Optimal Decay Rate for a Degenerate Hyperbolic-Parabolic Coupled System

Han Z, Yu K, Zuazua E (2026)

Publication Language: English

Publication Status: Accepted

Publication Type: Unpublished / Preprint

Future Publication Type: Article in Edited Volumes

Publication year: 2026

Publisher: SIAM J. Control and Optimization

Open Access Link: https://dcn.nat.fau.eu/wp-content/uploads/han-yu-zuazua-final.pdf

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 α and β, with α,β ∈[0,1). For smooth initial data, we establish that the system exhibits polynomial stabilization to zero as t→∞, with an explicit polynomial decay rate of order (2−α)(2−β) 2(1−α)+β , determined by the degeneracy exponents α and β. 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 degenerate heat-wave coupled systems over the full range of exponents α,β ∈[0,1). 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.

Involved external institutions

Tianjin University of Science and Technology / 天津科技大学 CN China (CN) Nankai University (NKU) / 南开大学 CN China (CN)

How to cite

APA:

Han, Z., Yu, K., & Zuazua, E. (2026). Optimal Decay Rate for a Degenerate Hyperbolic-Parabolic Coupled System. (Unpublished, Accepted).

MLA:

Han, Zhongjie, Kai Yu, and Enrique Zuazua. Optimal Decay Rate for a Degenerate Hyperbolic-Parabolic Coupled System. Unpublished, Accepted. 2026.

BibTeX: Download