Journal of the
Korean Mathematical Society JKMS

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