An $$\mathrm{S}$$ & $$\mathrm{K}$$ mixed condition for symmetric hyperbolic systems with non-symmetric relaxations

Abstract

We establish asymptotic stability results and provide explicit decay rate for linear symmetric hyperbolic systems in terms of the dissipative structure. In particular, we are concerned with the model systems which have a non-symmetric relaxation term and therefore their decay estimates and energy estimates are of regularity-loss type. We present a kind of S & K mixed criterion which is applicable also to weakly dissipative models including the Timoshenko–Cattaneo system, for which any already known criterion did not work.