  This paper is concerned with a parabolic-elliptic Keller-Segel system where
both diffusive and chemotactic coefficients (motility functions) depend on the
chemical signal density. This system was originally proposed by Keller and
Segel in \cite{KS-1971-JTB2} to describe the aggregation phase of {\it
Dictyostelium discoideum} cells in response to the secreted chemical signal
cyclic adenosine monophosphate (cAMP), but the available analytical results are
very limited by far. Considering system in a bounded smooth domain with Neumann
boundary conditions, we establish the global boundedness of solutions in any
dimensions with suitable general conditions on the signal-dependent motility
functions, which are applicable to a wide class of motility functions. The
existence/nonexistence of non-constant steady states is studied and abundant
stationary profiles are found. Some open questions are outlined for further
pursues. Our results demonstrate that the global boundedness and profile of
stationary solutions to the Keller-Segel system with signal-dependent
motilities depend on the decay rates of motility functions, space dimensions
and the relation between the diffusive and chemotactic motilities, which makes
the dynamics immensely wealthy.
