Decay rate of solutions of a wave equation with damping and external force (Q5947216)

The author considers the following nonlinear wave equation in a bounded domain \(\Omega\subset \mathbb R^n\): NEWLINE\[NEWLINE \begin{cases} u_{tt}-m(\|\nabla_x u\|_{L^2(\Omega)}^2)\Delta_x u+|u_t|^{q-2}u_t=f(t,x),\quad (t,x)\in \mathbb R_+\times\Omega,\\ u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),\;u\big|_{\partial\Omega}=0,\end{cases} \tag{1} NEWLINE\]NEWLINE where \(u=u(t,x)\) is an unknown function, \(f(t,x)\) is a given external force which vanishes as \(t\to\infty\), \(q\geq 2\), and \(m(z)\) is a given function which satisfies one of the following conditions: NEWLINE\[NEWLINE \begin{aligned} &1)\text{ nondegenerate case: } \quad m(z)=az^\gamma+b,\quad \gamma\geq 0, \quad a\geq 0,\;b>0;\\ &2)\text{ degenerate case: }\quad m(z)=az^\gamma,\quad a>0,\quad \gamma\geq 0. \end{aligned} NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEFor both of the choices mentioned above, the author obtaines upper bounds for the decay rate of the solution \(u(t,x)\) of (1) as \(t\to\infty\) in terms of the decay rate of the external force \(f\) and the parameters \(q\) and \(\gamma\).