  This paper establishes the global uniform-in-time boundedness of solutions to
the following Keller-Setel system with signal-dependent diffusion and
chemotaxis \begin{equation}\left\{ \begin{array}{ll}
u_t=\nabla\cdot(\gamma(v)\nabla u - u\phi(v)\nabla v),\quad & x\in \Omega,
t>0,\\ v_t = d\Delta v- v+u,\quad & x\in \Omega, t>0
\end{array}\right.\end{equation} in a bounded domain
$\Omega\subset\mathbb{R}^N(N\leq4)$ with smooth boundary, where the
density-dependent motility functions $\gamma(v)$ and $\phi(v)$ denote the
diffusive and chemotactic coefficients, respectively. The model was originally
proposed by Keller and Segel in \cite{Keller-1} to describe the aggregation
phase of Dictyostelium discoideum cells, where the two motility functions
satisfy a proportional relation $\chi(v)=(\alpha -1)\gamma'(v)$ with $\alpha>0$
denoting the ratio of effective body length (i.e. distance between receptors)
to the step size. The major technical difficulty in the analysis is the
possible degeneracy of diffusion. In this work, we show that if $\gamma(v)>0$
and $\phi(v)>0$ are smooth on $[0,\infty)$ and satisfy $$\inf_{v\geq0}
\frac{d\gamma(v)}{v\phi(v)(v\phi(v)+d-\gamma(v))_+}>\frac{N}{2},$$ then the
above Keller-Segel system subject to Neumann boundary conditions admits
classical solutions uniformly bounded in time. The main idea of proving our
results is the estimates of a weighted functional $\int_\Omega u^{p}v^{-q}dx$
for $p>\frac{N}{2}$ by choosing a suitable exponent $p$ depending on the
unknown $v$, by which we are able to derive a uniform $L^\infty$-norm of $v$
and hence rule out the diffusion degeneracy.
