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Consider 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.

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