On solutions of nondegenerate wave equations with nonlinear damping terms

Abstract

Let $\Omega $ be a bounded domain in $\Bbb R^N$ with smooth boundary $\partial \Omega$. In this paper, we consider the existence and energy decay of solutions of the following problem: \begin{align} & u_{tt}(t,x)-(a+b (\|\nabla u(t,x)\|_2^2+\|\nabla v(t,x)\|_2^2)^\gamma) \Delta u(t,x)+\beta \Delta^2 u(t,x) \nonumber \\ & \quad+\delta|u_t(t,x)|^{\rho}u_t(t,x) =\mu|u(t,x)|^{\alpha}u(t,x), \quad x \in \Omega,\ t \in [0, T], \nonumber \\ & v_{tt}(t,x)-(a+b (\|\nabla u(t,x)\|_2^2+\|\nabla v(t,x)\|_2^2)^\gamma ) \Delta v(t,x)+\beta \Delta^2 v(t,x) \nonumber \\ & \quad+\delta|v_t(t,x)|^{\rho}v_t(t,x) =\mu|v(t,x)|^{\alpha}v(t,x), \quad x \in \Omega,\ t \in [0, T], \tag*{(1.1)} \\ & u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x), \quad x \in \Omega, \nonumber \\ & v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x), \quad x \in \Omega, \nonumber \\ & u|_{\partial \Omega}=v|_{\partial \Omega}=0, \nonumber \end{align} where $T >0$, $\alpha > 0$, $\rho \geq 0$, $\delta >0$, $\mu \in \Bbb R$, $a+b \geq 0$, $b \geq 0$, $\gamma \geq 1 $ and $$ \|\nabla u\|^2_2 =\sum_{i=1}^N \int_\Omega|\frac{\partial u}{\partial x_i}(t,x)|^2dx, \quad u_t=\frac{\partial u}{\partial t}\quad\hbox{and}\quad \Delta u=\sum_{i=1}^N \frac{\partial^2u}{\partial x_i^2}. $$