  We study the global strong solutions to a 3-dimensional parabolic-hyperbolic
Keller-Segel model with initial data close to a stable equilibrium with
perturbations belonging to $L^2(\mathbb R^3)\times H^1(\mathbb{R}^3)$. We
obtain global well-posedness and decay property. Furthermore, if the mean value
of initial cell density is smaller than a suitabale constant, then the chemical
concentration decays exponentially to zero as $t$ goes to infinity. Proofs of
the main results are based on an application of Fourier analysis method to
uniform estimates for a linearized parabolic-hyperbolic system and also based
on the smoothing effect of the cell density as well as the damping effect of
the chemical concentration.
