Global existence, uniform boundedness, and stabilization in a chemotaxis system with density-suppressed motility and nutrient consumption

TL;DR

This paper studies a complex chemotaxis system involving cell movement, chemoattractant, and nutrient, establishing conditions for global solutions, boundedness, and convergence to steady states, with implications for biological modeling.

Contribution

It introduces new conditions on motility functions ensuring global existence, boundedness, and convergence in a chemotaxis model with nutrient consumption.

Findings

Established well-posedness for a broad class of motility functions.

Identified growth conditions ensuring uniform boundedness of solutions.

Proved exponential convergence to steady state under specific decay conditions.

Abstract

Well-posedness and uniform-in-time boundedness of classical solutions are investigated for a three-component parabolic system which describes the dynamics of a population of cells interacting with a chemoattractant and a nutrient. The former induces a chemotactic bias in the diffusive motion of the cells and is accounted for by a density-suppressed motility. Well-posedness is first established for generic positive and non-increasing motility functions vanishing at infinity. Growth conditions on the motility function guaranteeing the uniform-in-time boundedness of solutions are next identified. Finally, for sublinearly decaying motility functions, convergence to a spatially homogeneous steady state is shown, with an exponential rate for consumption rates behaving linearly near zero.