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

Abstract

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.