On solutions of quasilinear wave equations with nonlinear damping terms

Abstract

In this paper we consider the existence and asymptotic behavior of solutions of the following problem:

$$\begin{gathered} utt(t,x) - (\alpha + \beta {\kern 1pt} \parallel \,\nabla u(t,x)\,\parallel _2^2 + \beta \,\parallel \,\nabla u(t,x)\,\parallel _2^2 )\Delta u(t,x) + \delta \,|\,u_t (t,x)^{p - 1} u_t (t,x) \hfill \\ \;\;\; = \mu \,|\,u(t,x){\kern 1pt} \,|^{q - 1} u(t,x),\;\;\;x \in \Omega ,\;\;\;t \geqslant 0, \hfill \\ \end{gathered}$$

$$\begin{gathered} utt(t,x) - (\alpha + \beta {\kern 1pt} \parallel \,\nabla u(t,x)\,\parallel _2^2 + \beta \,\parallel \,\nabla {\kern 1pt} \nu (t,x)\,\parallel _2^2 )\Delta \nu (t,x) + \delta \,|\,\nu _t (t,x)^{p - 1} \nu _t (t,x) \hfill \\ \;\;\; = \mu \,|\,\nu (t,x){\kern 1pt} \,|^{q - 1} \nu (t,x),\;\;\;x \in \Omega ,\;\;\;t \geqslant 0, \hfill \\ \end{gathered}$$

$$u(0,x) = u_0 (x),\;\;u_t (0,x) = u_1 (x),\;\;x \in \Omega ,$$

$$v(0,x) = v_0 (x),\;\;v_t (0,x) = v_1 (x),\;\;x \in \Omega ,$$

$$u{\kern 1pt} |_{\;\partial \Omega } = v{\kern 1pt} |_{\;\partial \Omega } = 0$$

where q>1, q⩾1, δ>0, α>0, β⩾0, $\mu \in {\mathbb{R}}$ is the Laplacian in ${\mathbb{R}}^N$.