Blow-up Theory And Maximum Principle For Some Nonlinear Partial Differential Equations

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.