  This paper deals with classical solutions to the parabolic-parabolic system
\begin{align*} \begin{cases}
  u_t=\Delta (\gamma (v) u )
  &\mathrm{in}\ \Omega\times(0,\infty), \\[1mm]
  v_t=\Delta v - v + u
  &\mathrm{in}\ \Omega\times(0,\infty), \\[1mm] \displaystyle \frac{\partial
u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0
  &\mathrm{on}\ \partial\Omega \times (0,\infty), \\[1mm]
  u(\cdot,0)=u_0, \ v(\cdot,0)=v_0 &\mathrm{in}\ \Omega, \end{cases}
\end{align*} where $\Omega$ is a smooth bounded domain in $\mathbf{R}^n$($n
\geq 3$), $\gamma (v)=v^{-k}$ ($k>0$) and the initial data $(u_0,v_0)$ is
positive and regular. This system has striking features similar to those of the
logarithmic Keller--Segel system. It is established that classical solutions of
the system exist globally in time and remain uniformly bounded in time if $k
\in (0,n/(n-2))$, independently the magnitude of mass. This constant $n/(n-2)$
is conjectured as the optimal range guaranteeing global existence and
boundedness in the corresponding logarithmic Keller--Segel system. We will
derive sufficient estimates for solutions through some single evolution
equation that some auxiliary function satisfies. The cornerstone of the
analysis is the refined comparison estimate for solutions, which enables us to
control the nonlinearity of the auxiliary equation.
