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

TL;DR

We study global boundedness for the fully parabolic chemotaxis system with singular sensitivity in $n\\ge3$. The authors introduce the energy functional $F_{\\lambda}$, augmented by a gradient-term, to obtain time-uniform bounds and to circumvent convexity requirements on the domain. They prove that there exists $\\chi_0>\\sqrt{\\frac{2}{n}}$ such that for all $\\chi\\in(0,\\chi_0)$ the system admits a unique global positive classical solution with $\\sup_{t>0}\\|u(\\cdot,t)\\|_{L^{\\infty}(\\Omega)}<\\infty$; the explicit bound $\\chi_0\\ge\\sqrt{\\frac{2(1+\\delta)}{n}}$ is provided. These results show that diffusion and signal degradation can prevent blow-up at higher dimensions even under singular sensitivity, extending previous bounds and removing the convexity restriction on the domain.

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 = 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.

Theorems & Definitions (28)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof