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In this paper we consider the initial Neumann boundary value problem for adegenerate Keller--Segel model which features a signal-dependent non-increasingmotility function. The main obstacle of analysis comes from the possibledegeneracy when the signal concentration becomes unbounded. In the currentwork, we are interested in boundedness and exponential stability of theclassical solution in higher dimensions. With the aid of a Lyapunov functionaland a delicate Alikakos--Moser type iteration, we are able to establish atime-independent upper bound of the concentration provided that the motilityfunction decreases algebraically. Then we further prove the uniform-in-timeboundedness of the solution by constructing of an estimation involving aweighted energy. Finally, thanks to the Lyapunov functional again, we prove theexponential stabilization toward the spatially homogeneous steady states. Ourboundedness result improves those in \cite{Anh19,FJ20a} and the exponentialstabilization is obtained for the first time.

Boundedness and Exponential Stabilization in a Parabolic-Elliptic Keller--Segel Model with Signal-dependent Motilities for Local Sensing Chemotaxis