Stability for Semilinear Wave Equation in an Inhomogeneous Medium with Frictional Localized Damping and Acoustic Boundary Conditions
Marcelo M. Cavalcanti, V. N. Domingos Cavalcanti, C.L. Frota, André Vicente
Abstract
This paper is concerned with the study of local decay rates of the energy associated to a semilinear wave equation in an inhomogeneous medium with frictional localized damping. The problem is considered in $\Omega\subset \mathbb{R}^n$, an open, bounded, and connected set, $n\geq 2$, with smooth boundary $\Gamma=\Gamma_0\cup\Gamma_1$ such that $\overline{\Gamma_0}\cap \overline{\Gamma_1}=\emptyset$. On $\Gamma_0$ we consider the homogeneous Dirichlet conditions while on $\Gamma_1$ we consider the acoustic boundary conditions with source term and nonlinear frictional dissipation. To prove the main result we used the microlocal analysis tools of Burq and Gérard [Contrôle Optimal des équations aux dérivées partielles, 2001] combined with Lasiecka and Tataru [Differential Integral Equations, 6 (1993), pp. 507--533] arguments and a construction of an appropriate damping region which can be considered with measure as small as desired, however totally distributed on $\Omega$.