STABILIZATION FOR WAVE EQUATION WITH LOCALIZED KELVIN-VOIGT DAMPING ON CUBOIDAL DOMAIN: A DEGENERATE CASE

In this paper, we study the stabilization issue for a multidimensional wave equation with localized Kelvin-Voigt damping on a cuboidal domain, in which the damping region does not satisfy the geometric control condition (GCC). The variable damping coefficient is assumed to be degenerate near the interface. We prove that the system is polynomially stable with a decay rate depending on the degree of the degeneration \gamma. A relationship between the decay order and \gamma is identified. In particular, this decay rate is consistent with the optimal one for the corresponding system with constant damping coefficient (i.e., \gamma = 0) obtained in [K. Yu and Z.-J. Han, SIAM J. Control Optim., 59 (2021), pp. 1973-1988]. Moreover, it is the first result on the decay rates of the solutions to multidimensional wave equations with localized degenerate Kelvin-Voigt damping when GCC is not satisfied.