On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical range

Yu.V. Egorov; V.A. Galaktionov; V.A. Kondratiev; S.I. Pohozaev

doi:10.1016/S1631-073X(02)02567-0

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. 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

Comptes Rendus. Mathématique, Volume 335 (2002) no. 10, pp. 805-810.

Résumé
Abstract

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

We study the asymptotic behaviour of global bounded solutions of the Cauchy problem for the semilinear 2mth order parabolic equation u_t=−(−Δ)^mu+|u|^p in R^N×R₊, where m>1, p>1, with bounded integrable initial data u₀. We prove that in the supercritical Fujita range p>p_F=1+2m/N any small global solution with nonnegative initial mass, $\int u_{0} dx ⩾ 0$ , exhibits as t→∞ the asymptotic behaviour given by the fundamental solution of the linear parabolic operator (unlike the case $p \in] 1, p_{F}]$ 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 ${p_{l} = 1 + 2 m / (l + N), l = 0, 1, 2, ...}$ , where p₀=p_F, are discussed.

Reçu le : 2002-06-19
Accepté le : 2002-07-23
Publié le : 2002-10-08

DOI : 10.1016/S1631-073X(02)02567-0

Affiliations des auteurs :

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

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     author = {Yu.V. Egorov and V.A. Galaktionov and V.A. Kondratiev and S.I. Pohozaev},
     title = {On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical range},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {805--810},
     publisher = {Elsevier},
     volume = {335},
     number = {10},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02567-0},
     language = {en},
}

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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/

Bibliographie
Cité par

[1] S. Cui Local and global existence of solutions to semilinear parabolic initial value problems, Nonlin. Anal. TMA, Volume 43 (2001), pp. 293-323

[2] Yu.V. Egorov; V.A. Galaktionov; V.A. Kondratiev; S.I. Pohozaev On the necessary conditions of existence to a quasilinear inequality in the half-space, C. R. Acad. Sci. Paris, Volume 330 (2000), pp. 93-98

[3] S.D. Eidelman Parabolic Systems, North-Holland, Amsterdam, 1969

[4] V.A. Galaktionov; S.P. Kurdyumov; A.A. Samarskii On asymptotic “eigenfunctions” of the Cauchy problem for a nonlinear parabolic equation, Math. USSR Sbornik, Volume 54 (1986), pp. 421-455

[5] V.A. Galaktionov, S.I. Pohozaev, Existence and blow-up for higher-order semilinear parabolic equations: majorizing order-preserving operators, Indiana Univ. Math. J., to appear

[6] V.A. Galaktionov; J.L. Vazquez Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach, J. Funct. Anal, Volume 100 (1991), pp. 435-462

[7] A. Gmira; L. Véron Large time behaviour of the solutions of a semilinear parabolic equation in R^N, J. Differential Equations, Volume 53 (1984), pp. 258-276

[8] S. Kamin; L.A. Peletier Large time behaviour of solutions of the heat equation with absorption, Ann. Sc. Norm. Pisa Cl. Sci. (4), Volume 12 (1984), pp. 393-408

[9] I. Gohberg; S. Goldberg; M.A. Kaashoek Classes of Linear Operators, Vol. 1, Operator Theory: Advances and Applications, 49, Birkhäuser, Basel, 1990

[10] A. Lunardi Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995

[11] L.A. Peletier; W.C. Troy Spatial Patterns. Higher Order Models in Physics and Mechanics, Birkhäuser, Boston, 2001

[12] A.A. Samarskii; V.A. Galaktionov; S.P. Kurdyumov; A.P. Mikhailov Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995

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