Well-posedness and decay of solution for a transmission problem in the presence of infinite history and varying delay

Abstract

In this article we consider a transmission problem in a bounded domain with a varying delay term and the presence of infinite history in the first equation. Under suitable assumptions on the weight of the damping and the weight of the delay, we prove the existence and the uniqueness of the solution using the semigroup theory. Also we show the exponential stability of the solution by introducing a suitable Lyaponov functional.