Stability of ODE blow-up for the energy critical semilinear heat equation

Charles Collot; Frank Merle; Pierre Raphaël

doi:10.1016/j.crma.2016.10.020

Partial differential equations

Stability of ODE blow-up for the energy critical semilinear heat equation
[Stabilité de l'explosion type EDO pour l'équation de la chaleur énergie critique]

Présenté par : Jean-Michel Coron

Charles Collot ¹ ; Frank Merle ^2,³ ; Pierre Raphaël ¹

¹ Laboratoire Jean-Alexandre-Dieudonné, Université de Nice–Sophia Antipolis, France
² Laboratoire Laga, Université de Cergy-Pontoise, France
³ IHES, Bures-sur-Yvette, France

Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 65-79.

Résumé
Abstract

Nous considérons l'équation de la chaleur énergie critique

\partial_{t} u = Δ u + | u |^{\frac{4}{d - 2}} u, x \in R^{d}

en dimension

d \geq 3

. Nous proposons une preuve auto-contenue de la stabilité du régime explosif de type EDO

{‖ u ‖}_{L^{\infty}} \sim κ {(T - t)}^{\frac{d - 2}{4}}, T > 0, κ : = {(\frac{d - 2}{4})}^{\frac{d - 2}{4}}

qui adapte au cas énergie critique la preuve de Fermanian, Merle, Zaag [4].

We consider the energy critical semilinear heat equation

\partial_{t} u = Δ u + | u |^{\frac{4}{d - 2}} u, x \in R^{d}

in dimension

d \geq 3

. We propose a self-contained proof of the stability of solutions u blowing-up in finite time with type-I ODE blow-up

{‖ u ‖}_{L^{\infty}} \sim κ {(T - t)}^{\frac{d - 2}{4}}, T > 0, κ : = {(\frac{d - 2}{4})}^{\frac{d - 2}{4}}

which adapts to the energy critical case the proof of Fermanian, Merle, Zaag [4].

Reçu le : 2016-06-27
Accepté le : 2016-10-24
Publié le : 2016-11-22

DOI : 10.1016/j.crma.2016.10.020

Affiliations des auteurs :

Charles Collot ¹ ; Frank Merle ^{2, 3} ; Pierre Raphaël ¹

¹ Laboratoire Jean-Alexandre-Dieudonné, Université de Nice–Sophia Antipolis, France
² Laboratoire Laga, Université de Cergy-Pontoise, France
³ IHES, Bures-sur-Yvette, France

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     author = {Charles Collot and Frank Merle and Pierre Rapha\"el},
     title = {Stability of {ODE} blow-up for the energy critical semilinear heat equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {65--79},
     publisher = {Elsevier},
     volume = {355},
     number = {1},
     year = {2017},
     doi = {10.1016/j.crma.2016.10.020},
     language = {en},
}

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Charles Collot; Frank Merle; Pierre Raphaël. Stability of ODE blow-up for the energy critical semilinear heat equation. Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 65-79. doi : 10.1016/j.crma.2016.10.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.10.020/

Bibliographie
Cité par

[1] H. Bahouri; J.-Y. Chemin; R. Danchin Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343, Springer Science Business, Media, 2011

[2] H. Brezis; T. Cazenave A nonlinear heat equation with singular initial data, J. Anal. Math., Volume 68 (1996) no. 1, pp. 277-304

[3] C. Collot, F. Merle, P. Raphaël, Dynamics near the ground state for the energy critical nonlinear heat equation in large dimension, preprint, 2016.

[4] C. Fermanian Kammerer; F. Merle; H. Zaag Stability of the blow-up profile of non-linear heat equations from the dynamical system point of view, Math. Ann., Volume 317 (2000) no. 2, pp. 347-387

[5] S. Filippas; M.A. Herrero; J.J. Velazquez Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity, Proc. R. Soc. Lond. A, Volume 456 (2000) no. 2004, pp. 2957-2982

[6] Y. Giga On elliptic equations related to self-similar solutions for nonlinear heat equations, Hiroshima Math. J., Volume 16 (1986) no. 3, pp. 539-552

[7] Y. Giga; R.V. Kohn Asymptotically self-similar blow-up of semilinear heat equations, Commun. Pure Appl. Math., Volume 38 (1985) no. 3, pp. 297-319

[8] Y. Giga; R.V. Kohn Characterizing blowup using similarity variables, Indiana Univ. Math. J., Volume 36 (1987), pp. 1-40

[9] Y. Giga; R.V. Kohn Nondegeneracy of blowup for semilinear heat equations, Commun. Pure Appl. Math., Volume 42 (1989) no. 6, pp. 845-884

[10] Y. Giga; S.Y. Matsui; S. Sasayama Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J., Volume 53 (2004) no. 2, pp. 483-514

[11] F. Merle; H. Zaag Optimal estimates for blowup rate and behavior for nonlinear heat equations, Commun. Pure Appl. Math., Volume 51 (1998) no. 2, pp. 139-196

[12] F. Merle; H. Zaag A Liouville theorem for vector-valued nonlinear heat equations and applications, Math. Ann., Volume 316 (2000) no. 1, pp. 103-137

[13] P. Quittner; P. Souplet Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Springer Science and Business Media, 2007

[14] R. Schweyer Type II blow-up for the four dimensional energy critical semi linear heat equation, J. Funct. Anal., Volume 263 (2012) no. 12, pp. 3922-3983

[15] F.B. Weissler Local existence and nonexistence for semilinear parabolic equations in Lp, Indiana Univ. Math. J., Volume 29 (1980) no. 1, pp. 79-102

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