  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.
