Boundedness criteria for a chemotaxis consumption model with gradient nonlinearities

Abstract

This work deals with the consumption chemotaxis problem$\{\begin{array}{cc}{u}_t=\Delta u-\chi \nabla \cdot u\nabla v+\lambda u-\mu u^2-c|\nabla u{|}^{\gamma },& \text{in}\Omega \times (0,T_{\text{max}}),\\ v_t=\Delta v-uv,& \text{in}\Omega \times (0,T_{\text{max}}),\end{array}$ in a bounded and smooth domain $\Omega \subset R^n$, $n\ge 3$, under homogeneous Neumann boundary conditions, for $\chi ,\lambda ,\mu ,c>0$, $T_{\text{max}}\in (0,\infty ]$ 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 $\gamma \in (\frac{2n}{n+1},2]$. 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.