Summary: Relevant to modeling starvation-driven species dispersal is the chemotaxis-consumption system, featuring singular signal-dependent motilities, given by \[ \begin{cases} u_t = \Delta (u^m v^{- \alpha}), \\ v_t = \Delta v - u v, \end{cases} \] which is considered under homogeneous boundary conditions in smoothly bounded domains \(\Omega \subset \mathbb{R}^n\), \(n \geq 1\), with \(m > 1\) and \(\alpha > 0\).
In the context of concurrent strengthening of diffusion and cross-diffusion in the first equation of this system, regulated by the motility function \(v^{- \alpha}\) and the porous-medium-type diffusion concerning species, with both the diffusion and cross-diffusion exhibiting potential singularities near \(\{ v = 0 \} \), it is shown that for all sufficiently regular initial data, when the species diffusion partially conforms to mild porous-medium-type behavior (i.e., \( \max \{ 1, \frac{ n - 2}{ 4} \} < m \leq \frac{ n}{ 2}\)) and the motility function \(v^{- \alpha}\) displays appropriately strong singularities (quantified by \(\alpha > \frac{ n - 2 m}{ 4 m - n + 2})\), the system admits globally defined weak solutions, whereas in situations characterized by pronounced porous-medium-type diffusion in the species (i.e., \( m > \frac{ n}{ 2}\)), not only can global weak solutions be constructed, but they are continuous and locally bounded, even in the presence of arbitrarily strong singular behavior in the motility function \(v^{- \alpha}\) in the vicinity of \(\{ v = 0 \} \).