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