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 $Ω\subset\R^n$, $n\geq 3$, under Neumann boundary conditions, for $χ,λ,μ,c>0$, $\tmax\in(0,\infty]$ 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 $γ\in\bigl(\frac{2n}{n+1},2\bigr]$. Moreover, we have the same result for $γ=\frac{2n}{n+1}$ and a condition that involves the parameters $c,μ,n,χ$ and the initial data.