Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2000; 37(3): 339-358

Printed May 1, 2000

Copyright © The Korean Mathematical Society.

On the existence of solutions of quasilinear wave equations with viscosity

Jong Yeoul Park and Jeong Ja Bae

Pusan National University

Abstract

Let $\Omega $ be a bounded domain in ${\mathbb R}^N$ with smooth boundary $\partial \Omega$. In this paper, we consider the existence of solutions of the following problem: $$ \begin{aligned} &u_{tt}(t,x)-\hbox{div} \{\sigma(|\nabla u(t,x)|^2)\nabla u(t,x) \}-\Delta u(t,x)-\Delta u_t(t,x)\\ &\quad+\delta|u_t(t,x)|^{p-1}u_t(t,x) =\mu|u(t,x)|^{q-1}u(t,x),\\ & \qquad\quad x \in \Omega,\quad t \in [0,T],\\ &u(t,x)|_{ \partial \Omega}=0,\\ &u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x), \quad x \in \Omega, \end{aligned}\tag{1.1} $$ where $q > 1$, $p \geq 1$, $\delta >0$, $\mu \in {\mathbb R}$, $\Delta$ the Laplacian in ${\mathbb R}^N$ and $\sigma= \sigma (v^2)$ is a positive function like as $\frac{1}{(1+v^2)^{1/2}}$.

Keywords: quasilinear wave equation, energy identity, Galerkin method, existence and uniqueness

MSC numbers: 35L70, 35L15, 74S05