## Differential and Integral Equations

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

Soichiro Katayama

#### 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.

#### Article information

Source
Differential Integral Equations, Volume 17, Number 9-10 (2004), 1043-1078.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060313

Mathematical Reviews number (MathSciNet)
MR2082459

Zentralblatt MATH identifier
1150.35492

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations

#### Citation

Katayama, Soichiro. Global and almost-global existence for systems of nonlinear wave equations with different propagation speeds. Differential Integral Equations 17 (2004), no. 9-10, 1043--1078. https://projecteuclid.org/euclid.die/1356060313