Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension (Q2101062)

The initial-boundary value problem with the homogeneous Neumann data for the chemotaxis-consumption system \begin{align*} u_t&=\Delta (u\phi(u)),\\ v_t&=\Delta v-uv, \end{align*} is considered in bounded smooth domains of \(\mathbb R^n\). A very weak-strong solution is constructed for diffusions with \(\phi(\xi)\sim \xi^{-\alpha}\). Regularity and long time asymptotic behavior of those solutions is studied for \(2\le n\le 5\), \(\alpha>\frac{n-2}{6-n}\), as well as for \(n=1\) under \(0<\phi\in C^3(0,\infty)\) assumption.