Global Existence and Uniform Boundedness in a Fully Parabolic Keller-Segel System with Non-monotonic Signal-dependent Motility

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 $N\geq1$, or decays at a certain slow rate at infinity for $N\geq2$. Our results remove the crucial non-increasing requirement on the motility function in some recent work \cite{JLZ22,FJ19b,FS22} 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 \cite{JLZ22,FJ19b,LJ21}.