【作者】 丁俊堂；

【导师】 李胜家；

【作者基本信息】 山西大学 ， 基础数学， 2006， 博士

【摘要】 在本文中,我们主要讨论两类非线性方面的内容,其一为非线性抛物方程的爆破理论,其二为非线性椭圆方程和非线性抛物方程的最大值原理,所使用的方法主要是辅助函数法、极值原理法、上下解法和凸函数法等。 全文共分为六章。 在第一章中,我们首先简述了非线性抛物方程爆破理论的研究进展,然后简要回顾了非线性椭圆方程和非线性抛物方程的最大值原理的研究情况,最后列出了本文所使用的一些重要定理。 在第二章中,考虑的问题是 （?） 这里D是R^N中的光滑有界区域,N≥2,在对a,b,f,g,σ和初值u₀（x）做合适的假设之下,给出了爆破解的存在性定理、“爆破时刻”的上界估计、“爆破率”的上估计、整体解的存在性定理、整体解的上估计获得的结果被用到一些a,b,f,g和σ是幂函数或指数函数的例子。 在第三章中,我们研究下列问题: （?） 这里D是R^N中的光滑有界区域,N≥2,q=|▽u|²,在对a,b,f,g,σ和初值u₀（x）做合适的假设之下,给出了爆破解的存在性定理、“爆破时刻”的上界估计、“爆破率”的上估计.获得的结果被用到一些a,6,f,g和σ是幂函数或指数函数的例子。 在第四章中,我们考虑了如下问题: （?）更多还原

【Abstract】 The purpose of this paper is to discuss two classes of nonlinear problems, one of which is the blow-up theory for nonlinear parabolic equations and the other is the maximum principles for nonlinear elliptic equations and nonlinear parabolic equations. The methods employed are mainly auxiliary function method, maximum principle method, super- and sub-solution method, convex function method and so on.This paper includes six chapters.In chapter 1, firstly, we provide a simple research summary of the blow-up theory for nonlinear parabolic equations. Secondly, we recall the study case of the maximum principles for nonlinear parabolic equations and nonlinear elliptic equations. Finially, we present some important theorems which are applied in this paper.In chapter 2, the type of problem under consideration iswhere D is a smooth bounded domain of R^N, N≥2. The existence theorems of blowup solutions, upper bound of "blow-up time", upper estimates of "blow-up rate", existence theorems of global solutions, and upper estimates of global solutions are given under suitable assumptions on a,b,f,g,σ, and initial data u₀（x）. The obtained results are applied to some examples in which a, b, f, g, and σ are power functions or exponential functions.In chapter 3, we study the following problem,where D is a smooth bounded domain of R^N, N≥2, q=|（?）u|². The existence theorems of blow-up solutions, upper bound of "blow-up time", and upper estimates of "blowup rate" are given under suitable assumptions on a,b,f,g,σ, and initial date u₀（x）.The obtained results are applied to some examples in which a, b, f, g, and σ are power functions or exponential functions.In chapter 4, we consider the following problem,where D is a smooth bounded domain of R^N, N≥ 2, q = | （?）u|². The existence theorems of smooth blow-up solutions and weak blow-up solutions, upper bound of "blow-up time", and upper estimates of "blow-up rate" are given under suitable assumptions on a, f, and initial date u₀（x）. The obtained results are applied to some examples in which a and f are power functions or exponential functions. In chaper 5, we research into the following three problems,where D is a smooth bounded domain of R^N, N≥2, q= |（?）u|². We obtain maximum principles for functions which are defined on solutions of the three problems respectively. By means of the maximum principles derived, the estimation of gradient q and the estimation of the solution u are given.In chaper 6, we research into the following three problems,where D is a smooth bounded domain of RN, N>2, g=|Vw|2. We obtain maximum principles for functions which are defined on solutions of the three problems respectively. By means of the maximum principles derived, the estimation of gradient q are given.更多还原