## Differential and Integral Equations

### Local regularity of non-resonant nonlinear wave equations

Kimitoshi Tsutaya

#### Abstract

We study the problem of minimal regularity required to ensure local well-posedness for systems of nonlinear wave equations with different propagation speeds in three space dimensions \begin{align} (\partial_t^2 -C_1^2\Delta)u &= F(u,v,\partial u,\partial v), \\ (\partial_t^2 -C_2^2\Delta)v &= G(u,v,\partial u,\partial v). \end{align} We prove that if $C_2 > C_1$ and $F$, $G$ have the form $\p u \cdot v$, then the problem is well-posed in $H^1$. Our proof is based on the same type of space-time estimates as those of Klainerman and Machedon.

#### Article information

Source
Differential Integral Equations, Volume 11, Number 2 (1998), 279-292.

Dates
First available in Project Euclid: 30 April 2013

https://projecteuclid.org/euclid.die/1367341071

Mathematical Reviews number (MathSciNet)
MR1741846

Zentralblatt MATH identifier
1004.35093

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

#### Citation

Tsutaya, Kimitoshi. Local regularity of non-resonant nonlinear wave equations. Differential Integral Equations 11 (1998), no. 2, 279--292. https://projecteuclid.org/euclid.die/1367341071