Multiple stable patterns for some reaction-diffusion equation in disrupted environments

Takanori Ide, Kazuhiro Kurata, Kazunaga Tanaka

    Research output: Contribution to journalArticle

    Abstract

    We study the existence of multiple positive stable solutions for -ε2Δu(x) = u(x)2(b(x) - u(x)) in Ω, ∂u/∂n(x) = 0 on ∂Ω. Here ε > 0 is a small parameter and b(x) is a piecewise continuous function which changes sign. These type of equations appear in a population growth model of species with a saturation effect in biology.

    Original languageEnglish
    Pages (from-to)93-116
    Number of pages24
    JournalDiscrete and Continuous Dynamical Systems
    Volume14
    Issue number1
    DOIs
    Publication statusPublished - 2006 Jan

    Fingerprint

    Piecewise continuous
    Population Growth
    Stable Solution
    Sign Change
    Population Model
    Growth Model
    Reaction-diffusion Equations
    Small Parameter
    Biology
    Saturation
    Continuous Function

    Keywords

    • Nonlinear elliptic equations
    • Pattern formation
    • Singular perturbation
    • Stable solutions
    • Variational methods

    ASJC Scopus subject areas

    • Mathematics(all)
    • Discrete Mathematics and Combinatorics
    • Applied Mathematics
    • Analysis

    Cite this

    Multiple stable patterns for some reaction-diffusion equation in disrupted environments. / Ide, Takanori; Kurata, Kazuhiro; Tanaka, Kazunaga.

    In: Discrete and Continuous Dynamical Systems, Vol. 14, No. 1, 01.2006, p. 93-116.

    Research output: Contribution to journalArticle

    @article{53cd859610284aafa4d72ff6728d9a10,
    title = "Multiple stable patterns for some reaction-diffusion equation in disrupted environments",
    abstract = "We study the existence of multiple positive stable solutions for -ε2Δu(x) = u(x)2(b(x) - u(x)) in Ω, ∂u/∂n(x) = 0 on ∂Ω. Here ε > 0 is a small parameter and b(x) is a piecewise continuous function which changes sign. These type of equations appear in a population growth model of species with a saturation effect in biology.",
    keywords = "Nonlinear elliptic equations, Pattern formation, Singular perturbation, Stable solutions, Variational methods",
    author = "Takanori Ide and Kazuhiro Kurata and Kazunaga Tanaka",
    year = "2006",
    month = "1",
    doi = "10.3934/dcds.2006.14.93",
    language = "English",
    volume = "14",
    pages = "93--116",
    journal = "Discrete and Continuous Dynamical Systems- Series A",
    issn = "1078-0947",
    publisher = "Southwest Missouri State University",
    number = "1",

    }

    TY - JOUR

    T1 - Multiple stable patterns for some reaction-diffusion equation in disrupted environments

    AU - Ide, Takanori

    AU - Kurata, Kazuhiro

    AU - Tanaka, Kazunaga

    PY - 2006/1

    Y1 - 2006/1

    N2 - We study the existence of multiple positive stable solutions for -ε2Δu(x) = u(x)2(b(x) - u(x)) in Ω, ∂u/∂n(x) = 0 on ∂Ω. Here ε > 0 is a small parameter and b(x) is a piecewise continuous function which changes sign. These type of equations appear in a population growth model of species with a saturation effect in biology.

    AB - We study the existence of multiple positive stable solutions for -ε2Δu(x) = u(x)2(b(x) - u(x)) in Ω, ∂u/∂n(x) = 0 on ∂Ω. Here ε > 0 is a small parameter and b(x) is a piecewise continuous function which changes sign. These type of equations appear in a population growth model of species with a saturation effect in biology.

    KW - Nonlinear elliptic equations

    KW - Pattern formation

    KW - Singular perturbation

    KW - Stable solutions

    KW - Variational methods

    UR - http://www.scopus.com/inward/record.url?scp=33644518410&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=33644518410&partnerID=8YFLogxK

    U2 - 10.3934/dcds.2006.14.93

    DO - 10.3934/dcds.2006.14.93

    M3 - Article

    VL - 14

    SP - 93

    EP - 116

    JO - Discrete and Continuous Dynamical Systems- Series A

    JF - Discrete and Continuous Dynamical Systems- Series A

    SN - 1078-0947

    IS - 1

    ER -