Boundedness criteria for a chemotaxis consumption model with gradient nonlinearities

TL;DR

This paper establishes boundedness criteria for solutions to a chemotaxis consumption model with gradient nonlinearities, identifying conditions on parameters and initial data that ensure global existence and uniform boundedness.

Contribution

It provides new boundedness criteria for the chemotaxis model with gradient nonlinearities, extending previous results to a wider range of nonlinearities and parameter conditions.

Findings

Unique, globally bounded classical solutions for certain gradient nonlinearities.

Boundedness holds for gamma in (n/(n+1), 2] and at the critical value with additional conditions.

Conditions depend on parameters c, mu, chi, initial data, and dimension n.

Abstract

This work deals with the consumption chemotaxis problem \begin{equation*} \begin{cases*} u_t = \Delta u - \chi \nabla \cdot u\nabla v + \lambda u - \mu u^2 - c \lvert \nabla u \rvert^\gamma, & \text{in $\Omega\times(0,\tmax)$}, v_t = \Delta v - uv, & \text{in $\Omega\times(0,\tmax)$}, \end{cases*} \end{equation*} in a bounded and smooth domain $\Omega\subset\R^n$, $n\geq 3$, under Neumann boundary conditions, for $\chi,\lambda,\mu,c>0$, $\tmax\in(0,\infty]$ 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 $\gamma\in\bigl(\frac{2n}{n+1},2\bigr]$. 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.