Global existence and minimal decay regularity for the Timoshenko system: The case of non-equal wave speeds

TL;DR

This paper establishes the global existence and optimal decay rates of solutions to the Timoshenko system with non-equal wave speeds in critical Besov spaces, overcoming previous limitations by developing a new frequency-localization decay inequality.

Contribution

It introduces a novel frequency-localization time-decay inequality and proves optimal decay rates for the Timoshenko system in critical Besov spaces, addressing challenges posed by regularity-loss.

Findings

Global solutions exist in critical Besov spaces.

Optimal decay rates are achieved despite weaker dissipative mechanisms.

The new decay inequality captures high-frequency integrability information.

Abstract

As a continued work of [18], we are concerned with the Timoshenko system in the case of non-equal wave speeds, which admits the dissipative structure of \textit{regularity-loss}. Firstly, with the modification of a priori estimates in [18], we construct global solutions to the Timoshenko system pertaining to data in the Besov space with the regularity $s=3/2$. Owing to the weaker dissipative mechanism, extra higher regularity than that for the global-in-time existence is usually imposed to obtain the optimal decay rates of classical solutions, so it is almost impossible to obtain the optimal decay rates in the critical space. To overcome the outstanding difficulty, we develop a new frequency-localization time-decay inequality, which captures the information related to the integrability at the high-frequency part. Furthermore, by the energy approach in terms of high-frequency and low-frequency decomposition, we show the optimal decay rate for Timoshenko system in critical Besov spaces, which improves previous works greatly.