Global well-posedness and flat-hump-shaped stationary solutions for degenerate chemotaxis systems with threshold density

In a smoothly bounded domain $ \Omega \subset \mathbb{R}^N $, a no-flux initial-boundary value problem for the degenerate chemotaxis system with volume-filling effects,$ \begin{align*} \begin{cases} u_t = \nabla \cdot (D(u, v) \nabla u - h(u, v) \nabla v), & \quad x\in \Omega, \ t>0, \\ v_t = \Delta v + g(u, v), & \quad x\in \Omega, \ t>0, \end{cases} \end{align*} $is considered under the assumption that $ D(1, s) = 0 $ and $ h(0, s) = h(1, s) = 0 $. Here, initial data $ u_0 $ and $ v_0 $ have suitable regularity and satisfy $ 0\le u_0\le 1 $ and $ v_0\ge 0 $ with $ \nabla v_0 \cdot \nu|_{\partial \Omega} = 0 $. It is proved that there exists a global weak solution such that $ 0\le u\le 1 $ and $ v\ge 0 $. Moreover, when $ D(r, s) = D(r) $ for all $ r\in[0, 1] $ and $ s\in[0, \infty) $ and additional conditions on $ D $, $ h $, and $ g $ are assumed, uniqueness of global weak solutions with the mass conservation law $ \int_\Omega u(x, t) \, dx = \int_\Omega u_0(x) \, dx $ is shown. Also, a flat-hump-shaped stationary solution is constructed in the one-dimensional setting.