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

Takashi Narazaki, Kenji Nishihara

    Research output: Contribution to journalArticle

    12 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

    Keywords

    • Asymptotic profile
    • Damped wave equation
    • Slowly decaying data

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

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