Study On Solution Of Plate Equation With Non-local Source Terms

Plate equation is a kind of basic and important mathematical and physical equation.With the development of science and technology,the plate equation with non-local source term has become an active topic in partial differential equation.This thesis mainly discusses a class of plate equation with non-local source term.Including,with nonlinear damping term and degenerated damping term.This kind of problem plays an important role in the research of plate equation.This paper is divided into four chapters.The first chapter is the introduction,which introduces the research background and research status of non-local source plate equations with nonlinear and degenerate damping terms.The second chapter introduces the notations and inequalities used in this thesis.In the third chapter,it analyzed that the properties of the solutions of plate equation with nonlinear damping term.First,the uniqueness of local solutions is proved by the fixed point theorem.Next,with the use of the modified energy and the continuity theorem we prove the existence and uniqueness of the global solution.Then,under certain conditions,we prove the decay estimate.Finally,when the initial energy is negative or bounded in some set,the solution blows up in finite time.In chapter 4,the plate equations with degenerate damping terms are studied.By using the fixed point,we prove the existence of the generalized solution,when k satisfies some conditions,the generalized solution becomes the weak solution.And then we obtain that the solution blows up in finite time when the initial energy is negative or the initial energy is positive and bounded.Finally,the asymptotic stability of the weak solution is proved for the initial energy is bounded.