Global existence and optimal decay rates for the Timoshenko system: the case of equal wave speeds

TL;DR

This paper establishes global existence and optimal decay rates for solutions to the Timoshenko system with equal wave speeds, using Besov spaces and a novel decay framework that avoids traditional spectral analysis.

Contribution

It introduces a new decay analysis method for the Timoshenko system with non-symmetric dissipation, applicable in Besov spaces, and provides optimal decay estimates.

Findings

Global solutions exist for initial data in Besov spaces.

Optimal decay rates of solutions and derivatives are achieved.

A new decay framework reduces reliance on spectral analysis.

Abstract

This work first gives the global existence and optimal decay rates of solutions to the classical Timoshenko system on the framework of Besov spaces. Due to the \textit{non-symmetric} dissipation, the general theory for dissipative hyperbolic systems ([30]) can not be applied to the Timoshenko system directly. In the case of equal wave speeds, we construct global solutions to the Cauchy problem pertaining to data in the spatially Besov spaces. Furthermore, the dissipative structure enables us to give a new decay framework which pays less attention on the traditional spectral analysis. Consequently, the optimal decay estimates of solution and its derivatives of fractional order are shown by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As a by-product, the usual decay estimate of $L^{1}(\mathbb{R})$-$L^{2}(\mathbb{R})$ type is also shown.