The energy decay rate of a transmission system governed by the degenerate wave equation with drift and under heat conduction with the memory effect

In this paper, we investigate the stabilization of the transmission problem of the degenerate wave equation and the heat equation under the Coleman–Gurtin heat conduction law or Gurtin–Pipkin law with the memory effect. We investigate the polynomial stability of this system when employing the Coleman–Gurtin heat conduction, establishing a decay rate of type $t^{-4}$. Next, we demonstrate exponential stability in the case when Gurtin–Pipkin heat conduction is applied.