[Une onde progressive peut-elle connecter deux équilibres instables ? Le cas de lʼéquation de Fisher non-locale]
Nous étudions dans cette Note les propriétés des solutions de type ondes progressives pour lʼéquation de Fisher non-locale. Lʼexistence de telles solutions a été prouvée récemment dans Berestycki et al. (2009) [3] mais leur comportement asymptotique était encore mal compris. Nous développons ici une nouvelle méthode dʼapproximation numérique montrant que certaines ondes progressives connectent les deux états dʼéquilibre homogènes 0 et 1, ce qui est surprenant puisque 0 est dynamiquement instable et 1 est instable au sens de Turing.
This Note investigates the properties of the traveling waves solutions of the nonlocal Fisher equation. The existence of such solutions has been proved recently in Berestycki et al. (2009) [3] but their asymptotic behavior was still unclear. We use here a new numerical approximation of these traveling waves which shows that some traveling waves connect the two homogeneous steady states 0 and 1, which is a striking fact since 0 is dynamically unstable and 1 is unstable in the sense of Turing.
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@article{CRMATH_2011__349_9-10_553_0,
author = {Nadin, Gr\'egoire and Perthame, Beno{\^\i}t and Tang, Min},
title = {Can a traveling wave connect two unstable states? {The} case of the nonlocal {Fisher} equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {553--557},
publisher = {Elsevier},
volume = {349},
number = {9-10},
year = {2011},
doi = {10.1016/j.crma.2011.03.008},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.03.008/}
}
TY - JOUR AU - Nadin, Grégoire AU - Perthame, Benoît AU - Tang, Min TI - Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation JO - Comptes Rendus. Mathématique PY - 2011 SP - 553 EP - 557 VL - 349 IS - 9-10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.03.008/ DO - 10.1016/j.crma.2011.03.008 LA - en ID - CRMATH_2011__349_9-10_553_0 ER -
%0 Journal Article %A Nadin, Grégoire %A Perthame, Benoît %A Tang, Min %T Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation %J Comptes Rendus. Mathématique %D 2011 %P 553-557 %V 349 %N 9-10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.03.008/ %R 10.1016/j.crma.2011.03.008 %G en %F CRMATH_2011__349_9-10_553_0
Nadin, Grégoire; Perthame, Benoît; Tang, Min. Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation. Comptes Rendus. Mathématique, Tome 349 (2011) no. 9-10, pp. 553-557. doi : 10.1016/j.crma.2011.03.008. http://www.numdam.org/articles/10.1016/j.crma.2011.03.008/
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