Global and almost-global existence for systems of nonlinear wave equations with different propagation speeds

Abstract

We consider a system of nonlinear wave equations $$ ({\partial}_t^2-c_i^2 \Delta_x)u_i=F_i(u, {\partial} u, {\partial}_x {\partial} u) \text{ in $(0, \infty)\times \mathbb R^3$} $$ for $ i=1, \dots, m $, where $ F=(F_1, \dots, F_m) $ is a smooth function satisfying $$ F(u, {\partial} u, {\partial}_x {\partial} u)=O(|u|^3+|{\partial} u|^2+|{\partial}_x {\partial} u|^2) \quad \text{near the origin,} $$ $ u=(u_1, \dots, u_m) $, while $ {\partial} u $ and $ {\partial}_x {\partial} u $ represent the first and second derivatives of $ u $, respectively. We assume $ 0 < c_1\le c_2 \le \cdots \le c_m $. In this paper, we show global existence of classical solutions to the above system with small initial data under the ``null condition'' for systems with different propagation speeds. We also show ``almost-global'' existence for the above system for the case where the null condition is not satisfied.