### Abstract

The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping U_{tt} + u_{t} -div (a(∇_{u}) ∇_{u}) = 0, and show that, at least when n ≤ 3, they tend, as t → +∞, to those of the nonlinear parabolic equation v_{t} - div (a( ∇_{v}) ∇_{v}) = 0, in the sense that the norm ∥u(.,t) - v(., t)∥L∞(R^{n}) 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 language | English |
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Pages (from-to) | 63-70 |

Number of pages | 8 |

Journal | Chinese Annals of Mathematics. Series B |

Volume | 21 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2000 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

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*Chinese Annals of Mathematics. Series B*, vol. 21, no. 1, pp. 63-70. https://doi.org/10.1007/BF02731959

**On the diffusion phenomenon of quasilinear hyperbolic waves.** / Yang, Han; Milani, Albert.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the diffusion phenomenon of quasilinear hyperbolic waves

AU - Yang, Han

AU - Milani, Albert

PY - 2000

Y1 - 2000

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]).

UR - http://www.scopus.com/inward/record.url?scp=51649163984&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51649163984&partnerID=8YFLogxK

U2 - 10.1007/BF02731959

DO - 10.1007/BF02731959

M3 - Article

AN - SCOPUS:51649163984

VL - 21

SP - 63

EP - 70

JO - Chinese Annals of Mathematics. Series B

JF - Chinese Annals of Mathematics. Series B

SN - 0252-9599

IS - 1

ER -