Blow-up And Global Existence For Two Classes Of Nonlinear Parabolic Problems

The purpose of this paper is to discuss two problems, one of which is the blow-up solu-tions, global existence and exponential decay estimates for a class of second order parabolic problems with Dirichlet boundary conditions, and the other is the blow-up solutions for a class of nonlinear parabolic problems with homogeneous Neumann boundary conditions. The methods employed are mainly auxiliary function method, maximum principle method and the first-order differential inequality technique.This paper includes three chapters.In chapter 1, firstly, we provide a simple research summary of the global existence and blow-up problems for nonlinear parabolic problems. Then we present maximum principles and comparison principles for nonlinear parabolic problems, which are applied in this paper.In chapter 2, the type of problem under consideration is: where D(?)RN is a bounded convex domain with smooth boundary (?)D, N≥2,0< T<+∞. By constructing auxiliary functions and using maximum principles, the sufficient conditions for the existence of the blow-up solution, the sufficient conditions for the global existence of the solution, an upper bound for the "blow-up time", and some explicit expo-nential decay bounds for the solution and its derivatives are obtained under some suitable assumptions on k, g, f and initial date uo.In chapter 3, we study the blow-up solutions for the following nonlinear parabolic problems with homogeneous Neumann boundary conditions: where D(?)RN is assumed to be bounded, starshaped, convex in two orthogonal directions and with smooth boundary (?)D, N≥2. By using comparison principles and the first-order differential inequality techniques, the sufficient conditions for the existence of the blow-up solutions, an upper bound of the blow-up time, and a lower bound of the blow-up time are obtained under some appropriate assumptions on the functions k,ρ, f and initial data uo.