## Journal of the Mathematical Society of Japan

### On global smooth solutions to the initial-boundary value problem for quasilinear wave equations in exterior domains

Mitsuhiro NAKAO

#### Abstract

We consider the initial-boundary value problem for the standard quasilinear wave equation:

$u_{tt}-\mathrm{div}\{\sigma(|\nabla_{u}|^{2})\nabla_{u}\}+a(x)u_{t}=0 in \Omega\times[0,\infty)$ in $\Omega\times[0,\infty)$

$u(x,0)=u_{0}(x)$ and $u_{t}(x,0)=u_{1}(x)$ and $u|_{\partial\Omega}=0$

where $\Omega$ is an exterior domain in $R^{N}, \sigma(v)$ is a function like $\sigma(v)=1/\sqrt{1+v}$ and $a(x)$ is a nonnegative function. Under two types of hypotheses on $a(x)$ we prove existence theorems of global small amplitude solutions. We note that $a(x)u_{t}$ is required to be effective only in localized area and no geometrical condition is imposed on the boundary $\partial\Omega$.

#### Article information

Source
J. Math. Soc. Japan, Volume 55, Number 3 (2003), 765-795.

Dates
First available in Project Euclid: 3 October 2007

https://projecteuclid.org/euclid.jmsj/1191419002

Digital Object Identifier
doi:10.2969/jmsj/1191419002

Mathematical Reviews number (MathSciNet)
MR1978222

Zentralblatt MATH identifier
1030.35124

#### Citation

NAKAO, Mitsuhiro. On global smooth solutions to the initial-boundary value problem for quasilinear wave equations in exterior domains. J. Math. Soc. Japan 55 (2003), no. 3, 765--795. doi:10.2969/jmsj/1191419002. https://projecteuclid.org/euclid.jmsj/1191419002