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Exponential Decay Rates in Quasi-Linear Hyperbolic Heat Conduction

We study different exponential decay rates that appear in quasi-linear symmetric hyperbolic systems describing heat conduction models in the context of extended thermodynamics. In normal conditions they are described using two different time scales. We show, after a study of the Cauchy problem for these systems, that for initial data close enough to equilibrium, the solutions exist globally in time and decay exponentially with the shortest relaxation time to the classical (Fourier's theory) solutions; then they continue decaying exponentially to equilibrium with the larger decaying time.