Differential and Integral Equations

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

Soichiro Katayama

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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.


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