几类非线性抛物方程（组）解的性质研究

发布时间：2018-01-03 17:19

本文关键词：几类非线性抛物方程（组）解的性质研究　出处：《江苏大学》2016年博士论文　论文类型：学位论文

【摘要】：非线性抛物方程作为偏微分方程中的一类重要方程,在物理学、化学和生物学等学科都有其体现,在渗流理论、相变理论、图象处理等领域也都有其具体的数学模型.因此,对非线性抛物方程解的性质研究可解释诸多的自然现象,而且也能丰富偏微分方程的理论.本文主要研究几类非线性退化抛物方程(组)解的性质,讨论了具有不同源项的抛物方程(组)在不同边界值条件下解的局部存在性、解的整体存在性、解在有限时刻爆破及解的渐近行为.全文分为八章:第一章概述本文所研究的背景及其相关工作,并简要介绍了本文的主要内容.第二章研究了一类源项为局部型与范数型乘积的退化抛物方程在齐次Dirichlet边界条件下的解的性质.运用正则化方法证明了非负弱解的局部存在性,根据比较原理得到了解在有限时刻爆破的充分条件,并建立了爆破解的精确爆破速率估计.第三章研究了一类具范数型源项的非线性退化抛物方程组在齐次Dirichlet边界条件下解的整体存在性和有限时刻爆破的问题.利用上下解方法和比较原理,建立了解整体存在和有限时刻爆破的准则.第四章应用与上章类似的方法和技巧,研究了一类n元退化抛物方程组解的问题,它可看作是关于单个方程讨论的推广,并得到了临界爆破指标.第五章研究了一类具有非局部化源和非局部非线性边界条件的非散度型退化抛物方程解的爆破问题.讨论了方程的扩散系数、边界条件中的权函数及非线性指标对方程解性质的影响,给出了解整体存在和在有限时刻爆破的条件.对特殊情形,建立了爆破解的爆破速率估计.第六章将上一章的结果推广到方程组的情形.类似单个方程的研究,通过构造上下解,证明了非负解整体存在和不存在的结果,并给出了特殊情形下爆破速率的估计.第七章,研究了一类带范数型源和非局部边界条件的退化抛物方程组解爆破和整体存在的条件.证明了边界条件上的加权函数对解爆破与否起到了关键的作用,同时给出了爆破速率估计.在本文的最后一章,研究了一类具pL-范数反应项的退化抛物方程组在正边值条件下解的爆破问题.确定了非负解的爆破准则和整体存在性.结果表明,在确定解的爆破中,正的边界值起到了关键作用.
[Abstract]:As a kind of important equations in partial differential equations, nonlinear parabolic equations are embodied in physics, chemistry and biology, in seepage theory and phase transition theory. Image processing and other fields also have their specific mathematical models. Therefore, the study of the properties of nonlinear parabolic equations can explain many natural phenomena. And it can enrich the theory of partial differential equation. In this paper, we mainly study the properties of solutions of some nonlinear degenerate parabolic equations. In this paper, we discuss the local existence and global existence of solutions for parabolic equations (systems) with different homologous terms under different boundary value conditions. This paper is divided into eight chapters: chapter 1 summarizes the background and related work of this paper. In chapter 2, we study the properties of solutions of a class of degenerate parabolic equations with local and norm product under homogeneous Dirichlet boundary conditions. The local existence of nonnegative weak solutions is proved by the method. According to the principle of comparison, the sufficient conditions for the solution to blow up at finite time are obtained. In chapter 3, we study the global existence and finite time of solutions for a class of nonlinear degenerate parabolic equations with norm-type source terms under homogeneous Dirichlet boundary conditions. The problem of blasting. Using the method of upper and lower solutions and the principle of comparison. In Chapter 4th, the problem of solutions to a class of n-variable degenerate parabolic equations is studied by applying the methods and techniques similar to those in the previous chapter. It can be seen as a generalization of the discussion of a single equation. In chapter 5th, we study the solution of a class of nondivergent degenerate parabolic equations with nonlocal source and nonlocal nonlinear boundary conditions, and discuss the diffusion coefficient of the equation. The influence of the weight function and nonlinear index in boundary condition on the properties of the solution of the equation is given. The conditions for the global existence of the solution and the blow-up at finite time are given. In Chapter 6th, the results of the previous chapter are extended to the case of equations. Similar to the study of single equations, the global existence and non-existence of non-negative solutions are proved by constructing upper and lower solutions. The estimate of blasting rate in special cases is given. Chapter 7th. A class of degenerate parabolic equations with norm source and nonlocal boundary conditions is studied. It is proved that the weighting function on the boundary condition plays a key role in the solution blasting or not. At the same time, the blasting rate estimation is given. In the last chapter of this paper. In this paper, we study the blow-up problem for a class of degenerate parabolic equations with pL- norm reaction term under positive boundary value conditions. The blasting criterion and global existence of nonnegative solutions are determined. Positive boundary values play a key role.
【学位授予单位】：江苏大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.26

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