Recently, a time-decay framework L2(Rn)∩B ˙2,∞-s(Rn)(s>0) has been given by  for linearized dissipative hyperbolic systems, which allows to pay less attention to the traditional spectral analysis. However, owing to interpolation techniques, those decay results for nonlinear hyperbolic systems hold true only in higher dimensions (n≥ 3), and the analysis in low dimensions (say, n= 1, 2) was left open. We try to give a satisfactory answer in the current work. First of all, we develop new time-decay properties on the frequency-localization Duhamel principle, and then it is shown that the classical solution and its derivatives of fractional order decay at the optimal algebraic rate in dimensions n= 1, 2, by using a new technique which is the so-called "piecewise Duhamel principle" in localized time-weighted energy approaches compared to . Finally, as direct applications, explicit decay statements are worked out for some relevant examples subjected to the same dissipative structure, for instance, damped compressible Euler equations, the thermoelasticity with second sound, and Timoshenko systems with equal wave speeds.
- Critical Besov spaces
- Decay estimates
- Duhamel principle
- Hyperbolic systems of balance laws
ASJC Scopus subject areas