Roles of density-related diffusion and signal-dependent motilities in a chemotaxis–consumption system

This study examines an initial-boundary value problem involving the system

$$\begin{aligned} \left\{ \begin{array}{l} u_t = \Delta \big (u^m\phi (v)\big ), \\[1mm] v_t = \Delta v-uv. \\[1mm] \end{array} \right. \qquad (\star ) \end{aligned}$$

in a smoothly bounded domain \(\Omega \subset \mathbb {R}^n\) with no-flux boundary conditions, where \(m, n\ge 1\). The motility function \(\phi \in C^0([0,\infty )) \cap C^3((0,\infty ))\) is positive on \((0,\infty )\) and satisfies

$$\begin{aligned} \liminf _{\xi \searrow 0} \frac{\phi (\xi )}{\xi ^{\alpha }}>0 \qquad \hbox { and }\qquad \limsup _{\xi \searrow 0} \frac{|\phi '(\xi )|}{\xi ^{\alpha -1}}<\infty , \end{aligned}$$

for some \(\alpha >0\). Through distinct approaches, we establish that, for sufficiently regular initial data, in two- and higher-dimensional contexts, if \(\alpha \in [1,2m)\), then \((\star )\) possesses global weak solutions, while in one-dimensional settings, the same conclusion holds for \(\alpha >0\), and notably, the solution remains uniformly bounded when \(\alpha \ge 1\). Furthermore, for the one-dimensional case where \(\alpha \ge 1\), the bounded solution additionally possesses the convergence property that

$$\begin{aligned} u(\cdot ,t)\overset{*}{\rightharpoonup }\ u_{\infty } \ \ \hbox {in } L^{\infty }(\Omega ) \hbox { and } v(\cdot ,t)\rightarrow 0 \ \ \hbox { in }\,\,W^{1,\infty }(\Omega ) \qquad \hbox {as } t\rightarrow \infty , \end{aligned}$$

with \(u_{\infty }\in L^{\infty }(\Omega )\). Further conditions on the initial data enable the identification of admissible initial data for which \(u_{\infty }\) exhibits spatial heterogeneity.