The Keller–Segel system on bounded convex domains in critical spaces

Abstract

AbstractConsider the classical Keller–Segel system on a bounded convex domain $$\varOmega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 . In contrast to previous works it is not assumed that the boundary of $$\varOmega $$ Ω 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.