Global boundedness of solutions in a parabolic-parabolic chemotaxis system with singular sensitivity

We consider a parabolic-parabolic Keller-Segel system of chemotaxis model with singular sensitivity $u_t=Δu-χ\nabla\cdot(\frac{u}{v}\nabla v)$, $v_t=kΔv-v+u$ under homogeneous Neumann boundary conditions in a smooth bounded domain $Ω\subset\mathbb{R}^n$ $(n\geq2)$, with $χ,k>0$. It is proved that for any $k>0$, the problem admits global classical solutions, whenever $χ\in\big(0,-\frac{k-1}{2}+\frac{1}{2}\sqrt{(k-1)^2+\frac{8k}{n}}\big)$. 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.

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