Boundedness criteria for a chemotaxis consumption model with gradient nonlinearities

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, & $\text{in }\mathrm{\Omega }\times (0,\text{tmax})$, v_t = Δv - uv, & $\text{in }\mathrm{\Omega }\times (0,\text{tmax})$, \end{cases*} \end{equation*} in a bounded and smooth domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 3$, under Neumann boundary conditions, for $\chi ,\lambda ,\mu ,c>0$, $\text{tmax}\in (0,\mathrm{\infty }]$ 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 $\gamma \in (\frac{2n}{n+1},2]$. Moreover, we have the same result for $\gamma =\frac{2n}{n+1}$ and a condition that involves the parameters $c,\mu ,n,\chi$ and the initial data.