Existence And Uniqueness Of Global Solutions For A Kind Of Vivcoelastic Wave Equation

Partial differential equations are differential equations which associate with unknown functions, partial derivatives and independent variables. People have studied them for a long time. Most of nonlinear partial differential equations are applied to simplify the engineering problems in the field of natural sciences. The wave equations with nonhomogeneous boundary conditions are important parts of the nonlinear partial differential equations. The main research is based on the existence of local solutions, the existence of global solutions, regularity and the energy decay estimate.In this paper, we will use Faedo-Galerkin method to study a class of viscoelastic wave equation with nonhomogeneous boundary conditions in the generalized function space. uu-uxx+∫tok(t-s)uxx(s)ds+h(u,ut)=f(x,t)(x,t)∈(0,1)x(0,T) In the initial conditions u(x,0)=uo(x), ut(x,0)=u1(x) And non-homogeneous boundary conditions ux(0,t)＋η1u(0,t)＝E(t),Ux(1,t)＋η2u(1,t)＝M(t) WhereE(t),M(t),uo(x),u1(x),f(x,t),k(·),h(u,ut) are given functions.This article will be divided into the following four-part to study:1. We will make some sum-up and comments of the development and research of nonlinear partial differential equation.2. We will give some concepts and lemmas for this paper.3. We will prove the existence and uniqueness of weak solutions by Faedo-Galerkin method.4. We will prove the existence and uniqueness of strong solutions by Faedo-Galerkin method.