Improvement of criteria for global boundedness in a minimal parabolic–elliptic chemotaxis system with singular sensitivity

This article deals with the following singular parabolic–elliptic chemotaxis system (0.1)$\left(\begin{array}{cc}{u}_t=\Delta u-\chi \nabla \cdot (\frac{u}{v}\nabla v),& x\in \Omega ,\\ 0=\Delta v-\alpha v+\mu u,& x\in \Omega ,\end{array}\right)$under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset R^N$ with $N\ge 3$, where parameters $\chi ,\alpha$ and $\mu$ are positive constants. Fujie, Winkler, and Yokota Fujie(2015) in 2014 and Fujie and Senba Fujie(2016) in 2016 proved that system (0.1) has a unique globally bounded classical solution when $\alpha =\mu =1$ and (0.2)$N=2\text{or}\chi <\frac{2}{N}\text{with}N\ge 3,$which has remained a critical point for over a decade. However, this article presents a new perspective and shows that assumption (0.2) does not actually constitute a turning point for global classical solutions. Among others, we prove that for all suitable smooth initial data and all $\alpha ,\mu >0$, the problem $(0.1)$ possesses a global classical solution that is uniformly bounded if (0.3)$\chi <\frac{2}{N}+\frac{2N-1}{2N^3}\cdot \sqrt{\frac{N}{2N+2}}\text{with}N\ge 3.$We remark that the current study has enhanced the result (0.2) found in Fujie(2015) and has developed a systematic technique to generate further improvements on the assumption (0.3). Future advancements are intentionally left open for the reader’s consideration.