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 $$\{\begin{array}{l}{u}_t=\Delta (u^m\varphi (v)),\\ v_t=\Delta v-uv,\end{array}(\star )$$ in smoothly bounded domains $\Omega \subset {\mathbb{R}}^n$, where $m>1$ and $n\ge 2$. It is shown that if $\varphi \in C^3([0,\infty ))$ 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.

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