The Initial-boundary Value Problem For A Class Of Nonlinear Fourth Order Parabolic Equations

The initial boundary value problem of a kind of fourth order parabolic equation(?) is considered in this paper,where T?(0,?],?(?)Rn is a bounded domain with smooth enough boundary,and u0 ? H2(?),1/|?|??u0dx=0,u0(?)0.Such a problem describes a diffusion of liquid surface.The threshold result of global existence and non-existence for the sign-changing weak solutions is given.Further,the conditions under which the global solutions extinct in a finite time and the asymptotic behavior of non-extinct solutions is studied.The uniqueness and smoothness of the global weak solution are also considered.The background are introduced,the main results of this paper are also given in the first chapter.The potential wells and their properties in the form of lemmas are studied in the second chapter.In the third chapter,some prior estimates are given with the help of potential well method.Then the existence of global weak solutions is obtained obviously by Galerkin method.And the uniqueness of global weak solution is obtained in using Sobolev inequalities and Gron,all inequality in three dimensions case.The asymptotic behaviors of the weak solution are also studied.The smoothness of the global weak solution are considered in the fourth chapter.It is proved that the smoothness of the global weak solution u(t)increases with the smoothness of the initial value u0.It is proved that the weak solution will blow up in the L2(0,T;L2(?))by constructing unstable sets in the fifth chapter.