Sharp decay rates of degenerate hyperbolic-parabolic coupled system: Rectangular domain vs one-dimensional domain

Abstract

In this work, we study the long time behavior of a coupled PDEs' system consisting of one wave equation and one degenerate heat equation in two connected regions. The coupling between these two different components occurs at the interface with certain transmission conditions. For the system on rectangular domain, by frequency domain analysis, the explicit polynomial decay rate of the solutions to the system is obtained, which only depends on the degree of degeneration of the diffusion coefficient of the heat equation near the interface line. Specifically, a clear relationship between the decay rate and the degeneration parameter of the diffusion coefficient is identified. Moreover, the explicit decay rate of the solutions to such a system on one-dimensional domain is also further estimated. Especially, these two obtained decay rates are consistent with the optimal ones for the systems on rectangular domain and one-dimensional domain with constant-diffusion coefficient, obtained by Batty et al. [7] and Zhang and Zuazua [23], respectively. From this view, the obtained decay rates can all be considered as sharp ones for such kind of degenerate systems.