Initial-boundary Value Problems For Two-dimensional Nonlinear Integro-differential Equations With Memory Term

Nonlinear developing equation is a kind of representation of manynonlinear problems in mathematics. Now, integro-differential equations are topicproblem, especially, those with memory term have been attached highlyimportance to many mathematician.In this paper, we just consider the initial-boundary value problems of aclass of two-dimensional nonlinear integro-differential equation, simultaneouslyconsidering the damped term and memory termThe details will go as follows.Firstly, the current study situation about nonlinear integro-differentialequations is introduced;Secondly, we put forward some important definition and lemma, simultaneity explain some marks;Thirdly, we prove the existence and uniqueness of the weak solution of thevariational problem of the problem (1)-(3)by means of the Galerkin method;Fourthly, we prove the existence of the strongly solution of the problem(1)-(3)and the exponential decay of the energy;Fifthly, we study the existence of the classical solution of the problem(1)-(3).