Blow-up Analysis Of Solutions To Some Nonlocal Parabolic Equations

Nonlocal parabolic equations have important applications in physical problems such as thermoelasticity theory and thermistors.In this thesis,we consider the blow-up phe-nomena of solutions for some typical nonlocal parabolic equations.By constructing some suitable auxiliary functions,using the differential inequality technique and the embedding theorems in Sobolev spaces,we obtain a criterion to guarantee the existence of global solution or blow-up solution.When blow-up does occur,the estimate of upper of the blow-up time are derived.By using the embedding theorems in Sobolev spaces,we get an estimate of the lower bound.Since the upper and lower bounds of the blow-up time can give a controllable time,the research in this thesis has practical significance.Moreover,some examples are given to illustrate our abstract results.This thesis is divided into the following four chapters.In Chapter 1,we introduce relevant historical backgrounds,significance and latest advances of the problem of global existence or blow up of solutions to the nonlinear parabolic equations.We also show some important lemmas that we used in the thesis.In Chapter 2,we consider the following blow-up problem of the solution to the reac-tion diffusion equations with nonlocal boundary conditions:where D is a bounded convex region in Rn(n?2),and the boundary(?)D is smooth.By constructing some auxiliary functions,using differential inequality technique and Sobolev inequalities,we derive that the solution blows up at some finite time.Further,upper and lower bounds of the blow-up time are obtained.The purpose of Chapter 3 is to deal with the blow-up problems of the following p-Laplacian parabolic equations with nonlocal boundary conditions:where p>2,D(?)Rn(n?2)is a bounded convex region,and the boundary(?)D is smooth.With the help of differential inequality technique and Sobolev inequalities,we prove that the blow-up does occur on some certain conditions of the data.In addition,we obtain upper bounds and lower bounds of the blow-up time.In Chapter 4,We investigate the following nonlinear parabolic equations with non-local source and nonlinear boundary conditions:where p and ?1 are nonnegative constants,m,l,?2,and r are some positive constants,D(?)Rn(n?2)is a bounded convex region with smooth boundary(?)D.By making use of differential inequality technique,the embedding theorems in Sobolev spaces and constructing some auxiliary functions,we derive that blow-up solution and global existence on some appropriate conditions on the data.Furthermore,an upper bound and a lower bound for the blow-up time are obtained.