Asymptotic Behavior of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy

We study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law inmspace dimensions, under the Shizuta-Kawashima condition. We show that these solutions approach constant equilibrium state in theL p-norm at a rateO(t −m 2 (1−1 p)), ast→∞, for p∈[min{m,2},∞]. Moreover, we can show that we can approximate, with a faster order of convergence, the conservative part of the solution in terms of the linearized hyperbolic operator form≥2, and by a parabolic equation, in the spirit of Chapman-Enskog expansion in every space dimension. The main tool is given by a detailed analysis of the Green function for the linearized problem