On solutions of quasilinear wave equations with nonlinear damping terms

Park, Jong Yeoul ; Bae, Jeong Ja

Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 565-585.

Résumé

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 $\mathbb R^N$.

Détail

Classification : 35B35, 35L15, 35L70, 65M60
Keywords: quasilinear wave equation; existence and uniqueness; asymptotic behavior; Galerkin method