EVENTUAL SMOOTHNESS AND EXPONENTIAL STABILIZATION OF GLOBAL WEAK SOLUTIONS TO SOME CHEMOTAXIS SYSTEMS

In this paper we develop a unified approach to study the eventual smoothness and exponential stabilization of global weak solutions of two different chemotaxis systems. One is a Keller-Segel system with consumption of chemo-attractants recently studied in [Y. Tao and M. Winkler, J. Differential Equations, 252 (2012), pp. 2520-2543] and the other is a chemo-repulsion system studied in [T. Cieslak, P. Laurencot, and C. Morales-Rodrigo, Banach Center Publ., 81 (2008), pp. 105-117]. For both systems in dimension three, we prove the existence of weak solutions that become regular after certain time T > 0 and obtain the exponential convergence rate toward spatially homogeneous steady states. Our method relies on the stability of constant steady states of these chemotaxis systems in the corresponding scaling-invariant spaces. For the first system, we improve the results in Tao and Winkler and [M. Winkler, Trans. Amer. Math. Soc., 369 (2017), pp. 3067-3125] in a sense that the convexity assumption on the domain is removed and, moreover, exponential stabilization with an optimal convergence rate is obtained for the first time, while for the second system, our result is completely new. In addition, we provide an alternative proof for the chemo-repulsion system via an energy method by deriving delicate higher-order estimates.