Nonlinear Viscoelastic Wave Equation (group) Of The Nature Of The Solution

Along with science’s and technology’s development, various non-linear prob-lem have aroused people’s widespread attention. The boundary problem of non-linear partial differential equation (equations) originate from applied mathemat-ics, physics, control theory and other applied disciplines, it is presently one of the most active research topics in nonlinear science field, and it is hot topic in recent years, and it is a quite important field of the partial differential equations.In this paper, using the multiplier method, auxiliary function and the poten-tial well method, we study the properties of the solutions for nonlinear viscoelastic wave equation(equations).The thesis is divided into there chapters.In chapter1, we introduce the research current situation and the main results of the discussing problem in this paper.In chapter2, we consider the nonlinear viscoelastic wave equation with acoustic boundary conditions. We prove that the energy will grow up as an exponential function as time goes to infinity, provide that the initial energy is negative.In chapter3, we consider a system of two coupled nonlinear viscoelastic equations with Dirichlet boundary condition. By using the auxiliary functional and using the potential well method, we show that the energy functional uniform decay (with exponential and polynomial rates) under suitable assumptions.