Boundedness criteria for a chemotaxis consumption model with gradient nonlinearities

• DOI: 10.48550/arxiv.2408.14250
• Publication Date: 2024-08-26
• Authors: Alessandro Columbu
• Keywords: Consumption

This work deals with the consumption chemotaxis problem \begin{equation*} \begin{cases*} u_t = \Delta u - \chi \nabla \cdot u\nabla v + \lambda \mu u^2 c \lvert \rvert^\gamma, & \text{in $\Omega\times(0,\tmax)$}, v_t uv, \end{cases*} \end{equation*} in a bounded and smooth domain $\Omega\subset\R^n$, $n\geq 3$, under Neumann boundary conditions, for $\chi,\lambda,\mu,c>0$, $\tmax\in(0,\infty]$ $u_0,v_0$ positive initial data certain regularity. We will show that has unique uniformly classical solution $\gamma\in\bigl(\frac{2n}{n+1},2\bigr]$. Moreover, we have same result $\gamma=\frac{2n}{n+1}$ condition involves parameters $c,\mu,n,\chi$ data.