Global existence and asymptotic behavior of smooth solutions to a coupled hyperbolic-parabolic system

In this paper, we are concerned with a model arising from biology, which is a coupled hyperbolic-parabolic system. Firstly, we prove global existence of smooth solutions to the Cauchy problem without any smallness assumption on the initial data. Secondly, we prove both global existence and asymptotic behavior of smooth solutions, provided the initial data are of small H1H1-norm energy but possibly large HsHs-norm energy. Finally, if the Hs∩L1Hs∩L1-norm of initial data are sufficiently small, we also establish decay rates of the global smooth solutions. These results are obtained by constructing a new nonnegative convex entropy and combining spectral analysis with energy methods.