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 \(\begin{aligned} \left\{ \begin{array}{l} u_t = \Delta \big (u^m\phi (v)\big ), \\[1mm] v_t = \Delta v-uv. \\[1mm] \end{array} \right. \qquad (\star ) \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t=\mathrm{\Delta }(u^m\varphi (v)),\hfill \\ [1mm]v_t=\mathrm{\Delta }v-uv.\hfill \\ [1mm]\hfill \end{array}\right)(\star )\end{array}$$ in a smoothly bounded domain \(\Omega \subset \mathbb {R}^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with no-flux boundary conditions, where \(m, n\ge 1\) $m,n\ge 1$ . The motility function \(\phi \in C^0([0,\infty )) \cap C^3((0,\infty ))\) $\varphi \in C^0([0,\infty ))\cap C^3((0,\infty ))$ is positive on \((0,\infty )\) $(0,\infty )$ and satisfies \(\begin{aligned} \liminf _{\xi \searrow 0} \frac{\phi (\xi )}{\xi ^{\alpha }}>0 \qquad \hbox { and }\qquad \limsup _{\xi \searrow 0} \frac{|\phi '(\xi )|}{\xi ^{\alpha -1}}<\infty , \end{aligned}\) $$\begin{array}{c}\hfill \underset{\xi \searrow 0}{lim\; inf}\frac{\varphi (\xi )}{{\xi }^{\alpha }}>0\text{and}\underset{\xi \searrow 0}{lim\; sup}\frac{|{\varphi }^{\prime }(\xi )|}{{\xi }^{\alpha -1}}<\infty ,\end{array}$$ for some \(\alpha >0\) $\alpha >0$ . Through distinct approaches, we establish that, for sufficiently regular initial data, in two- and higher-dimensional contexts, if \(\alpha \in [1,2m)\) $\alpha \in [1,2m)$ , then \((\star )\) $(\star )$ possesses global weak solutions, while in one-dimensional settings, the same conclusion holds for \(\alpha >0\) $\alpha >0$ , and notably, the solution remains uniformly bounded when \(\alpha \ge 1\) $\alpha \ge 1$ . Furthermore, for the one-dimensional case where \(\alpha \ge 1\) $\alpha \ge 1$ , the bounded solution additionally possesses the convergence property that \(\begin{aligned} u(\cdot ,t)\overset{*}{\rightharpoonup }\ u_{\infty } \ \ \hbox {in } L^{\infty }(\Omega ) \hbox { and } v(\cdot ,t)\rightarrow 0 \ \ \hbox { in }\,\,W^{1,\infty }(\Omega ) \qquad \hbox {as } t\rightarrow \infty , \end{aligned}\) $$\begin{array}{c}\hfill u(·,t)\stackrel{\ast }{\rightharpoonup }{u}_{\infty }\text{in}{L}^{\infty }(\mathrm{\Omega })\text{and}v(·,t)\to 0\text{in}{W}^{1,\infty }(\mathrm{\Omega })\text{as}t\to \infty ,\end{array}$$ with \(u_{\infty }\in L^{\infty }(\Omega )\) $u_{\infty }\in L^{\infty }(\mathrm{\Omega })$ . Further conditions on the initial data enable the identification of admissible initial data for which \(u_{\infty }\) $u_{\infty }$ exhibits spatial heterogeneity.