The Life Span Of The Classical Solution For Nonlinear Wave Equations In 3 Space Dimensions

The present Ph.D. dissertation deals with the Cauchy problem for nonlinear wave equations with different propagation speeds and the initial boundary value problem to nonlinear wave equation outside of star-shaped obstacle with small initial data in three space dimensions.Since the classical work of John [7], lots of efforts has been made to study the lifespan(the maximal existence time of unique local classical solutions) of classical solution to nonlinear wave equation with small cauchy data. Most of those efforts were devoted to study cauchy problem of the nonlinear wave equation by means of the Lorentz invariance of the wave operator (?)_t²-â–³(see [7,8,9,14-18,22-26]) etc. This paper is devoted to study the lifespan of the classical solution for the nonlinear wave equations with different propagation speeds and the initial boundary problem of the nonlinear wave equation which do not have Lorentz invariance any more.The first part of this dissertation is devoted to study the lifespan for the multiple propagation speeds systems of nonlinear wave equations with the nonlinear term explicitly depending on the unknown function u. To prove our result, we use the generalized energy method of Klainerman [15]. In the single speed case, the key of this method is to use the Lorentz invariance of the wave operator (?)_t²-â–³. In the multiple speed case, the Lorentz invariance do not hold, so additional technical difficulties arise. To compensate the lack of Lorentz boosts, in [18], klainerman and Sideris developed some weighted estimates of the seconde order derivatives of the solution to handle the case that the nonlinear term do not explicitly depend on the unknown function itself. When the nonlinear term explicitly depend on the unknown function itself, some more estimates are necessary. In the first Chapter of this dissertation, the author will develop the weighted L² estimates of the first order derivatives of the solution to handle the case that the nonlinear term explicitly depend on the unknown function itself.In the second part of this dissertation, the author will study the lifespan for the nonlinear wave equation outside of star-shaped obstacle with the nonlinear term explicitly depending on the unknown function u. The main method of our proof is still to use the generalized energy method of Klainerman [15], however, for the case that the problem outside of a star-shaped obstacle, the Lorentz invariance do not hold either, at the same time, another difficulty we encounter in the obstacle case is related to the scaling operator .In the Minkowski space case, Lpreserve the equation ((?)_t²-â–³)u = 0, in the obstacle case, that the Dirichlet boundary conditions are not preserved by this operator. When we deal with the boundary conditions, the coefficient became large on the obstacle as t goes to infinity if we use L twice. To overcome this difficulty, Keel, Smith and Sogge [19,21] developed some weighted L_t,x² estimates for the first order derivatives of unknown function u and use those estimates to handle the case that the nonlinear term do not explicitly depending on the unknown function u. In the second part of this paper, the author will develop the weighted L_t,x² estimates for the unknown function itself, and by the same frame, we can handle the case that the nonlinear term explicitly depending on the unknown function u.This dissertation is organized as follows. In the first Chapter, we will introduce the research history of nonlinear wave equation. The main results of the dissertation will also be stated in chapter 1.In the second Chapter, the author will develop some new weighted estimates and obtain the lower bound of lifespan for the multiple speed case with the nonlinear term depending explicitly with the unknown function u itself.In the third Chapter, the author will show some estimates for the linear wave equation both in the Minkowski space and for the case outside of star-shaped obsta- cle. And in the forth Chapter, the author will obtain the lifespan for the nonlinear wave equation outside of star-shaped obstacle with the nonlinear term explicitly depending on the unknown function u itself.