Bounded solutions in three-dimensional cross-degenerate nutrient-taxis systems

In a smoothly bounded domain $ \Omega \subset \mathbb{R}^n $, $ n\le 3 $, we consider the Neumann initial-boundary problem for the cross-degenerate chemotaxis-consumption system$ \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \nabla \cdot (u^ \alpha v\nabla u) - \chi \nabla\cdot (u^{ \alpha+1} v\nabla v) + \ell uv, \\ v_t = \Delta v - uv, \end{array} \right. \qquad \qquad (\star) \end{eqnarray*} $with $ \alpha\ge 0 $, which was proposed in [Nature 578, 588-592 (2020)] for $ \alpha = 0 $ to explore the role of motility in boosting bacterial diversity, and in [Physica A 392, 5644-5662 (2013)] for $ \alpha = 1 $ to explain the formation of complex patterns in populations of bacteria in nutrient-poor environments.It is shown that for any given suitably regular initial data $ (u_0, v_0) $ with $ u_0\ge 0 $ and $ v_0>0 $, there exists $ \delta = \delta(u_0, v_0, \alpha)>0 $ such that whenever $ \chi\in [0, \delta] $ and $ \ell\in [0, \delta] $, the problem admits a global bounded solution, weak in the case when $ \alpha>0 $ and classical if $ \alpha = 0 $.