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

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.

Automated summary

This study investigates the well-posedness and uniform-in-time boundedness of a three-component parabolic system describing the dynamics of cell populations interacting with a chemoattractant and a nutrient. The results show that under certain conditions, solutions converge to a spatially homogeneous steady state with an exponential rate for consumption rates behaving linearly near zero. Additionally, growth conditions on the motility function are identified to ensure the uniform-in-time boundedness of solutions.