Application of the Moser–Trudinger inequality in the construction of global solutions to a strongly degenerate migration model - Winkler, Michael (Bulletin of mathematical sciences, 2023)
A no-flux initial-boundary value problem for the cross-diffusion system u t = Δ ( u ϕ ( v ) ) , v t = Δ v − u v is considered in smoothly bounded domains Ω ⊂ ℝ n with n ≤ 2 . It is shown that whenever ϕ ∈ C 0 ( [ 0 , ∞ ) ) is positive on ( 0 , ∞ ) and such that lim inf ξ ↘ 0 ϕ ( ξ ) ξ α > 0 ( ⋆ ) for some α > 0 , for all suitably regular positive initial data a global very weak solution, particularly preserving mass in its first component, can be constructed. This extends previous results which either concentrate on non-degenerate analogs, or are restricted to the special case α = 1 . To appropriately cope with the considerably stronger cross-degeneracies thus allowed through ( ⋆ ) when α is large, in its core part the analysis relies on the use of the Moser–Trudinger inequality in controlling the respective diffusion rates ϕ ( v ) from below.