Initial-Boundary Value Problems For The Nonlinear Viscous-Elastic Plate Equatons

Nonlinear developing equation is a kind of representation of many nonlinear problems in mathematics. Now, viscous-elastic equations are topic problem, especially, those with memory term have been attached highly importance to many mathematician.In this paper, we just consider the initial-boundary value problems of a class of two-dimensional nonlinear viscous-elastic equation, simultaneously considering the damped term and memory termu(0,y,t)=u(1,y,t)=u(x,0,t)=u(x,1,t)=0 ,u_xx(0,y,t)=u_xx(1,y,t)=u_yy(x,0,t)=u_yy(x,1,t)=0,(x,y;t)∈Γ×(0,T) (2)u(x,y;0)=u₀ (x, y), (?)(x, y;0)=u₁(x, y),(x,y)∈Ω(3)The details will go as follows.Firstly, the current study situation about nonlinear viscous-elastic equations is introduced;Secondly, we put forward some important definition and lemma, simultaneity explain some marks;Thirdly, we prove the existence and uniqueness of the solution of the initial-boundary value problem (1)-(3)by means of the Galerkin method;Fourthly, we study the existence of the classical solution of the initial-boundary value problem (1)-(3).