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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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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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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