Optimality of the decay rate for degenerate heat-wave coupled system

This study investigates the optimal decay rate of a degenerate coupled heat-wave system within two interconnected bounded intervals. In Han et al. (2023), the authors delved into the long-time behavior of such a coupled system and established that the solutions to this system converge to zero with a decay rate of t − 2−α/1−α , where α ∈ [0, 1) is the degree of degeneration of the diffusion coefficient in the heat equation near the interface. However, the optimality of this derived decay rate remained unresolved. The present work aims to affirm the optimality of this explicit decay rate through a meticulous asymptotic spectral analysis of the system operator by the properties of Bessel functions, complemented by an examination of the frequency characteristics of the semigroup.