On the long-time limit of positive solutions to the degenerate logistic equation

We study the long-time behavior of positive solutions to the problem u _t - Δu = au - b(x)u^p in (0, ∞) × Ω, Bu = 0 on (0, ∞) × ∂Ω, where a is a real parameter, 6 ≥ 0 is in C^μ(Ω̄) and p > 1 is a constant, Ω is a C^2+μ bounded domain in R^N (N ≥ 2), the boundary operator B is of the standard Dirichlet, Neumann or Robyn type. Under the assumption that Ω̄_o := {x ∈ Ω : b(x) = 0} has non-empty interior, is connected, has smooth boundary and is contained in ω, it is shown in [8] that when a ≥ λ ^d ₁(Ω_o), for any fixed x ∈ Ω̄_o, lim_t→∞ u(t, x) = ∞, and for any fixed x ∈ Ω̄ \ Ω̄_o, lim̄_t→∞u (t, x) ≤ Ū_a(x), lim _t→∞u(t, x) ≥ U_a(x), where U_a and Ū_a denote respectively the minimal and maximal positive solutions of the boundary blow-up problem -Δu = au - b(x)u^p in Ω \ Ω̄_o, Bu = 0 on ∂Ω, u = ∞ on ∂Ω_o- The main purpose of this paper is to show that, under the above assumptions, lim u(t, x) = U_a(x), ∀x ∈ Ω̄\Ω̄_o. t→∞ This proves a conjecture stated in [8]. Some extensions of this result are also discussed.