The full Keller–Segel model is well-posed on nonsmooth domains**This work is dedicated to Prof Willi Jäger.

Abstract

In this paper we prove that the full Keller–Segel system, a quasilinear strongly coupled reaction-crossdiffusion system of four parabolic equations, is well-posed in the sense that it always admits an unique local-in-time solution in an adequate function space, provided that the initial values are suitably regular. The proof is done via an abstract solution theorem for nonlocal quasilinear equations by Amann and is carried out for general source terms. It is fundamentally based on recent nontrivial elliptic and parabolic regularity results which hold true even on rather general nonsmooth spatial domains. For space dimensions 2 and 3, this enables us to work in a nonsmooth setting which is not available in classical parabolic systems theory. Apparently, there exists no comparable existence result for the full Keller–Segel system up to now. Due to the large class of possibly nonsmooth domains admitted, we also obtain new results for the ‘standard’ Keller–Segel system consisting of only two equations as a special case.