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 $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ 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 $\ddot{I}‡,r,\hat{I}¼,\hat{I}\pm ,{\hat{I}}^2$ 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 $\hat{I}¼$ is sufficiently large.