Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation

Shoshana Kamin; Philip Rosenau

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2004)

  • Volume: 15, Issue: 3-4, page 271-280
  • ISSN: 1120-6330

Abstract

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We study the behaviour of the solutions of the Cauchy problem u t = u m x ⁢ x + u ⁢ 1 - u m - 1 , x ∈ R , t > 0 ⁢ u ⁢ 0 , x = u 0 ⁢ x , u 0 ⁢ x ≥ 0 , and prove that if initial data u 0 ⁢ x decay fast enough at infinity then the solution of the Cauchy problem approaches the travelling wave solution spreading either to the right or to the left, or two travelling waves moving in opposite directions. Certain generalizations are also mentioned.

How to cite

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Kamin, Shoshana, and Rosenau, Philip. "Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 15.3-4 (2004): 271-280. <http://eudml.org/doc/252421>.

@article{Kamin2004,
abstract = {We study the behaviour of the solutions of the Cauchy problem $$u\_\{t\} = (u^\{m\})\_\{xx\}+u(1-u^\{m-1\}), \quad x \in \mathbb\{R\}, \quad t > 0 \quad u(0,x)=u\_\{0\}(x), \quad u\_\{0\}(x) \ge 0,$$ and prove that if initial data $u_\{0\}(x)$ decay fast enough at infinity then the solution of the Cauchy problem approaches the travelling wave solution spreading either to the right or to the left, or two travelling waves moving in opposite directions. Certain generalizations are also mentioned.},
author = {Kamin, Shoshana, Rosenau, Philip},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Asymptotic behaviour of solutions; nonlinear diffusion; reaction-diffusion equation; traveling waves},
language = {eng},
month = {12},
number = {3-4},
pages = {271-280},
publisher = {Accademia Nazionale dei Lincei},
title = {Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation},
url = {http://eudml.org/doc/252421},
volume = {15},
year = {2004},
}

TY - JOUR
AU - Kamin, Shoshana
AU - Rosenau, Philip
TI - Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2004/12//
PB - Accademia Nazionale dei Lincei
VL - 15
IS - 3-4
SP - 271
EP - 280
AB - We study the behaviour of the solutions of the Cauchy problem $$u_{t} = (u^{m})_{xx}+u(1-u^{m-1}), \quad x \in \mathbb{R}, \quad t > 0 \quad u(0,x)=u_{0}(x), \quad u_{0}(x) \ge 0,$$ and prove that if initial data $u_{0}(x)$ decay fast enough at infinity then the solution of the Cauchy problem approaches the travelling wave solution spreading either to the right or to the left, or two travelling waves moving in opposite directions. Certain generalizations are also mentioned.
LA - eng
KW - Asymptotic behaviour of solutions; nonlinear diffusion; reaction-diffusion equation; traveling waves
UR - http://eudml.org/doc/252421
ER -

References

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