Existence And Energy Estimates Of Solution For A Viscoelastic Equation With Nonlinear Boundary Dissipation

In this paper we consider the existence and uniqueness and the energy decay forms of solution for viscoelastic equations with nonlinear boundary dissipation.The existence and uniqueness of the solution are proved by using Galerkin approximation method,we also give the variational form of the solution.The energy decay rate for the solution of equations are obtained by using integral inequalities and Sobolev trace theorem.In chapter one,we introduce the physical background and the existing study results of viscoelastic equation.In chapter two,we present some function spaces,definitions,theorems and inequalities which will be used in this paper.In chapter three,firstly,we transform the partial differential problem into a Cauchy problem by using Galerkin method and get existence for the solution of the equation according to the ordinary differential equation theory.Secondly,we get the boundedness of the solution by making priori estimation for the solution.Then the convergence of the solution is obtained by using Aubin-lions lemma.Finally,we prove the uniqueness for the solution of the equation.Because of the introduction of nonlinear boundary,the initial terms should be specialized to overcome the difficulties brought by the boundary in the process of proof.In chapter four,firstly,we define the equation of the energy functional.Then in order to get the energy decay rate,we need to construct some auxiliary functionals and get their derivative also relevant integral inequalities.Finally,we prove the equation of energy decay form is exponential decay by using the functional equivalence and Gronwall lemma.