J. Korean Math. Soc. 1998; 35(2): 465-490
Printed June 1, 1998
Copyright © The Korean Mathematical Society.
Jong Yeoul Park and Jeong Ja Bae
Pusan National University and Pusan National University
In this paper, we consider the existence and asymptotic behavior of solutions of the following problem: $$ \align &u_{tt}(t,x)-(\|\nabla u(t,x)\|_2^2+\|\nabla v(t,x)\|_2^2)^\gamma \Delta u(t,x) +\delta|u_t(t,x)|^{p-1}u_t(t,x)\\ &\quad=\mu|u(t,x)|^{q-1}u(t,x), \quad x \in \Omega,\quad t \in [0, T],\\ &v_{tt}(t,x)-(\|\nabla u(t,x)\|_2^2+\|\nabla v(t,x)\|_2^2)^\gamma \Delta v(t,x) +\delta|v_t(t,x)|^{p-1}v_t(t,x)\\ &\quad=\mu|v(t,x)|^{q-1}v(t,x), \quad x \in \Omega,\quad t \in [0, T],\\ &u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x), \quad x \in \Omega,\\ &v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x), \quad x \in \Omega,\\ &u|_{\partial \Omega}=v|_{\partial \Omega}=0 \endalign $$ where $T >0$, $q > 1$, $p \geq 1$, $\delta >0$, $\mu \in R$, $\gamma \geq 1$ and $\Delta$ is the Laplacian in $R^N$.
Keywords: existence and uniqueness, asymptotic behavior, degenerate wave equation, Galerkin method
MSC numbers: 35L70, 35L15, 65M60
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