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

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 = Δu - χ\nabla \cdot \left( \dfrac{u}{v} \nabla v \right), \ v_t = Δv - v + u, \end{cases} \end{equation*} posed in a bounded domain $Ω\subset \mathbb{R}^n$ with $n \geq 3$, admits a global bounded classical solution provided that $χ\in (0,χ_0)$ with $χ_0 > \sqrt{\frac{2}{n}}$ can be determined explicitly. This result extends several existing works, which established global boundedness under the more restrictive condition $χ< \sqrt{\frac{2}{n}}$, and shows that this threshold is not an optimal upper bound for preventing blow-up.

💡 Research Summary

The paper studies the fully parabolic chemotaxis system with singular sensitivity
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