Boundedness criteria for a chemotaxis consumption model with gradient nonlinearities

Abstract

This work deals with the consumption chemotaxis problem \begin{equation*} \begin{cases*} u_t = Δu - χ\nabla \cdot u\nabla v + λu - μu^2 - c \lvert \nabla u \rvert^γ, & \text{in $Ω\times(0,\tmax)$}, v_t = Δv - uv, & \text{in $Ω\times(0,\tmax)$}, \end{cases*} \end{equation*} in a bounded and smooth domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$, $n\ge 3$, under Neumann boundary conditions, for $\ddot{I}‡,\hat{I}»,\hat{I}¼,c>0$, $\tmax\in(0,\infty]$ and for $u_0,v_0$ positive initial data with a certain regularity. We will show that the problem has a unique and uniformly bounded classical solution for ${\hat{I}}^3\in (\frac{2n}{n+1},2]$. Moreover, we have the same result for ${\hat{I}}^3=\frac{2n}{n+1}$ and a condition that involves the parameters $c,\hat{I}¼,n,\ddot{I}‡$ and the initial data.