L2-estimates of solutions for damped wave equations with space-time dependent damping term

Jiayun Lin, Kenji Nishihara, Jian Zhai

    Research output: Contribution to journalArticle

    17 Citations (Scopus)

    Abstract

    In this paper, we consider the damped wave equation with space-time dependent potential b (t, x) and absorbing semilinear term | u |ρ - 1 u. Here, b (t, x) = b0 (1 + | x |2)- frac(α, 2) (1 + t)- β with b0 > 0, α, β ≥ 0 and α + β ∈ [0, 1). Based on the local existence theorem, we obtain the global existence and the L2 decay rate of the solution by using the weighted energy method. The decay rate coincides with the result of Nishihara [K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, preprint] in the case of β = 0 and coincides with the result of Nishihara and Zhai [K. Nishihara, J. Zhai, Asymptotic behaviors of time dependent damped wave equations, preprint] in the case of α = 0.

    Original languageEnglish
    Pages (from-to)403-422
    Number of pages20
    JournalJournal of Differential Equations
    Volume248
    Issue number2
    DOIs
    Publication statusPublished - 2010 Jan 15

    Fingerprint

    Damped Wave Equation
    Damping Term
    Wave equations
    Damping
    Space-time
    Decay Rate
    Semilinear
    Dependent
    Estimate
    Local Existence
    Energy Method
    Term
    Absorbing
    Existence Theorem
    Global Existence
    Asymptotic Behavior
    Decay

    Keywords

    • Damped wave equation
    • Decay rate
    • Global existence
    • Weighted energy method

    ASJC Scopus subject areas

    • Analysis

    Cite this

    L2-estimates of solutions for damped wave equations with space-time dependent damping term. / Lin, Jiayun; Nishihara, Kenji; Zhai, Jian.

    In: Journal of Differential Equations, Vol. 248, No. 2, 15.01.2010, p. 403-422.

    Research output: Contribution to journalArticle

    Lin, Jiayun ; Nishihara, Kenji ; Zhai, Jian. / L2-estimates of solutions for damped wave equations with space-time dependent damping term. In: Journal of Differential Equations. 2010 ; Vol. 248, No. 2. pp. 403-422.
    @article{eb343fd8ac784f289764cf418f5fb4a3,
    title = "L2-estimates of solutions for damped wave equations with space-time dependent damping term",
    abstract = "In this paper, we consider the damped wave equation with space-time dependent potential b (t, x) and absorbing semilinear term | u |ρ - 1 u. Here, b (t, x) = b0 (1 + | x |2)- frac(α, 2) (1 + t)- β with b0 > 0, α, β ≥ 0 and α + β ∈ [0, 1). Based on the local existence theorem, we obtain the global existence and the L2 decay rate of the solution by using the weighted energy method. The decay rate coincides with the result of Nishihara [K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, preprint] in the case of β = 0 and coincides with the result of Nishihara and Zhai [K. Nishihara, J. Zhai, Asymptotic behaviors of time dependent damped wave equations, preprint] in the case of α = 0.",
    keywords = "Damped wave equation, Decay rate, Global existence, Weighted energy method",
    author = "Jiayun Lin and Kenji Nishihara and Jian Zhai",
    year = "2010",
    month = "1",
    day = "15",
    doi = "10.1016/j.jde.2009.09.022",
    language = "English",
    volume = "248",
    pages = "403--422",
    journal = "Journal of Differential Equations",
    issn = "0022-0396",
    publisher = "Academic Press Inc.",
    number = "2",

    }

    TY - JOUR

    T1 - L2-estimates of solutions for damped wave equations with space-time dependent damping term

    AU - Lin, Jiayun

    AU - Nishihara, Kenji

    AU - Zhai, Jian

    PY - 2010/1/15

    Y1 - 2010/1/15

    N2 - In this paper, we consider the damped wave equation with space-time dependent potential b (t, x) and absorbing semilinear term | u |ρ - 1 u. Here, b (t, x) = b0 (1 + | x |2)- frac(α, 2) (1 + t)- β with b0 > 0, α, β ≥ 0 and α + β ∈ [0, 1). Based on the local existence theorem, we obtain the global existence and the L2 decay rate of the solution by using the weighted energy method. The decay rate coincides with the result of Nishihara [K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, preprint] in the case of β = 0 and coincides with the result of Nishihara and Zhai [K. Nishihara, J. Zhai, Asymptotic behaviors of time dependent damped wave equations, preprint] in the case of α = 0.

    AB - In this paper, we consider the damped wave equation with space-time dependent potential b (t, x) and absorbing semilinear term | u |ρ - 1 u. Here, b (t, x) = b0 (1 + | x |2)- frac(α, 2) (1 + t)- β with b0 > 0, α, β ≥ 0 and α + β ∈ [0, 1). Based on the local existence theorem, we obtain the global existence and the L2 decay rate of the solution by using the weighted energy method. The decay rate coincides with the result of Nishihara [K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, preprint] in the case of β = 0 and coincides with the result of Nishihara and Zhai [K. Nishihara, J. Zhai, Asymptotic behaviors of time dependent damped wave equations, preprint] in the case of α = 0.

    KW - Damped wave equation

    KW - Decay rate

    KW - Global existence

    KW - Weighted energy method

    UR - http://www.scopus.com/inward/record.url?scp=70349783543&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=70349783543&partnerID=8YFLogxK

    U2 - 10.1016/j.jde.2009.09.022

    DO - 10.1016/j.jde.2009.09.022

    M3 - Article

    AN - SCOPUS:70349783543

    VL - 248

    SP - 403

    EP - 422

    JO - Journal of Differential Equations

    JF - Journal of Differential Equations

    SN - 0022-0396

    IS - 2

    ER -