Finite time extinction for solutions to fast diffusion stochastic porous media equations

Viorel Barbu; Giuseppe Da Prato; Michael Röckner

doi:10.1016/j.crma.2008.11.018

Probability Theory

Finite time extinction for solutions to fast diffusion stochastic porous media equations
[Extinction en temps fini pour les solutions des équations des milieu poreux avec diffusion rapide]

Présenté par : Paul Malliavin

Viorel Barbu ¹ ; Giuseppe Da Prato ² ; Michael Röckner ^3,⁴

¹ Institute of Mathematics “Octav Mayer”, 700506 Iasi, Romania
² Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy
³ Faculty of Mathematics, University of Bielefeld, Germany
⁴ Department of Mathematics and Statistics, Purdue University, USA

Comptes Rendus. Mathématique, Volume 347 (2009) no. 1-2, pp. 81-84.

Résumé
Abstract

Nous prouvons l'extinction avec une probabilité strictement positive pour les solutions des équations des milieux poreux avec diffusion rapide.

We prove that the solutions to fast diffusion stochastic porous media equations have finite time extinction with strictly positive probability.

Reçu le : 2008-08-26
Accepté le : 2008-11-27
Publié le : 2008-12-18

DOI : 10.1016/j.crma.2008.11.018

Affiliations des auteurs :

Viorel Barbu ¹ ; Giuseppe Da Prato ² ; Michael Röckner ^{3, 4}

¹ Institute of Mathematics “Octav Mayer”, 700506 Iasi, Romania
² Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy
³ Faculty of Mathematics, University of Bielefeld, Germany
⁴ Department of Mathematics and Statistics, Purdue University, USA

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     author = {Viorel Barbu and Giuseppe Da Prato and Michael R\"ockner},
     title = {Finite time extinction for solutions to fast diffusion stochastic porous media equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {81--84},
     publisher = {Elsevier},
     volume = {347},
     number = {1-2},
     year = {2009},
     doi = {10.1016/j.crma.2008.11.018},
     language = {en},
}

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Viorel Barbu; Giuseppe Da Prato; Michael Röckner. Finite time extinction for solutions to fast diffusion stochastic porous media equations. Comptes Rendus. Mathématique, Volume 347 (2009) no. 1-2, pp. 81-84. doi : 10.1016/j.crma.2008.11.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.11.018/

Bibliographie
Cité par

[1] V. Barbu; G. Da Prato; M. Röckner Existence and uniqueness of nonnegative solutions to the stochastic porous media equation, Indiana Univ. Math. J., Volume 57 (2008), pp. 187-212

[2] V. Barbu, G. Da Prato, M. Röckner, Existence of strong solutions for stochastic porous media equation under general monotonicity conditions, Ann. Probab., in press

[3] V. Barbu, G. Da Prato, M. Röckner, Stochastic porous media equations and self-organized criticality, Comm. Math. Phys., in press

[4] J. Berryman; C. Holland Stability of the separable solution for fast diffusion, Arch. Rational Mech. Anal., Volume 74 (1980) no. 4, pp. 379-388

[5] J. Ren; M. Röckner; F.Y. Wang Stochastic generalized porous media and fast diffusion equations, J. Differential Equations, Volume 238 (2007) no. 1, pp. 118-152

[6] M. Röckner, F.Y. Wang, Non-monotone stochastic generalized porous media equations, J. Differential Equations, in press

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