Global Existence and Convergence Rates for the Strong Solutions in H2 to the 3D Chemotaxis Model

Abstract

We are concerned with a 3D chemotaxis model arising from biology, which is a coupled hyperbolic-parabolic system. We prove the global existence of a strong solution when $H^{\mathrm{2}}$-norm of the initial perturbation around a constant state is sufficiently small. Moreover, if additionally, $L^1$-norm of the initial perturbation is bounded; the optimal convergence rates are also obtained for such a solution. The proofs are obtained by combining spectral analysis with energy methods.