Global boundedness in the higher-dimensional fully parabolic chemotaxis with weak singular sensitivity and logistic source

We consider the following chemotaxis system under homogeneous Neumann boundary conditions in a smooth, open, bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with $n\ge 3$: \begin{equation*} \begin{cases} u_t = Δu - χ\nabla \cdot \left( \frac{u}{v^k} \nabla v \right) + ru - μu^2, & $\text{in }$ Ω\times (0,T_{\rm max}), v_t = Δv - αv + βu, & $\text{in }$ Ω\times (0,T_{\rm max}), \end{cases} \end{equation*} where $k\in (0,1)$, and $\chi ,r,\mu ,\alpha ,\beta$ are positive parameters. In this paper, we demonstrate that for suitably smooth initial data, the problem admits a unique nonnegative classical solution that remains globally bounded in time when $\mu$ is sufficiently large.