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

In a smoothly bounded domain Ω ⊂ RN, a no-flux initial-boundary value problem for the degenerate chemotaxis system with volume-filling effects,align* cases uₜ = ∇ · (D(u, v) ∇ u - h(u, v) ∇ v), & x∈ Ω, \ t>0, \\ vₜ = Δ v + g(u, v), & x∈ Ω, \ t>0, cases align*is considered under the assumption that $ D(1, s) = 0 $ and $ h(0, s) = h(1, s) = 0 $. Here, initial data u₀ and v₀ have suitable regularity and satisfy 0≤ u₀≤ 1 and v₀≥ 0 with ∇ v₀ · ν|∂ Ω = 0. It is proved that there exists a global weak solution such that 0≤ u≤ 1 and v≥ 0. Moreover, when $ D(r, s) = D(r) $ for all r∈[0, 1] and s∈[0, ∞) and additional conditions on D, h, and g are assumed, uniqueness of global weak solutions with the mass conservation law ∫_Ω u(x, t) \, dx = ∫_Ω u₀(x) \, dx is shown. Also, a flat-hump-shaped stationary solution is constructed in the one-dimensional setting.