研究具有科学意义和潜在的应用价值.本文将对几类非线性反应扩散方程解的整体存在性、大时间性态及其定常问题解的稳定性作一些探讨.主要内容安排如下：
第一章简述非线性反应扩散方程及其定常问题的研究背景及本文的主要工作.
第二章,我们讨论如下Robin边值问题：运用上下解方法和临界点理论证明了问题(1)的多解性.具体地,当f(x)满足某些条件时,存在一个正数βf*使得当β∈(0,βf*)时,问题(1)没有正解；当β≥βf*时,问题(1)至少有两个正解,且有唯一极小解.
第三章,我们研究一类带Robin边界条件的非线性反应扩散方程整体解的存在性及大时间渐近性态.我们考虑对于任意给定的初值,其对应的整体解当时间趋于无穷大时是否趋于某个平衡态.运用整体解的先验估计和所对应的平衡态问题的解的相交性,证明了这类问题的极小平衡解是稳定的,而不同于极小平衡解的其它平衡解是其整体解存在与否的初值门槛.
在第四章中,我们研究了Rn中有界闭区域上非线性齐次和非齐次反应扩散方程组Dirichlet问题、Robin问题解的性态,将第三章中关于单个方程的门槛结果推广到方程组的门槛结果,证明了当非线性指标满足一定条件时,方程组整体解存在和不存在的结果,得到了方程组整体解存在与否的门槛结果.
第五章对一类空间分布不均匀的捕食模型进行探讨,建立了正平衡解上下界的精确估计,从而,运用拓扑度理论及椭圆方程的比较原理证明了非常数正平衡解的存在性和不存在性.
In this thesis, we study the properties of solution for nonlinear reaction-diffusion equation and its stationary problem. Nonlinear reaction-diffusion equa-tion involved in a number of issues from the physics, chemistry and mathematical model of the biological field, with a strong practical background. Thus, it has scientific significance and potential applications of putting forward all kinds of problem for nonlinear reaction-diffusion equation in mathematics and other natu ral sciences. We will study on the existence of global solution, la rge time behav-ior for several classes of nonlinear reaction-diffusion equation and the stability of solutions for stationary problem of those nonlinear reaction-diffusion equation in this article. The main contents are organized as follows:
In chapter1, we state the background and development of nonlinear reaction-diffusion equation and its stationary problem, and the main work of this article.
In chapter2, we consider the following Robin problem: Using upper and lower solution method, variation method, we prove the multiplic-ity of solution for problem (1). Specifically, suppose f(x) satisfies some condition, there exist a positive number βf*such that problem (1) has no positive solution when βε (0,βf*), and has at least two positive solutions when β≥βf*, of which there is a unique minimal solution.
Chapter3deal with the existence of global solutions and large time asymptotic behavior of a class of nonlinear reaction-diffusion equation with Robin boundary condition. We concern for any given initial value, whether global solutions tends to an equilibrium state or not when the time tends to infinity. By the a priori estimate of global solution and the intersection property for solutions of stationary problem, we obtain that the minimal stationary solution is stable, whereas, any other stationary solution is an initial datum threshold for the existence and non-existence of its global solutions.
In chapter4. we study the behavior of solutions for Dirichlet problem of non-linear homogeneous and non-homogeneous reaction-diffusion equations, the nonlin-ear reaction-diffusion equations with Robin boundary condition in bounded closed region of R" respectively. We generalize threshold results of a single equation in chapter3to the threshold of equations and obtain a threshold result when the nonlinear index meet certain conditions.
Chapter5concerns a class of uneven spatial distribution of the predator-prey model, we prove the accurate a priori estimate of the positive equilibrium solution of the upper and lower bounds. Thus, the existence and nonexistence of noncon-stant positive equilibrium solution by topological degree theory and comparison principle of elliptic equations.
第一章简述非线性反应扩散方程及其定常问题的研究背景及本文的主要工作.
第二章,我们讨论如下Robin边值问题：运用上下解方法和临界点理论证明了问题(1)的多解性.具体地,当f(x)满足某些条件时,存在一个正数βf*使得当β∈(0,βf*)时,问题(1)没有正解；当β≥βf*时,问题(1)至少有两个正解,且有唯一极小解.
第三章,我们研究一类带Robin边界条件的非线性反应扩散方程整体解的存在性及大时间渐近性态.我们考虑对于任意给定的初值,其对应的整体解当时间趋于无穷大时是否趋于某个平衡态.运用整体解的先验估计和所对应的平衡态问题的解的相交性,证明了这类问题的极小平衡解是稳定的,而不同于极小平衡解的其它平衡解是其整体解存在与否的初值门槛.
在第四章中,我们研究了Rn中有界闭区域上非线性齐次和非齐次反应扩散方程组Dirichlet问题、Robin问题解的性态,将第三章中关于单个方程的门槛结果推广到方程组的门槛结果,证明了当非线性指标满足一定条件时,方程组整体解存在和不存在的结果,得到了方程组整体解存在与否的门槛结果.
第五章对一类空间分布不均匀的捕食模型进行探讨,建立了正平衡解上下界的精确估计,从而,运用拓扑度理论及椭圆方程的比较原理证明了非常数正平衡解的存在性和不存在性.
In this thesis, we study the properties of solution for nonlinear reaction-diffusion equation and its stationary problem. Nonlinear reaction-diffusion equa-tion involved in a number of issues from the physics, chemistry and mathematical model of the biological field, with a strong practical background. Thus, it has scientific significance and potential applications of putting forward all kinds of problem for nonlinear reaction-diffusion equation in mathematics and other natu ral sciences. We will study on the existence of global solution, la rge time behav-ior for several classes of nonlinear reaction-diffusion equation and the stability of solutions for stationary problem of those nonlinear reaction-diffusion equation in this article. The main contents are organized as follows:
In chapter1, we state the background and development of nonlinear reaction-diffusion equation and its stationary problem, and the main work of this article.
In chapter2, we consider the following Robin problem: Using upper and lower solution method, variation method, we prove the multiplic-ity of solution for problem (1). Specifically, suppose f(x) satisfies some condition, there exist a positive number βf*such that problem (1) has no positive solution when βε (0,βf*), and has at least two positive solutions when β≥βf*, of which there is a unique minimal solution.
Chapter3deal with the existence of global solutions and large time asymptotic behavior of a class of nonlinear reaction-diffusion equation with Robin boundary condition. We concern for any given initial value, whether global solutions tends to an equilibrium state or not when the time tends to infinity. By the a priori estimate of global solution and the intersection property for solutions of stationary problem, we obtain that the minimal stationary solution is stable, whereas, any other stationary solution is an initial datum threshold for the existence and non-existence of its global solutions.
In chapter4. we study the behavior of solutions for Dirichlet problem of non-linear homogeneous and non-homogeneous reaction-diffusion equations, the nonlin-ear reaction-diffusion equations with Robin boundary condition in bounded closed region of R" respectively. We generalize threshold results of a single equation in chapter3to the threshold of equations and obtain a threshold result when the nonlinear index meet certain conditions.
Chapter5concerns a class of uneven spatial distribution of the predator-prey model, we prove the accurate a priori estimate of the positive equilibrium solution of the upper and lower bounds. Thus, the existence and nonexistence of noncon-stant positive equilibrium solution by topological degree theory and comparison principle of elliptic equations.
引文
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