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 laws in m space dimensions, under the Shizuta-Kawashima condition. We show that these solutions approach a constant equilibrium state in the L p -norm at a rate O(t -(m/2)(1-1/ p) ) as t -> ? for p ? [min{m, 2}, ?]. Moreover, we can show that we can approxi- mate, with a faster order of convergence, the conservative part of the solution in terms of the linearized hyperbolic operator for m >= 2, and by a parabolic equa- tion, 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.