  Very recently, a new decay framework has been given by [51] for linearized
dissipative hyperbolic systems satisfying the Kawashima-Shizuta condition on
the framework of Besov spaces, which allows to pay less attention on the
traditional spectral analysis. However, owing to interpolation techniques, the
analysis remains valid only for nonlinear systems in higher dimensions
$(n\geq3)$ and the corresponding case of low dimensions was left open, which
provides the main motivation of this work. Firstly, we develop new time-decay
properties on the frequency-localization Duhamel principle. Furthermore, it is
shown that the classical solution and its derivatives of fractional order decay
at the optimal algebraic rate in dimension 1, by time-weighted energy
approaches in terms of low-frequency and high-frequency decompositions.
Finally, as applications, decay results for several concrete systems subjected
to the same dissipative structure as general hyperbolic systems, for instance,
damped compressible Euler equations, thermoelasticity with second sound and
Timoshenko systems with equal wave speeds, are also obtained.
