Uniform Decay Rates for the Wave Equation with Nonlinear Damping Locally Distributed in Unbounded Domains with Finite Measure

This paper is concerned with the study of the uniform decay rates of the energy associated with the wave equation with nonlinear damping locally distributed $u_{tt} -\Delta u +a(x)\,g(u_t) =0 ~\hbox{ in }\Omega \times (0,\infty)$ subject to Dirichlet boundary conditions where $\Omega \subset \mathbb{R}^n$ $n\geq 2$ is an unbounded open set with finite measure and unbounded smooth boundary $\partial \Omega=\Gamma$. The function $a(x)$, responsible for the localized effect of the dissipative mechanism, is assumed to be nonnegative and essentially bounded and, in addition, $a(x) \geq a_0 >0~\hbox{ a.e.\ in } ~\omega,$ where $\omega=\omega' \cup \{x \in \Omega;||x|| > R\}$ ($R>0$) and $\omega'$ is a neighborhood in $\Omega$ of the closure of $\partial\Omega \cap B_R$, where $B_R=\{x \in \Omega;||x|| < R\}$.