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

Frequency-localization Duhamel principle and its application to the optimal decay of dissipative systems