The Existence Of A Class Of Nonlinear Viscoelastic Wave Equations With The Asymptotic Expansion

With the continuous development of science and technology, all kinds of nonlinear problem have aroused people's wide attention. Nonlinear partial differentialequation stems from applied mathematics, physics, control theory and other disciplines in a wide range of applications, it is presently one of the most active topics in the field of nonlinear science, the initial-boundary value problem of nonlinear viscoelastic partial differential equation is a hot topic these years, and it is a quite important field of the partial differential research.In this dissertation, we study the existence and asymptotic expansion of the following nonlinear viscoelastic wave equation,where F(u,u_t) = a(x)|u|^p-2u + b(x)|u_t|^q-2u_t, p≥2,q≥2;η₀≥0,η₁≥0 are constants, and k, a(x),b(x), f, h₀, h₁,(?) are given functions.This dissertation is divided into three chapters.The first chapter is our introduction. We introduce the importance of the nonlinear viscoelastic wave equation, as well as some previous research results.In the second chapter, we use Faedo-Galerkin method to prove the exsistence and the uniqueness of the solution, which including four parts, that is FaedoGalerkinapproximation, a priori eatimate, limiting process and uniqueness of the solution.We prove the asymptotic expansion of the solution of the nonlinear viscoelasticwave problem with respect to two parameters (η₀,η₁)in the third chapter.