On the diffusion phenomenon of quasilinear hyperbolic waves

Han Yang, Albert Milani

    Research output: Contribution to journalArticle

    4 Citations (Scopus)

    Abstract

    The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping Utt + ut -div (a(∇u) ∇u) = 0, and show that, at least when n ≤ 3, they tend, as t → +∞, to those of the nonlinear parabolic equation vt - div (a( ∇v) ∇v) = 0, in the sense that the norm ∥u(.,t) - v(., t)∥L∞(Rn) of the difference u - v decays faster than that of either u or v. This provides another example of the diffusion phenomenon of nonlinear hyperbolic waves, first observed by Hsiao, L. and Liu Taiping (see [1, 2]).

    Original languageEnglish
    Pages (from-to)63-70
    Number of pages8
    JournalChinese Annals of Mathematics. Series B
    Volume21
    Issue number1
    DOIs
    Publication statusPublished - 2000

    Fingerprint

    Quasilinear Hyperbolic Equation
    Nonlinear Parabolic Equations
    Asymptotic Behavior of Solutions
    Damping
    Decay
    Tend
    Norm

    All Science Journal Classification (ASJC) codes

    • Mathematics(all)
    • Applied Mathematics

    Cite this

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    abstract = "The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping Utt + ut -div (a(∇u) ∇u) = 0, and show that, at least when n ≤ 3, they tend, as t → +∞, to those of the nonlinear parabolic equation vt - div (a( ∇v) ∇v) = 0, in the sense that the norm ∥u(.,t) - v(., t)∥L∞(Rn) of the difference u - v decays faster than that of either u or v. This provides another example of the diffusion phenomenon of nonlinear hyperbolic waves, first observed by Hsiao, L. and Liu Taiping (see [1, 2]).",
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    On the diffusion phenomenon of quasilinear hyperbolic waves. / Yang, Han; Milani, Albert.

    In: Chinese Annals of Mathematics. Series B, Vol. 21, No. 1, 2000, p. 63-70.

    Research output: Contribution to journalArticle

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    T1 - On the diffusion phenomenon of quasilinear hyperbolic waves

    AU - Yang, Han

    AU - Milani, Albert

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    N2 - The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping Utt + ut -div (a(∇u) ∇u) = 0, and show that, at least when n ≤ 3, they tend, as t → +∞, to those of the nonlinear parabolic equation vt - div (a( ∇v) ∇v) = 0, in the sense that the norm ∥u(.,t) - v(., t)∥L∞(Rn) of the difference u - v decays faster than that of either u or v. This provides another example of the diffusion phenomenon of nonlinear hyperbolic waves, first observed by Hsiao, L. and Liu Taiping (see [1, 2]).

    AB - The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping Utt + ut -div (a(∇u) ∇u) = 0, and show that, at least when n ≤ 3, they tend, as t → +∞, to those of the nonlinear parabolic equation vt - div (a( ∇v) ∇v) = 0, in the sense that the norm ∥u(.,t) - v(., t)∥L∞(Rn) of the difference u - v decays faster than that of either u or v. This provides another example of the diffusion phenomenon of nonlinear hyperbolic waves, first observed by Hsiao, L. and Liu Taiping (see [1, 2]).

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