Differential Harnack Inequalities Of Solutions To A Class Of Semilinear Parabolic Equations

This paper studies a class of semilinear parabolic equations,using the maximum principle to obtain the differential Harnack inequalities.In this paper,we give a method to prove that the solutions blow up in finite time.Using the differential Harnack inequalities proves the solutions of the semilinear parabolic equations blowing up in finite time and integrating the differential Harnack estimates along the space-time curve obtains the classical Harnack inequalities.This paper is divided into four chapters.The main contents of each chapter are summarized as follows:The first chapter mainly introduces the development background of classical Harnack inequalities and the theoretical source of this paper,and gives the theorems and formulas that can be used in this paper.In chapter 2,we study the differential Harnack estimations of the solutions of ft=?f+t?fq and classical Harnack inequalities.In this chapter,we mainly discuss the differential Harnack inequalities of semilinear parabolic equations when the coefficient function is t?,and apply the differential Harnack inequality to prove that the solutions of the equations explode in finite time and obtain the classical Harnack inequalities.In chapter 3,we consider the differential Harnack estimates of solutions of ft=?f+e?t fq and classical Harnack inequalities.In this chapter,we mainly consider the differential Harnack inequalities of semilinear parabolic equations when the coefficient function is e?t,and use this inequality to prove that the solutions of the equations explode in finite time and obtain the classical Harnack inequalities.The last chapter mainly explores the differential Harnack inequalities corresponding to the equations ft=?f+h(t)fq.In this chapter we explore the classical Harnack inequalities of the solution of the equations when the coefficient function h(t)is the general case.