Comptes Rendus
no. 10
On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical range
[Sur les asymptotiques des solutions globales des équations paraboliques sémi-linéaires d'ordre supérieur dans le cas surcritique]
Yu.V. Egorov1 ; V.A. Galaktionov2 ; V.A. Kondratiev3 ; S.I. Pohozaev4
1 Laboratoire des mathématiques pour l'industrie et la physique, UMR 5640, Université Paul Sabatier, UFR MIG, 118, route de Narbonne, 31062, Toulouse cedex 4, France
2 University of Bath, Department of Math. Sciences, Claverton Down, BA2 7AY, Bath, UK
3 Mehmat. Faculty, Lomonosov State Univer., Vorob'evy Gory, 119899 Moscow, Russia
4 Steklov Math. Institute, Gubkina 8, GSP-1, Moscow, Russia
Comptes Rendus. Mathématique, Volume 335 (2002) no. 10, pp. 805-810.

On considère le comportement asymptotique des solutions globales bornées du problème de Cauchy pour l'équation parabolique sémi-linéaire d'ordre 2m ut=−(−Δ)mu+|u|p in RN×R+, u(x,0)=u0∈X=L1(RN)∩L∞(RN), où m>1, p>1. On vérifie que dans le cas surcritique de Fujita p>pF=1+2m/N toute petite solution globale avec la donnée initiale vérifiant ∫u0dx⩾0, montre le comportement asymptotique quand t→∞ défini par la solution fondamentale de l'opérateur linéaire parabolique, à la différence du cas p∈]1,pF] quand la solution peut exploser pour la donnée initiale arbitrairement petite. Le spectre discret des pistes possibles et la suite correspondante des exponents critiques {pl=1+2m/(l+N),l=0,1,2,...}, où p0=pF, sont descriptes.

We study the asymptotic behaviour of global bounded solutions of the Cauchy problem for the semilinear 2mth order parabolic equation ut=−(−Δ)mu+|u|p in RN×R+, where m>1, p>1, with bounded integrable initial data u0. We prove that in the supercritical Fujita range p>pF=1+2m/N any small global solution with nonnegative initial mass, ∫u0dx⩾0, exhibits as t→∞ the asymptotic behaviour given by the fundamental solution of the linear parabolic operator (unlike the case p∈]1,pF] where solutions can blow-up for any arbitrarily small initial data). A discrete spectrum of other possible asymptotic patterns and the corresponding monotone sequence of critical exponents {pl=1+2m/(l+N),l=0,1,2,...}, where p0=pF, are discussed.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02567-0

Yu.V. Egorov 1 ; V.A. Galaktionov 2 ; V.A. Kondratiev 3 ; S.I. Pohozaev 4

1 Laboratoire des mathématiques pour l'industrie et la physique, UMR 5640, Université Paul Sabatier, UFR MIG, 118, route de Narbonne, 31062, Toulouse cedex 4, France
2 University of Bath, Department of Math. Sciences, Claverton Down, BA2 7AY, Bath, UK
3 Mehmat. Faculty, Lomonosov State Univer., Vorob'evy Gory, 119899 Moscow, Russia
4 Steklov Math. Institute, Gubkina 8, GSP-1, Moscow, Russia 
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Yu.V. Egorov; V.A. Galaktionov; V.A. Kondratiev; S.I. Pohozaev. On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical range. Comptes Rendus. Mathématique, Volume 335 (2002) no. 10, pp. 805-810. doi : 10.1016/S1631-073X(02)02567-0. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02567-0/

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