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

$$\begin{aligned} \left\{ \begin{array}{l} u_t = \Delta \big (u^m\phi (v)\big ), \\ v_t = \Delta v -uv, \end{array} \right. \qquad \qquad (*) \end{aligned}$$

in smoothly bounded domains $\Omega \subset \mathbb {R}^n$, where $m>1$ and $n\ge 2$. 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.