Long Time Behavior Of The Solutions Of Two Types Of Heat Flow Equations

In this paper,we study the long time behavior of the solution of the two dimensional harmonic heat equation and the Yang-Mills heat equation.The first chapter is an introduction,which briefly introduces the research background and the related research results of the two-dimensional harmonic map heat flow equation and the four-dimensional Yang-Mills heat flow equation.And also,the main research models and results are given.The second chapter is the theoretical basis.First of all,under the given conditions,the long time behavior of the solutions of the two kinds of equations is given.Then,we give some lemmas that needed for the research model.In the third chapter,we study the long time behavior of the solutions of the two kinds of equations by changing the initial conditions or the correlated real valued functions.Firstly,the solution of the Yang-Mills heat flow equation is considered.By means of the structural solution,it is proved that the solution of the equation will not blow-up in a finite time when the parameter l ? 2.Then,the solution of the heat equation of the two-dimensional harmonic map is considered.By means of constructing the upper and lower solutions,we prove that the solution of the equation will blow-up in a finite time when the parameter l ? 2;and the solution of the equation will not blow-up in a finite time when the parameter l ? 4.Finally,this paper makes a summary of the contents of this study,and the corresponding expansion is given.There are a lot of researches on the heat equation and the Yang-Mills heat equation,however,the change of the parameters and the replacement of the function can be further studied.