On some sharper boundedness conditions in the higher-dimensional chemotaxis-consumption model

TL;DR

This paper extends the boundedness conditions for solutions in higher-dimensional chemotaxis-consumption models, showing that larger chemotactic sensitivities still yield bounded solutions in dimensions five and above.

Contribution

It improves existing results by allowing larger chemotactic sensitivity constants in dimensions five and higher, ensuring bounded solutions under broader conditions.

Findings

Bounded solutions exist for larger hi in dimensions  and above.

The results generalize previous bounds established for lower dimensions.

Enhanced boundedness conditions contribute to understanding chemotaxis models in higher dimensions.

Abstract

For the classical zero-flux chemotaxis-consumption model \begin{equation*} u_t= \Delta u - \chi \nabla \cdot (u \nabla v) \quad \textrm{and}\quad v_t=\Delta v- uv, \quad \text{ with } (x,t)\in \Omega \times (0,T_{max}), \end{equation*} $\Omega$ being a bounded and smooth domain of $\mathbb{R}^n$, $n\geq 3$, $\chi$ some positive number and $T_{max} \in (0,\infty]$, the following was established in a paper by Tao: for every sufficiently regular initial data $u(x,0)=u_0(x)\geq 0$ and $v(x,0)=v_0(x) \geq 0$, there is $\chi(\lVert v_0 \rVert_{L^\infty(\Omega)})$ such that for all $0<\chi\leq \chi(\lVert v_0 \rVert_{L^\infty(\Omega)})$, the initial-boundary value problem has a unique classical solution in $\Omega \times (0,\infty)$ which is bounded. In this paper, whenever $n\geq 5$, we obtain the same claim for larger values of the constant $\chi(\|v_0\|_{L^{\infty}(\Omega)})$.