Global boundedness in a chemotaxis-growth system with weak singular sensitivity in any dimensional setting

This paper concerns with the following parabolic–elliptic chemotaxis-growth system with weak singular sensitivity (1)$\left(\begin{array}{cc}{u}_t=\Delta u-\chi \nabla \cdot (\frac{u}{v^k}\nabla v)+ru-\mu u^2,& x\in \Omega ,\\ 0=\Delta v-\alpha v+\beta u,& x\in \Omega ,\end{array}\right)$under no-flux boundary conditions in a smoothly bounded domain $\Omega \subset R^n$ with $n\ge 3,$ the parameters $\chi ,r,\mu ,\alpha ,\beta >0$ are positive constants and $k\in (0,1).$

In the past few years, there has been considerable interest in exploring whether logistic kinetics can sufficiently guarantee the global existence and boundedness of classical solutions or prevent finite-time blow-up in various chemotaxis models. In particular, significant results have been reported by numerous authors regarding system Eq. (1). For instance, a recent study (Kurt 2025) demonstrated that Eq. (1) admits a globally bounded classical solution provided that $\mu$ is sufficiently large and $k<\frac{1}{2}+\frac{1}{n}$ with $n\ge 2.$

It is then natural to ask whether the boundedness of classical solutions of Eq. (1) can be established independently of the condition connecting $k$ and $n$ or considering a milder condition for $\mu$. This paper addresses this question by providing an extension of the upper bound for $k$ and a weaker condition for $\mu$ and proves that for all suitably smooth initial data and any given $k\in (0,1),$ Eq. (1) possesses a globally bounded classical solution if $\mu$ is suitably large such that $\mu >\left(\begin{array}{cc}\beta {\chi }^{\frac{1}{1-k}}{2}^{\frac{k}{1-k}}+4^{\frac{2-k}{1-k}}\beta {\left(\chi k\right)}^{\frac{1}{1-k}},& n=3,\\ \beta {\chi }^{\frac{1}{1-k}}{\left(\frac{n}{2}\right)}^{\frac{k}{1-k}}+2\beta {\left(\chi k\right)}^{\frac{1}{1-k}}{(2+\frac{4}{n-2})}^{\frac{1}{2}}{n}^{\frac{1}{1-k}},& n\ge 4.\end{array}\right)$