Long Time Behavior Of The Solutions Of The Two-dimensional Harmonic Map Heat Flow Equation

This paper studies the long time behavior of the solution of the two dimensional harmonic heat equation,and mainly discusses the content as follows: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 also,the main research models and results are given.The second chapter is the theoretical basis.First defined the upper and lower solutions and the basic knowledge of the general principle of upper and lower solution,and then gives the lemma to prove this research model needed method.In the third chapter,we study the long time behavior of the solutions of the kind of equations by changing the initial conditions or the correlated real valued functions.Firstly,the solution of the two-dimensional harmonic map heat flow equation is considered.By means of constructing the upper and lower solutions,we prove that the solution of the equation will blowup in a finite time when the parameter l = 1;and the solution of the equation will not blow-up in a finite time when the parameter l ≥ 6.Finally,this paper makes a simple summary of the models of this study,and put forward about the change of the parameter l and equation contains function replacement can be further research.