The Study Of Global Solutions Of The Equations Of Kirchhoff Type And The Beam Equations

This article investigates the properties of solutions of the wave equations of Kirchhoff type with nonlinear which arises from small amplitude vibrations of an elastic string and the properties of solutions of the beam equations. It contains the problems of local existence of weak solutions, global existence and finite time blow-up of the first equation and decay rate of solutions of the second equation. There are three parts in this thesis: Chapter 2 consider the initial-boundary value problemwhere Q is a bounded demain in RN with smooth boundary <3Q ; with We show the local existence, the golbalexistence and the decay rate of the solutions of that Kirchhoff equation. Chapter 3 consider the initial-boundary value problemutt -where Q is a bounded demain in RN with smooth boundary 3Q ; with M(s) is a nonnegative C1 -function for s>0 satisfyingwith We show that under certain conditions the solution blow up in finite time. Chapter 4 consider the initial-boundary value problemand 2.We show the decay rate of the solutions of the equation.To the first equation, the Banach contraction mapping theorem is used to show the local existence of the solutions, we use potintial well methodto prove the global existence and the decay rate of the solutions,to the blow-up of the solution we use the energy method. And, to the second equation, we use a so-called energy perturbation method to show decay rate of the solution.