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Well-posedness and uniform-in-time boundedness of classical solutions areinvestigated for a three-component parabolic system which describes thedynamics of a population of cells interacting with a chemoattractant and anutrient. The former induces a chemotactic bias in the diffusive motion of thecells and is accounted for by a density-suppressed motility. Well-posedness isfirst established for generic positive and non-increasing motility functionsvanishing at infinity. Growth conditions on the motility function guaranteeingthe uniform-in-time boundedness of solutions are next identified. Finally, forsublinearly decaying motility functions, convergence to a spatially homogeneoussteady state is shown, with an exponential rate for consumption rates behavinglinearly near zero.

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