The Existence And The Uniqueness Of Some Semi-Linear Weakly Hyperbolic Equations

The Cauchy problem for the following semi-linear weakly hyperbolic equations is discussed in this paper.The existence and the uniqueness of classical solutions is studied.Firstly, the linear weakly hyperbolic equation is studied by using the property of L~p spaece and Sobolev spaece . Secondly, on the base of above estimates, a function space is defined and some important inequalities are used to estimate the nonlinear items. Finally, with the help of the Banach's fixed point theorem , the existence and uniqueness of classical solutions of the semi-linear weakly hyperbolic equation are proved.This article is organized as follows, in Chapter 1, the background and current deve -lopment for wave equations are introduced. Then the main results as their principal results are given in Chapter 2 . The relative lemmas for the proof of the main Theore -ms are introduced in Chapter 3 and the two principal results are proved in Chapter 4.