On a Keller--Segel System with Density-Suppressed Motility, Indirect Signal Production, and External Sources

TL;DR

This paper studies a Keller--Segel model with density-dependent motility and external sources, proving global existence and boundedness of solutions, and identifying a critical mass phenomenon in 2D cases.

Contribution

It establishes global classical solutions for a broad class of non-increasing motility functions, including cases with external sources, and characterizes conditions for boundedness and blow-up.

Findings

Global existence of classical solutions in arbitrary dimensions.

External damping with superlinear growth ensures bounded solutions.

Critical mass phenomenon in 2D with exponential motility function.

Abstract

This paper investigates an initial-Neumann boundary value problem for a Keller--Segel system with parabolic-parabolic-ODE coupling. The model incorporates a signal-dependent, non-increasing motility function that, through indirect signal production, captures a self-trapping effect suppressing cellular movement at high densities. We establish the global existence of classical solutions in arbitrary spatial dimensions for a broad class of non-increasing motility functions, both with and without external source terms. Furthermore, we demonstrate that any external damping source exhibiting superlinear growth ensures uniform-in-time boundedness. Conversely, in the absence of such damping, solutions may become unbounded as time tends to infinity. More precisely, in the two-dimensional homogeneous case with the exponentially decaying motility function $\gamma(v) = e^{-v}$, a critical mass phenomenon emerges: classical solutions remain uniformly bounded for subcritical initial mass, while supercritical initial masses can lead to infinite-time blow-up. Our analysis relies on the construction of carefully designed auxiliary functions along with refined comparison methods and iteration arguments.