Boundedness in a taxis–consumption system involving signal-dependent motilities and concurrent enhancement of density-determined diffusion and cross-diffusion

Abstract

This paper is concerned with the migration-consumption taxis system involving signal-dependent motilities $$\left\{ \begin{array}{l} u_t = \Delta \big(u^m\phi(v)\big), \\[1mm] v_t = \Delta v-uv, \end{array} \right. \qquad \qquad (\star)$$ in smoothly bounded domains $\Omega\subset\mathbb{R}^n$, where $m>1$ and $n\ge2$. It is shown that if $\phi\in C^3([0,\infty))$ is strictly positive on $[0,\infty)$, for all suitably regular initial data an associated no-flux type initial-boundary value problem possesses a globally defined bounded weak solution, provided $m>\frac{n}{2}$, which is consistent with the restriction imposed on $m$ in corresponding signal production counterparts of $(\star)$ so as to establish the similar result.