Large time behavior for the hyperbolic-parabolic coupled system with the regularity-loss structure

TL;DR

This paper investigates the long-term behavior of a hyperbolic-parabolic coupled system derived from thermoelastic models, revealing new asymptotic profiles, decay estimates, and a simplified wave model with Riesz potential dissipation.

Contribution

It introduces novel large time asymptotic profiles with regularity-loss structure and establishes optimal decay estimates for the system's solutions.

Findings

Derived new large time asymptotic profiles with regularity-loss structure.

Established optimal decay estimates with higher regularity requirements.

Identified the wave equation with Riesz potential dissipation as an approximate model for the system.

Abstract

This paper considers the hyperbolic-parabolic coupled system, arising from the generalized thermoelastic coupled system, in the whole space $\mathbb{R}^n$. We study some qualitative properties for an energy term by diagonalization procedures, and for the solution by the WKB analysis. Particularly, we derive new large time asymptotic profiles with the regularity-loss structure (from the biharmonic parabolic equation and the diffusion wave equation with the Riesz potential operator) and optimal decay estimates with suitable higher regularities for the Cauchy data. Finally, we discover that the wave equation with the Riesz potential dissipation is a large time approximated model of our hyperbolic-parabolic coupled system.