The Keller\u2013Segel system on bounded convex domains in critical spaces

Abstract

Consider the classical Keller–Segel system on a bounded convex domain varOmega subset {mathbb {R}}^3. In contrast to previous works it is not assumed that the boundary of varOmega is smooth. It is shown that this system admits a local, strong solution for initial data in critical spaces which extends to a global one provided the data are small enough in this critical norm. Furthermore, it is shown that this system admits for given T-periodic and sufficiently small forcing functions a unique, strong T-time periodic solution.