# Article

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Keywords:
quasilinear wave equation; existence and uniqueness; asymptotic behavior; Galerkin method
Summary:
In this paper we consider the existence and asymptotic behavior of solutions of the following problem: $u_{tt}(t,x)-(\alpha +\beta \Vert \nabla u(t,x)\Vert _2^2 +\beta \Vert \nabla v(t,x)\Vert _2^2)\Delta u(t,x) +\delta |u_t(t,x)|^{p-1}u_t(t,x) \quad =\mu |u(t,x)|^{q-1}u(t,x), \quad x \in \Omega ,\quad t \ge 0, v_{tt}(t,x)-(\alpha +\beta \Vert \nabla u(t,x)\Vert _2^2+ \beta \Vert \nabla v(t,x)\Vert _2^2) \Delta v(t,x) +\delta |v_t(t,x)|^{p-1}v_t(t,x) \quad =\mu |v(t,x)|^{q-1}v(t,x), \quad x \in \Omega ,\quad t \ge 0, u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x), \quad x \in \Omega , v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x), \quad x \in \Omega , u|_{_{\partial \Omega }}=v|_{_{\partial \Omega }}=0$ where $q > 1$, $p \ge 1$, $\delta >0$, $\alpha > 0$, $\beta \ge 0$, $\mu \in \mathbb R$ and $\Delta$ is the Laplacian in $\mathbb R^N$.
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