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

In a smoothly bounded domain $Ω\subset \mathbb{R}^N$ $(N\in \mathbb{N})$, a no-flux initial-boundary value problem for the degenerate chemotaxis system with volume-filling effects, \begin{align*} u_t = \nabla \cdot (D(u,v) \nabla u - h(u,v) \nabla v), \quad v_t = Δv + g(u,v), \quad x\in Ω, \