Some Researche On Solutions Of A Degenerate Nonlinear Parabolic Equation With Variable Exponent Reaction Term

The degenerate nonlinear parabolic equation find in the diffusion phenomenon which exists in the nature,percolation theory,phase transformation theory,biochemistry andgroup dynamics.lt has attracted plenty of attention that a class of degenerate nonlinear parabolic equations which are used to describe the gas diffusion in porous media process For nearly four decades——a Newton percolation equation with nonlinear source term. Many mathematical workers from all over the world have studied on the properties of the solution of the equation.Such as J.L.Lions, V.A.Ga-laktionov, Mochizuki, R.Ferreira, Zhang Jibing, Li Fengjie,Chen Yazhe,Zhao Jun-ning,Song shiqin and so on.Expecially,there has been a lot of achievements in local existence of solutions,the solution blows-up in a finite time and estimate about blows-up rate.With the deepening study about the degenerate nonlinear parabolic equation and the diversity of background about mathematics.Recently, people began to pay attention to a Newton percolation equation which has the nonlinear source term with variable exponents.This class of equations find in the mathematical model which is called elect-rorheological fluid. This thesis,we study the properties about the blow-up of solutions,properties about gradient blow-up of solutions and the global existence of solution about these degenerate nonlinear parabolic equations with variable exponents reaction term.This thesis is divided into four chapters.In Chapter l,it’s the preliminary data.Firstly,we introduce the blow-up proper-ties of solutions,gradient blow-up properties of solutions and the global existence of solution about the Newton percolation equation.Secondly,some important inequal-ities are given by use of theorem proving in this paper.In Chapter 2,we study the blow-up properties of solutions of the Dirichlet prob- lem about the Newton percolation equation with variable exponents in gradient term such as ut- △um= f(x,u,|▽|q(x)).We consider the following three different situations about f(x,u,|▽|q(x)):And we present a sufficient conditions about the blow-up positive solutions of equations under various conditions.In Chapter 3, we study the percolation equation such as ut - △um=|▽|p(x)+ a(x)uq.We mainly study its gradient blow-up problem about the positive solutions of equations under nonhomogeneous boundary condition.In Chapter 4, we study the global existence of solution about a degenerate nonlinear parabolic equation by using the method of upper and lower solutions.