Well-posedness and exponential stability for a Klein-Gordon system with locally distributed viscoelastic dampings in a past-history framework

Abstract

Our main goal in the present work is to prove the well-posedness as well as the exponential stability for a strongly coupled Klein-Gordon system posed on a bounded domain of R2 with smooth boundary, subject to locally distributed viscoelastic effects driven by nonnegative functions a(x) and b(x) acting on Ω, where a=b=0 in a region A of Ω. Some difficulties imposed by the presence of integrals terms force us to consider a perturbed problem in order to obtain exponential decay rates. For this purpose, microlocal analysis tools combined with unique continuation property were used.