Existence,Nonexistence And Decay Rate Of Solution Of Nonlinear Viscoelastic Wave Equation With Density Function

In this master’s degree thesis,we mainly apply the basic theory of the infinite dimensional dynamical system,and combine the Faedo-Galerkin approximation method,the Banach compression mapping principle,the potential well method,the energy estimation and the priori estimation,by constructing the appropriate Lyapunov function to research the existence,uniqueness and energy decay rate of nonlinear viscoelastic wave equation with density function,where,a ∈R,ω>0,m≥ 1,p>1,n≥3,φ(x)is density function.The main content of this paper is composed of three parts:The first part,we review the basic concepts of dynamical system,as well as the abstract conclusions and common inequalities used in our problems.The second part,when m=1,we research the existence,uniqueness and energy decay rate of nonlinear viscoelastic wave equations with density function and linear damping.In this part,the first thing,combining the Faedo-Galerkin approximation method and Banach compression mapping principle,we obtain the local solution well-posedness results.This process is divided into two steps:firstly,the local solution of linear nonhomogeneous differential equation with density function and linear damping is obtained by using the FaedoGalerkin approximation method,secondly,the local solution of original equation is proved by Banach compression mapping principle.Then,the well-posedness of the global solution is proved by using the energy functional method,and then obtain the decay rate of energy.The third part,when m>1,we research the existence and uniqueness of the solution,the energy decay rate and the blow up to a nonlinear viscoelastic wave equations with density function and nonlinear damping.In this part,the potential well method introduced in article[13,17,18]is applied as a whole.It is divided into three parts:firstly,we obtain the well-posedness of the global solution by means of the Faedo-Galerkin approximation method and the energy estimation.Secondly,we obtain the decay rate of energy by constructing the proper Lyapunov function and applying prior estimation when 3<m<p.Finally,we obtain the blow up result of solution in finite time when 1<m<p.