The full Keller–Segel model is well-posed on nonsmooth domains

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 space dimensions 2 and 3 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. 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. AMS classification scheme numbers: 35A01, 35K45, 35K57, 35Q92, 92C17The full Keller-Segel model is well-posed on nonsmooth domains

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

“…Remark 1 It is interesting to compare Theorem 1 with the results obtained in [11] dealing with a much more complex geometrical situation. It shows that the method of time-weights combined with the special situation of bounded convex domains allows to improve the regularity index for the initial data by more than 1, meaning from B s q;r ðXÞ for s [ 3=q þ 1 and r [ 2ð1 À 3=qÞ À1 in [11] to B 3=q q;p ðXÞ.…”

“…Lipschitz domains. It was recently shown by Horstmann, Meinlschmidt and Rehberg [11] that under suitable conditions on the initial values and the geometry of the domain, one nevertheless obtains again the existence of a unique, strong, local solution to the Keller-Segel system.…”

“…The latter property is helpful to establish a priori estimates for the solution to (1.1) which are used to prove global existence results. Here we use the convexity of the domain to study the Keller-Segel system, similarly to [11], in domains with rough boundaries.…”

“…In contrast to [11] we investigate the Keller-Segel system within the framework of time-weighted Sobolev space allowing us to prove strong well-posedness results for initial data u 0 and v 0 lying in critical spaces. We note that the modified Keller-Segel system q;p ðXÞ Â B 3=q q;p ðXÞ are scaling invariant spaces for the modified Keller-Segel system in X & R 3 .…”

“…The situation of bounded convex domains is, of course, less general than the framework of Lipschitz domains as considered in [11], howewer, in the situation of bounded convex domains, several main ingredients of the approach for smooth domains, such as knowledge of the domain of the Neumann Laplacian, its maximal L p -regularity as well as the mixed derivative theorem carry over to the given situation. This allows allows us in particular to treat the case of initial data lying in critical Besov spaces as stated precisely in Theorem 1.…”