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This paper is concerned with global solvability of a fully parabolic systemof Keller--Segel-type involving non-monotonic signal-dependent motility. First,we prove global existence of classical solutions to our problem with genericpositive motility function under a certain smallness assumption at infinity,which however permits the motility function to be arbitrarily large within afinite region. Then uniform-in-time boundedness of classical solutions isestablished whenever the motility function has strictly positive lower andupper bounds in any dimension $N\geq1$, or decays at a certain slow rate atinfinity for $N\geq2$. Our results remove the crucial non-increasingrequirement on the motility function in some recent work\cite{JLZ22,FJ19b,FS22} and hence allow for both chemo-attractive andchemo-repulsive effect, or their co-existence in applications. The keyingredient of our proof lies in an important improvement of the comparisonmethod developed in \cite{JLZ22,FJ19b,LJ21}.

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