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.
Categories: math.AP