Global analysis of smooth solutions to a hyperbolic-parabolic coupled system

We investigate a model arising from biology, which is a hyperbolic-parabolic coupled system. First, we prove the global existence and asymptotic behavior of smooth solutions to the Cauchy problem without any smallness assumption on the initial data. Second, if the H ^s ∩ L ¹-norm of initial data is sufficiently small, we also establish decay rates of the global smooth solutions. In particular, the optimal L ² decay rate of the solution and the almost optimal L ² decay rate of the first-order derivatives of the solution are obtained. These results are obtained by constructing a new nonnegative convex entropy and combining spectral analysis with energy methods.