  We consider a parabolic-parabolic Keller-Segel system of chemotaxis model
with singular sensitivity $u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)$,
$v_t=k\Delta v-v+u$ under homogeneous Neumann boundary conditions in a smooth
bounded domain $\Omega\subset\mathbb{R}^n$ $(n\geq2)$, with $\chi,k>0$. It is
proved that for any $k>0$, the problem admits global classical solutions,
whenever
$\chi\in\big(0,-\frac{k-1}{2}+\frac{1}{2}\sqrt{(k-1)^2+\frac{8k}{n}}\big)$. The
global solutions are moreover globally bounded if $n\le 8$. This shows an exact
way the size of the diffusion constant $k$ of the chemicals $v$ effects the
behavior of solutions.
