Decay of a 3-D hyperbolic-parabolic system modeling chemotaxis

Abstract

We investigate a 3-D hyperbolic-parabolic system arising from biology. We prove the global existence of a solution when the H2 norm of initial data is small, but the higher order derivatives can be arbitrarily large. Furthermore, when the H−s norm (0 ≤ s < 3/2) or norm (0 < s ≤ 3/2) of initial data is finite, by a regularity interpolation trick, we prove the optimal decay rates of the solution. Particularly, the optimal decay rates of the higher order spatial derivatives of the solution are obtained. As an immediate byproduct, the Lp – L2 (1 ≤ p ≤ 2) type of the optimal decay rates follow without requiring the Lp norm of initial data is small.