  In this paper, we investigate the stabilization of transmission problem of
degenerate wave equation and heat equation under Coleman-Gurtin heat conduction
law or Gurtin-Pipkin law with 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.
