An improvement toward global boundedness in a fully parabolic chemotaxis with singular sensitivity in any dimension

Abstract

This paper deals with the problem of global solvability and boundedness of classical solutions to a fully parabolic chemotaxis system with singular sensitivity in any dimensional setting. In particular, We show that the system \begin{equation*} \begin{cases} u_t = \Delta u - \chi \nabla \cdot \left( \dfrac{u}{v} \nabla v \right), \\ v_t = \Delta v - v + u, \end{cases} \end{equation*} posed in a bounded domain $\Omega \subset \mathbb{R}^n$ with $n \geq 3$, admits a global bounded classical solution provided that $\chi \in (0,\chi_0)$ with $\chi_0 > \sqrt{\frac{2}{n}}$ can be determined explicitly. This result extends several existing works, which established global boundedness under the more restrictive condition $\chi < \sqrt{\frac{2}{n}}$, and shows that this threshold is not an optimal upper bound for preventing blow-up.