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

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

Comptes Rendus. Mathématique, Tome 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 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 $\int {\mathrm{u}}_0\mathrm{dx}⩾0$, montre le comportement asymptotique quand t→∞ défini par la solution fondamentale de l'opérateur linéaire parabolique, à la différence du cas $\mathrm{p}\in ]1,{\mathrm{p}}_F]$ quand la solution peut exploser pour la donnée initiale arbitrairement petite. Le spectre discret des pistes possibles et la suite correspondante des exponents critiques $\{p_l=1+2m/(l+N),l=0,1,2,...\}$, 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, $\int {\mathrm{u}}_0\mathrm{dx}⩾0$, exhibits as t→∞ the asymptotic behaviour given by the fundamental solution of the linear parabolic operator (unlike the case $\mathrm{p}\in ]1,{\mathrm{p}}_F]$ 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 $\{p_l=1+2m/(l+N),l=0,1,2,...\}$, where p₀=p_F, are discussed.

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

[2] Egorov, Yu.V.; Galaktionov, V.A.; Kondratiev, V.A.; Pohozaev, S.I. 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] Eidelman, S.D. Parabolic Systems, North-Holland, Amsterdam, 1969

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

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

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

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

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

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