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

Abstract

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

The authors were informed from one of their colleagues that a sharper result was already available in the literature