Initial Boundary Value Problems For Two Kinds Of Thermoelastic Coupled Beam Systems With Memory Terms

In this paper,we studied the initial boundary value problems for two kinds of thermoe-lastic coupled beam systems with memory terms.The specific contents of the study are as follows:The first chapter,This paper briefly introduces the research background and present situation of the whole dynamic behavior of elastic beam equations at home and abroad,as well as the main contents of this paper.The second chapter,gives some basic knowledge to be used in this paper,including the basic definition,the theorem and the common inequality.The third chapter,we study a class of thermoelastic coupled beam equations with mem-ory term,with initial value condition u(x,0)=u0(x),ut(x,0)=u1(x),?(x,0)??0(x)and under the boundary condition u(0,t)=u(l,t)=u(2)(0,t)=u(2)(l,t)=0,?(0,t)=?(l,t)=0.where x??,??(0,l),uo(x),ul(x),?0(x)are functions with a certain smoothness.Here u(x,t)is the transverse deflection of the beam,M are nonlinear functions,h,f sre the external force terms,?(x,t)is the temperature of the material,?,a are the thermal effectiveness coupling coefficient.The boundedness,existence and uniqueness of the weak solutions of a class of heatelastic coupled beam equations with thermal memory under homogeneous boundary conditions are studied by using Galerkin method.The four chapter,on the basis of the third chapter,boundary condition is unchanged,we study the existence uniqueness of strong solution.The five chapter,we study the global attractors of another class of thermoelastic coupled beam equations with memory terms,with initial value condition u(x,0)=u0(x),ut(x,0)=u1(x),?(x,0)??0(x).and under the boundary condition u(0,t)=u(l,t)=u(2)(0,t)=u(2)(l,t)=0,?(0,t)=?(l,t)=0.First,we apply the operator semigroups theorem and prove the existence and uniqueness of the weak solution of the beam system,then we also define a dynamics system,using a pri-ori estimation and some inequalities estimation to prove the existence of the system suction collection,and the system is dissipated.Finally,the asymptotic strictness of the system is proved by constructing the Lyapunov function,and the existence of the global absorber is verified under homogeneous boundary conditions and certain initial conditions in this heate-lastic coupled beam equations.