[Sur les asymptotiques des solutions globales des équations paraboliques sémi-linéaires d'ordre supérieur dans le cas surcritique]
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 , montre le comportement asymptotique quand t→∞ défini par la solution fondamentale de l'opérateur linéaire parabolique, à la différence du cas quand la solution peut exploser pour la donnée initiale arbitrairement petite. Le spectre discret des pistes possibles et la suite correspondante des exponents critiques , 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, , exhibits as t→∞ the asymptotic behaviour given by the fundamental solution of the linear parabolic operator (unlike the case 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 , where p0=pF, are discussed.
Accepté le :
Publié le :
Yu.V. Egorov 1 ; V.A. Galaktionov 2 ; V.A. Kondratiev 3 ; S.I. Pohozaev 4
@article{CRMATH_2002__335_10_805_0, 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}, }
TY - JOUR AU - Yu.V. Egorov AU - V.A. Galaktionov AU - V.A. Kondratiev AU - S.I. Pohozaev TI - On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical range JO - Comptes Rendus. Mathématique PY - 2002 SP - 805 EP - 810 VL - 335 IS - 10 PB - Elsevier DO - 10.1016/S1631-073X(02)02567-0 LA - en ID - CRMATH_2002__335_10_805_0 ER -
%0 Journal Article %A Yu.V. Egorov %A V.A. Galaktionov %A V.A. Kondratiev %A S.I. Pohozaev %T On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical range %J Comptes Rendus. Mathématique %D 2002 %P 805-810 %V 335 %N 10 %I Elsevier %R 10.1016/S1631-073X(02)02567-0 %G en %F CRMATH_2002__335_10_805_0
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/
[1] Local and global existence of solutions to semilinear parabolic initial value problems, Nonlin. Anal. TMA, Volume 43 (2001), pp. 293-323
[2] 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] Parabolic Systems, North-Holland, Amsterdam, 1969
[4] 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] Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach, J. Funct. Anal, Volume 100 (1991), pp. 435-462
[7] Large time behaviour of the solutions of a semilinear parabolic equation in RN, J. Differential Equations, Volume 53 (1984), pp. 258-276
[8] 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] Classes of Linear Operators, Vol. 1, Operator Theory: Advances and Applications, 49, Birkhäuser, Basel, 1990
[10] Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995
[11] Spatial Patterns. Higher Order Models in Physics and Mechanics, Birkhäuser, Boston, 2001
[12] Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995
Cité par Sources :
Commentaires - Politique