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

Yihong Du, Yoshio Yamada

    Research output: Contribution to journalArticle

    14 Citations (Scopus)

    Abstract

    We study the long-time behavior of positive solutions to the problem u t - Δu = au - b(x)up in (0, ∞) × Ω, Bu = 0 on (0, ∞) × ∂Ω, where a is a real parameter, 6 ≥ 0 is in Cμ(Ω̄) and p > 1 is a constant, Ω is a C2+μ bounded domain in RN (N ≥ 2), the boundary operator B is of the standard Dirichlet, Neumann or Robyn type. Under the assumption that Ω̄o := {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 1o), for any fixed x ∈ Ω̄o, limt→∞ u(t, x) = ∞, and for any fixed x ∈ Ω̄ \ Ω̄o, lim̄t→∞u (t, x) ≤ Ūa(x), lim t→∞u(t, x) ≥ Ua(x), where Ua and Ūa denote respectively the minimal and maximal positive solutions of the boundary blow-up problem -Δu = au - b(x)up in Ω \ Ω̄o, Bu = 0 on ∂Ω, u = ∞ on ∂Ωo- The main purpose of this paper is to show that, under the above assumptions, lim u(t, x) = Ua(x), ∀x ∈ Ω̄\Ω̄o. t→∞ This proves a conjecture stated in [8]. Some extensions of this result are also discussed.

    Original languageEnglish
    Pages (from-to)123-132
    Number of pages10
    JournalDiscrete and Continuous Dynamical Systems
    Volume25
    Issue number1
    DOIs
    Publication statusPublished - 2009 Sep

    Keywords

    • Asymptotic behavior
    • Boundary blow-up
    • Logistic equation

    ASJC Scopus subject areas

    • Discrete Mathematics and Combinatorics
    • Applied Mathematics
    • Analysis

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