Remarks on spreading and vanishing for free boundary problems of some reaction-diffusion equations

Yuki Kaneko, Kazuhiro Oeda, Yoshio Yamada

    Research output: Contribution to journalArticle

    6 Citations (Scopus)

    Abstract

    We discuss a free boundary problem for a di¤usion equation in a onedimensional interval which models the spreading of invasive or new species. Moreover, the free boundary represents a spreading front of the species and its dynamical behavior is determined by a Stefan-like condition. This problem has been proposed by Du and Lin (2010) and, recently, Kaneko and Yamada have studied a free boundary problem for a general reaction-di¤usion equation under Dirichlet boundary conditions. The main purpose of this paper is to define ‘‘spreading’’ and ‘‘vanishing’’ of species for a free boundary problem with general nonlinearity and study the underlying principle to determine the spreading or vanishing behavior as time tends to infinity. It will be proved that vanishing occurs if and only if the free boundary stays in a bounded interval, and that, when vanishing occurs, the population decreases exponentially to zero in large time.

    Original languageEnglish
    Pages (from-to)449-465
    Number of pages17
    JournalFunkcialaj Ekvacioj
    Volume57
    Issue number3
    Publication statusPublished - 2015 Jan 10

    Fingerprint

    Free Boundary Problem
    Reaction-diffusion Equations
    Free Boundary
    Interval
    Dynamical Behavior
    Dirichlet Boundary Conditions
    Nonlinearity
    Tend
    If and only if
    Decrease
    Zero
    Model

    Keywords

    • Comparison principle
    • Free boundary problem
    • Reaction-diffusion equation
    • Spreading and vanishing of species
    • Upper and lower solutions

    ASJC Scopus subject areas

    • Algebra and Number Theory
    • Analysis
    • Geometry and Topology

    Cite this

    Remarks on spreading and vanishing for free boundary problems of some reaction-diffusion equations. / Kaneko, Yuki; Oeda, Kazuhiro; Yamada, Yoshio.

    In: Funkcialaj Ekvacioj, Vol. 57, No. 3, 10.01.2015, p. 449-465.

    Research output: Contribution to journalArticle

    Kaneko, Yuki ; Oeda, Kazuhiro ; Yamada, Yoshio. / Remarks on spreading and vanishing for free boundary problems of some reaction-diffusion equations. In: Funkcialaj Ekvacioj. 2015 ; Vol. 57, No. 3. pp. 449-465.
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