The Initial Boundary Value Problem Of Fourth-order Wave Equation With Nonlinear Damping And Source Terms

In this paper,the initial-boundary value problem of nonlinear fourth-order wave equations with damping and source terms is studied on the base of Beam Equation:This paper consists of four chapters.The first chapter is an introduction,will give the physical meaning and research results of the equation.In the next chapter,we will prove the local existence of the mild solution of the fourth order wave equation with damping and source terms under certain conditions.When the effect of the nonlinear damping term is stronger than the source term,by using modified energy function,according to continuation principle,we will prove the existence and uniqueness of the global solution of the wave equation.In the third chapter,we will study how the nonlinear damping and source terms affect the blow-up of the problem.When the effect of the nonlinear damping term is weaker than the source term,by choosing the proper initial value and using the method of function energy and constructing stable set and unstable set, according to the potential well theory, the energy decay property of the global solution is proved,meanwhile,we will talk about the blow-up of solution when the initial value is in unstable set and the existence of global solution when the initial value is in the boundary of stable set or unstable set.The details are obtained as following:The main results of the second chapter are:Theorem 2.1 Letl0.E(t)≥0.then the energy decay property of the global solution of(P) is:Theorem 4.2 Letl