Well-posedness of a Parabolic-hyperbolic Keller-Segel System in the   Sobolev Space Framework
  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. 
Chao Deng and Tong Li
math.AP