Long-time Behavior Of Solutions For Two Classes Of Coupled Beam Equations

Beam equation is a kind of common partial differential equation.The properties of the solution and the existence of the attractor have always been the focus of research.In this paper,we study the initial boundary value problems for two kinds of coupled beam equations under homogeneous boundary conditions.By using Galerkin method,inequality technique,Sobolev space theory,the method of verifying compactness,the existence and uniqueness of the generalized solutions and the existence of global attractor for two kinds of coupled beam equations are proved.The structure of this paper is as followsThe first chapter introduces the research background,present situation of the beam equation and the main contents of this paper.The second chapter lists the basic definitions,important Lemmas and basic inequalities used in this paper.The third chapter researches existence and uniqueness of the solutions and existence of global attractor for a class of nonlinear coupled beam equations.Under the boundary condition u|(?)?=ux|(?)?=0,?|(?)?=0,Under the initial condition u(x,0)=u0(x),ut(x,0)=u1(x),?(x,0)=?0(x).Where ? =[0,L]is a bounded interval in R,p ? R,?,?,?>0.The fourth chapter studies existence and uniqueness of the solutions and existence of global attractor for a class of coupled beam equations with memory term and source term.Under the boundary condition u|(?)?=ux|(?)?=0,?|(?)?=0,Under the initial condition u(x,0)=u0(x),ut(x,0)=u1(x),?(x,0)=?0(x).Where ?=[0,L]is a bounded interval in R,?,?,k>0.The fifth chapter summarizes the full text and the prospect of further research on coupled beam equations is put forward.