Research On The Blow-up Problem Of Two Classes Of Nonlinear Parabolic Equations With Nonlinear Boundary Conditions

In this article,we mainly study two problems.The first problem mainly discusses the blow-up for a class of quasilinear parabolic equations with nonlinear boundary conditions.The second problem mainly discusses estimates of blow-up time for a class of porous media equations with gradient terms and nonlinear boundary conditions.Our research mainly relies on the construction of auxiliary functions and the use of first-order differential inequality technique.The full text is divided into three chapters.In chapter 1,we first briefly summarize the research background,significance and research progress of nonlinear parabolic equations at home and abroad,and then give the parabolic max-imum principles and the embedding theorems in Sobolev spaces.And some basic inequalities used in the article are provided.In chapter 2,we consider the following problem:(?) where D(?)Rn(n?2)is a bounded domain with smooth boundary(?).We establish some conditions on nonlinearities to guarantee the existence of blow-up solution or global solution by constructing auxiliary functions and using the parabolic maximum principles and the first-order differential inequality technique.Moreover,the upper estimate of the global solution,the upper estimate of the "blow-up rate",and the upper bound for the "blow-up time" are obtained.In chapter 3,we study the following problem:(?) where m>1,D(?)Rn(n?2)is a bounded convex domain with smooth boundary(?).We en-sure the existence of the blow-up solution by establishing some conditions.Further,the upper and lower bounds of blow-up time are obtained by using the first-order differential inequality technique and the embedding theorems in Sobolev spaces and establishing appropriate auxil-iary functions.