Blow-up And Stabilization Of Solutions For Two Classes Of Wave Equations

Partial differential equations is an important branch of mathematics.It arises in the natural science and engineering areas.Since it has a broad application background and important research value in biology,chemistry,physics and other fields of science,a large number of people have devoted much attention to these equations for a long time.In the last century,many mathematic scientists have proven some properties of different partial differential equations'solutions,such as existence,blow up,stabilization and so on.Espe-cially,some comprehensive research have been made about the properties of wave equation's solution.In this paper,we study the blow-up and stabilization of solutions for two classes of wave equations.This paper is mainly composed of three chapters.Chapter 1 is the introduction.We introduce the research status of the blow-up of solution for coupled viscoelastic wave equations and the stabilization for a one-dimensional wave equation.In chapter 2,we study the blow-up of the solutions of a class of coupled viscoelastic wave equations with strong damping term and dispersive term.By using convexity method,a finite time T blow-up result under certain conditions on the initial data and the relaxation function is given out.And the lower bounds of the blow-up time T is obtained by choosing the appropriate auxiliary function.In chapter 3,we study the stabilization of one-dimensional wave equation by the bound-ary displacement feedback.The stabilization of the system is proved by using operator semigroup theory and Riesz basis theory.