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

TL;DR

This paper proves the global existence and boundedness of solutions in a chemotaxis model with singular sensitivity, highlighting how the chemical diffusion rate influences solution behavior in bounded domains.

Contribution

It establishes conditions for global classical solutions and their boundedness in a chemotaxis system with singular sensitivity, depending on parameters and spatial dimension.

Findings

Global solutions exist for all time under specified parameter conditions.

Solutions are bounded when the spatial dimension is at most 8.

The diffusion constant of the chemical influences solution behavior.

Abstract

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.