We study the global solvability of the following chemotaxis system in an open bounded domain with smooth boundary \(\Omega \subset \mathbb {R}^n\) with \(n \ge 2\) \(\begin{aligned} {\left\{ \begin{array}{ll} u_t = \Delta (uv^{-\alpha })+ ru -\mu u^{\frac{n+2}{2}}\ln ^\gamma (u+e),\\ v_t = \Delta v -uv, \end{array}\right. } \end{aligned}\) where \(\alpha >0\) , \(r \in \mathbb {R}\) , \(\mu >0\) , and \(\gamma \ge 0\) . In a recent paper [19], it was proven that \(\gamma =0\) , \(\alpha \in (0,1)\) and \(n=2\) can ensure the existence of global solutions with an additional smallness assumption for initial data \(v_0\) . In this paper, we show that the smallness assumption can be removed when \(n=2\) and \(\alpha \in (0,1]\) . Moreover, we also demonstrate that solutions exist globally for any \(\alpha > 0\) when \(\gamma >\frac{(n-1)(n+2)}{2n}\) for any \(n \ge 2\) .
