Optimal Decay Rate for a Degenerate Hyperbolic-Parabolic Coupled System

Abstract.

We investigate the long-time asymptotic behavior of a one-dimensional degenerate hyperbolic-parabolic coupled PDE system modeling diffusion-wave interactions with degeneracy at the interface. The system consists of a degenerate heat-wave equation on a finite interval, where the degeneracy strengths of the diffusion and wave propagation are characterized respectively by the parameters [Formula: see text] and [Formula: see text], with [Formula: see text]. For smooth initial data, we establish that the system exhibits polynomial stabilization to zero as [Formula: see text], with an explicit polynomial decay rate [Formula: see text] determined by the degeneracy exponents [Formula: see text] and [Formula: see text]. A rigorous spectral analysis of the system operator further confirms the optimality of this decay rate. This constitutes the first proof of the sharp decay rate estimate for the degenerate heat-wave coupled systems over the full range of exponents [Formula: see text]. Our methodology combines frequency-domain techniques with asymptotic properties of Bessel functions and a detailed analysis of the underlying Sturm–Liouville structure. The results reveal the stabilizing role of degenerate diffusion and provide novel insights into the interplay between degeneracy and dissipation in hyperbolic-parabolic coupled systems.