Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application

TL;DR

This paper develops a new structural condition to analyze decay and stability in symmetric hyperbolic systems with non-symmetric relaxation, extending classical results to models with weaker, regularity-loss dissipative structures.

Contribution

It introduces a novel structural condition encompassing the Kawashima-Shizuta condition, enabling analysis of decay and stability in systems with non-symmetric relaxation.

Findings

Established decay rates for systems with non-symmetric relaxation.

Extended stability analysis to models with weaker dissipative structures.

Provided a framework for analyzing physical models like Timoshenko and Euler-Maxwell systems.

Abstract

This paper is concerned with the decay structure for linear symmetric hyperbolic systems with relaxation. When the relaxation matrix is symmetric, the dissipative structure of the systems is completely characterized by the Kawashima-Shizuta stability condition formulated in \cite{UKS84,SK85}, and we obtain the asymptotic stability result together with the explicit time-decay rate under that stability condition. However, some physical models which satisfy the stability condition have non-symmetric relaxation term (cf.~the Timoshenko system and the Euler-Maxwell system). Moreover, it had been already known that the dissipative structure of such systems is weaker than the standard type and is of the regularity-loss type (cf.~\cite{D,IHK08,IK08,USK,UK}). Therefore our purpose of this paper is to formulate a new structural condition which include the Kawashima-Shizuta condition, and to analyze the weak dissipative structure for general systems with non-symmetric relaxation.