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

Abstract

We investigate a model arising from biology, which is a hyperbolicparabolic 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∩L¹-norm of initial data is sufficiently small, we also establish decay rates of the global smooth solutions. In particular, the optimal L² decay rate of the solution and the almost optimal L² 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.

Keywords

Global analysis / hyperbolic-parabolic system / decay rate / convex entropy

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg