Study On The Stability And Related Properties Of A Class Of Degenerate Parabolic Equations

The study of nonlinear diffusion equation is a very important topic in the field of partial d-ifferential equations.The diffusion equations such as the evolutionary p-Laplacian equation and non Newtonian polytropic filtration equation appears in the fields of physics and chemistry.The nonlinear diffusion equation with degeneracy describes some physical realities more accurately.In this paper,we supplement the initial boundary value problem of a class of parabolic equation with boundary degenerate properties,and prove the existence and stability of the weak solution.In the first chapter of this paper,the background and research status of a class of partial differ-ential equations with boundary degeneracy are introduced,and the main conclusions of this paper are introduced.In the second chapter of this paper,the existence and uniqueness of weak solution for a class of polytropic filtration equation is discussed.The equation is ut = div(a(x)|?urn|p-2?um),(x,t)G?QT,and QT = ? ×(0,T),? is a bounded domain with a smooth boundary in RN.p is a constant,u0?L1(?),m(p-1)>1.The existence of the weak solution is proved by the regularization method and the Sobolev inequalities.When a(x)|(?)? = 0,it is proved that the uniqueness of the weak solution is independent of the boundary value condition.In the third chapter of this paper,the stability of the equation with a partial boundary value condition is proved.The equation isut=?i=1 n(ai(x)|(um)xi|pi(x)-2(um)xi)xi+f(x,t,u),(x,t)?QT,and p(x)? C1(?),a(x)is partially degenerate on the boundary.In the fourth chapter of this paper,the stability of the solution of a class of evolutionary p(x)-Laplacian equation ut = div(??|?u|p(x)-2 ?u)+ gi(x)Di(b(u)),(x,t)? QT,is proved under partial boundary value condition.In the two cases of ?<p--1 and ?>p--1,the stability of the equation is discussed.