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 $ \Omega \subset \mathbb{R}^n $ with $ n \geq 3 $:$ \begin{equation*} \begin{cases} u_t = \Delta u - \chi \nabla \cdot \left( \frac{u}{v^k} \nabla v \right) + ru - \mu u^2, & \text{in } \Omega \times (0, T_{\rm max}), \\ v_t = \Delta v - \alpha v + \beta u, & \text{in } \Omega \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.