Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption

Ryo Ikehata, Kenji Nishihara, Huijiang Zhao

    Research output: Contribution to journalArticle

    32 Citations (Scopus)

    Abstract

    We consider the Cauchy problem for the damped wave equation with absorption{A formula is presented} The behavior of u as t → ∞ is expected to be same as that for the corresponding heat equation φ{symbol}t - Δ φ{symbol} + | φ{symbol} |ρ - 1 φ{symbol} = 0, ( t, x ) ∈ R+ × RN. In the subcritical case 1 < ρ < ρc ( N ) : = 1 + 2 / N there exists a similarity solution wb ( t, x ) with the form t- 1 / ( ρ - 1 ) f ( x / sqrt(t) ) depending on b = lim| x | → ∞ | x |2 / ( ρ - 1 ) f ( | x | ) {greater than or slanted equal to} 0. Our first aim is to show the decay rates{A formula is presented} provided that the initial data without initial data size restriction spatially decays with reasonable polynomial order. The decay rates (**) are sharp in the sense that they are same as those of the similarity solution. The second aim is to show that the Gauss kernel is the asymptotic profile in the supercritical case, which has been shown in case of one-dimensional space by Hayashi, Kaikina and Naumkin [N. Hayashi, E.I. Kaikina, P.I. Naumkin, Asymptotics for nonlinear damped wave equations with large initial data, preprint, 2004]. We show this assertion in two- and three-dimensional space. To prove our results, both the weighted L2-energy method and the explicit formula of solutions will be employed. The weight is an improved one originally developed in [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].

    Original languageEnglish
    Pages (from-to)1-29
    Number of pages29
    JournalJournal of Differential Equations
    Volume226
    Issue number1
    DOIs
    Publication statusPublished - 2006 Jul 1

    Fingerprint

    Damped Wave Equation
    Asymptotics of Solutions
    Wave equations
    Cauchy Problem
    Absorption
    Similarity Solution
    Nonlinear Wave Equation
    Decay Rate
    Asymptotic Profile
    Energy Method
    Differential equations
    Damping
    Assertion
    Polynomials
    Heat Equation
    Critical Exponents
    Gauss
    Explicit Formula
    Decay
    Differential equation

    Keywords

    • Critical exponent
    • Explicit formula
    • Global asymptotics
    • Semilinear damped wave equation
    • Weighted energy method

    ASJC Scopus subject areas

    • Analysis

    Cite this

    Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption. / Ikehata, Ryo; Nishihara, Kenji; Zhao, Huijiang.

    In: Journal of Differential Equations, Vol. 226, No. 1, 01.07.2006, p. 1-29.

    Research output: Contribution to journalArticle

    Ikehata, Ryo ; Nishihara, Kenji ; Zhao, Huijiang. / Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption. In: Journal of Differential Equations. 2006 ; Vol. 226, No. 1. pp. 1-29.
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    AB - We consider the Cauchy problem for the damped wave equation with absorption{A formula is presented} The behavior of u as t → ∞ is expected to be same as that for the corresponding heat equation φ{symbol}t - Δ φ{symbol} + | φ{symbol} |ρ - 1 φ{symbol} = 0, ( t, x ) ∈ R+ × RN. In the subcritical case 1 < ρ < ρc ( N ) : = 1 + 2 / N there exists a similarity solution wb ( t, x ) with the form t- 1 / ( ρ - 1 ) f ( x / sqrt(t) ) depending on b = lim| x | → ∞ | x |2 / ( ρ - 1 ) f ( | x | ) {greater than or slanted equal to} 0. Our first aim is to show the decay rates{A formula is presented} provided that the initial data without initial data size restriction spatially decays with reasonable polynomial order. The decay rates (**) are sharp in the sense that they are same as those of the similarity solution. The second aim is to show that the Gauss kernel is the asymptotic profile in the supercritical case, which has been shown in case of one-dimensional space by Hayashi, Kaikina and Naumkin [N. Hayashi, E.I. Kaikina, P.I. Naumkin, Asymptotics for nonlinear damped wave equations with large initial data, preprint, 2004]. We show this assertion in two- and three-dimensional space. To prove our results, both the weighted L2-energy method and the explicit formula of solutions will be employed. The weight is an improved one originally developed in [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].

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