Asymptotic behavior of solutions for the damped wave equation with slowly decaying data

Takashi Narazaki, Kenji Nishihara

    Research output: Contribution to journalArticle

    11 Citations (Scopus)

    Abstract

    We consider the Cauchy problem for the damped wave equationut t - Δ u + ut = | u |ρ - 1 u, (t, x) ∈ R+ × RN and the heat equationφ{symbol}t - Δ φ{symbol} = | φ{symbol} |ρ - 1 φ{symbol}, (t, x) ∈ R+ × RN . If the data is small and slowly decays likely c1 (1 + | x |)- k N, 0 < k ≤ 1, then the critical exponent is ρc (k) = 1 + frac(2, k N) for the semilinear heat equation. In this paper it is shown that in the supercritical case there exists a unique time global solution to the Cauchy problem for the semilinear heat equation in any dimensional space RN, whose asymptotic profile is given byΦ0 (t, x) = under(∫, RN) frac(e- frac(| x - y |2, 4 t), (4 π t)N / 2) frac(c1, (1 + | y |2)k N / 2) d y provided that the data φ{symbol}0 satisfies lim| x | → ∞ 〈 x 〉k N φ{symbol}0 (x) = c1(≠ 0) . Even in the semilinear damped wave equation in the supercritical case a time global solution u with the data (u, ut) (0, x) = (u0, u1) (x) is shown in low dimensional spaces RN, N = 1, 2, 3, to have the same asymptotic profile Φ0 (t, x) provided that lim| x | → ∞ 〈 x 〉k N (u0 + u1) (x) = c1(≠ 0) . Those proofs are given by elementary estimates on the explicit formulas of solutions.

    Original languageEnglish
    Pages (from-to)803-819
    Number of pages17
    JournalJournal of Mathematical Analysis and Applications
    Volume338
    Issue number2
    DOIs
    Publication statusPublished - 2008 Feb 15

    Fingerprint

    Damped Wave Equation
    Asymptotic Behavior of Solutions
    Wave equations
    Asymptotic Profile
    Semilinear Heat Equation
    Global Solution
    Cauchy Problem
    Semilinear Wave Equation
    Damped
    Heat Equation
    Critical Exponents
    Hot Temperature
    Explicit Formula
    Likely
    Decay
    Estimate

    Keywords

    • Asymptotic profile
    • Damped wave equation
    • Slowly decaying data

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

    Cite this

    Asymptotic behavior of solutions for the damped wave equation with slowly decaying data. / Narazaki, Takashi; Nishihara, Kenji.

    In: Journal of Mathematical Analysis and Applications, Vol. 338, No. 2, 15.02.2008, p. 803-819.

    Research output: Contribution to journalArticle

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    N2 - We consider the Cauchy problem for the damped wave equationut t - Δ u + ut = | u |ρ - 1 u, (t, x) ∈ R+ × RN and the heat equationφ{symbol}t - Δ φ{symbol} = | φ{symbol} |ρ - 1 φ{symbol}, (t, x) ∈ R+ × RN . If the data is small and slowly decays likely c1 (1 + | x |)- k N, 0 < k ≤ 1, then the critical exponent is ρc (k) = 1 + frac(2, k N) for the semilinear heat equation. In this paper it is shown that in the supercritical case there exists a unique time global solution to the Cauchy problem for the semilinear heat equation in any dimensional space RN, whose asymptotic profile is given byΦ0 (t, x) = under(∫, RN) frac(e- frac(| x - y |2, 4 t), (4 π t)N / 2) frac(c1, (1 + | y |2)k N / 2) d y provided that the data φ{symbol}0 satisfies lim| x | → ∞ 〈 x 〉k N φ{symbol}0 (x) = c1(≠ 0) . Even in the semilinear damped wave equation in the supercritical case a time global solution u with the data (u, ut) (0, x) = (u0, u1) (x) is shown in low dimensional spaces RN, N = 1, 2, 3, to have the same asymptotic profile Φ0 (t, x) provided that lim| x | → ∞ 〈 x 〉k N (u0 + u1) (x) = c1(≠ 0) . Those proofs are given by elementary estimates on the explicit formulas of solutions.

    AB - We consider the Cauchy problem for the damped wave equationut t - Δ u + ut = | u |ρ - 1 u, (t, x) ∈ R+ × RN and the heat equationφ{symbol}t - Δ φ{symbol} = | φ{symbol} |ρ - 1 φ{symbol}, (t, x) ∈ R+ × RN . If the data is small and slowly decays likely c1 (1 + | x |)- k N, 0 < k ≤ 1, then the critical exponent is ρc (k) = 1 + frac(2, k N) for the semilinear heat equation. In this paper it is shown that in the supercritical case there exists a unique time global solution to the Cauchy problem for the semilinear heat equation in any dimensional space RN, whose asymptotic profile is given byΦ0 (t, x) = under(∫, RN) frac(e- frac(| x - y |2, 4 t), (4 π t)N / 2) frac(c1, (1 + | y |2)k N / 2) d y provided that the data φ{symbol}0 satisfies lim| x | → ∞ 〈 x 〉k N φ{symbol}0 (x) = c1(≠ 0) . Even in the semilinear damped wave equation in the supercritical case a time global solution u with the data (u, ut) (0, x) = (u0, u1) (x) is shown in low dimensional spaces RN, N = 1, 2, 3, to have the same asymptotic profile Φ0 (t, x) provided that lim| x | → ∞ 〈 x 〉k N (u0 + u1) (x) = c1(≠ 0) . Those proofs are given by elementary estimates on the explicit formulas of solutions.

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