A degenerate migration-consumption model in domains of arbitrary dimension

Abstract

In a smoothly bounded convex domain Ω ⊂ ℝn with n ≥ 1, a no-flux initial-boundary value problem for ut = Δ(uΦ(v)), vt = Δv-uv, is considered under the assumption that near the origin, the function Φ suitably generalizes the prototype given by Φ(ζ) = ζα, ζϵ [0, ζ 0]. By means of separate approaches, it is shown that in both cases α ϵ (0, 1) and α ϵ [1, 2] some global weak solutions exist which, inter alia, satisfy C(T) :=ess sup tϵ(0,T) ∫Ω u(·, t) ln u(·, t) < ∞ for all T > 0,with supT>0C(T) < ∞if α ϵ [1, 2].