Global existence and uniform boundedness in a fully parabolic Keller–Segel system with non-monotonic signal-dependent motility

Abstract

This paper is concerned with global solvability of a fully parabolic system of Keller–Segel-type involving non-monotonic signal-dependent motility. First, we prove global existence of classical solutions to our problem with generic positive motility function under a certain smallness assumption at infinity, which however permits the motility function to be arbitrarily large within a finite region. Then uniform-in-time boundedness of classical solutions is established whenever the motility function has strictly positive lower and upper bounds in any dimension N≥1, or decays at a certain slow rate at infinity for N≥2. Our results remove the crucial non-increasing requirement on the motility function in some recent work [10,13,16] and hence allow for both chemo-attractive and chemo-repulsive effect, or their co-existence in applications. The key ingredient of our proof lies in an important improvement of the comparison method developed in [10,16,18].