Indirect stability of a multidimensional coupled wave equations with one locally boundary fractional damping

In this work, we consider a system of multidimensional wave equations coupled by velocities with one localized fractional boundary damping. First, using a general criteria of Arendt–Batty, by assuming that the boundary control region satisfy some geometric conditions, under the equality speed propagation and the coupling parameter of the two equations is small enough, we show the strong stability of our system in the absence of the compactness of the resolvent. Our system is not uniformly stable, in general, since it is the case of the interval. Hence, we look for a polynomial decay rate for smooth initial data for our system by applying a frequency domain approach combining with a multiplier method. Indeed, by assuming that the boundary control region satisfy some geometric conditions, the waves propagate with equal speed, and the coupling parameter term is small enough, we establish a polynomial energy decay rate for smooth solutions, which depends on the order of the fractional derivative.