The Existence Of Solutions For The Initial-Boundary Value Problem Of The Dual-Temperature Heat Equation

In this paper, we consider the initial boundary value problem for a class of semilinear double temperature heat equation. is a bounded domain and f∈C with f(u)u≥0. And the existence and nonexistence of global solutions are established.This thesis mainly consists of six parts:In section one,there is an outline and an introduction of this work. It simply introduces the background of partial differential equations and the history of nonlinear evolution equations. And the main results that we have known on the problem are given.In section two,we introduce the potential well and its properties. In this part, we first give the concept of potential well and the definition of the family of potential wells. Then we give and prove some theorems and lemmas on potential well.In section three and four, we study the existence of the global weak solutions and strong solutions of question (1)—(3) respectively. In these two parts, we mainly prove the existence theorems of the global weak solutions and strong solutions,and then we prove some L2 norm estimates of solutions and their derivatives before the proof of theorem 4.4.The fifth part are about the invariable sets of solutions and the vacuum isolating of solutions. And the vacuum isolating was first shown by Liu Yacheng in 2003.The sixth part is about the blow-up of solutions for problem (1)—(3).In this part, by using integral estimate method and eigenfuction method, we prove the finite time blow-up for solutions of the problem.