A Class Of Thin-film Equations With Logarithmic Nonlinearity

The main purpose of this paper is to study the following initial boundary value problem for a thin-film equation with logarithmic nonlinearity(?)where c Rn(n ? 1)is a bounded smooth domain and u0 ?H01Q(?)?H2(?).This thesis is divided into four chapters.In Chapter 1,we first describe the background of the problem under consideration and briefly recall some related works obtained both in China and aboard.Then we state our problem and the methods and techniques that will be used.In the first part of Chapter 2,we introduce some necessary lemmas as preliminaries.While in the second part,we define some functionals and sets related to the potential well method and investigate their basic properties in the solution space.In Chapter 3,local ex-istence of weak solutions is obtained by using Galerkin's approximation.Then we show that weak solutions exist globally when the initial data are in the potential well.The decay estimates of the global solutions are also derived.In Chapter 4,we obtain the existence of solutions that blow up at infinity,by using the modified logarithmic Sobolev inequality and the potential well method.Our main results can be summarized in the following three theorems:Based on the relationship between the initial energy(?)(?)W+ =?u?H01(?)?H2(?:J(u)?d,I(u)?)0},W+= {u?H01(?)?H2(?:J(u)?d,I(u)?0}.we obtain the following conclusions:Theorem 1.Letuo? H01(?)?H2(?).Then there exists a positive constant r such that problem(1.1)admits a weak solution u(x,t)in ?×(0,T0).Furthermore,u(x,t)satisfies the energy identity(?)Theorem 2.Let u0 ? W+.Then problem(1.1)admits a global weak solution u(x,t)such that u(x,t)? W+ for t ?[0,?).Moreover,we have the following decay rates:(i)If J(u0)<M,then we have(?)(?)(?)Theorem 3.Let u0 ? W-with J(u0)? M and u(x,t)be a local weak solution to problem(1.1).Then u(x,t)blows up at +?,i.e.(?)d?M is the depth of the potential well.