Comptes Rendus
no. 7
Partial Differential Equations
Gradient bounds for solutions of semilinear parabolic equations without Bernstein's quadratic condition
[Estimations du gradient des solutions d'équations paraboliques semi-linéaires sans la condition quadratique de Bernstein]

Présenté par : Haïm Brézis

Jean-Philippe Bartier1, 2 ; Philippe Souplet2, 3
1 Ceremade, UMR CNRS 7534, Université Paris IX – Dauphine, place de Lattre de Tassigny, 75775 Paris cedex 16, France
2 Laboratoire de mathématiques appliquées, UMR CNRS 7641, Université de Versailles, 45, avenue des États-Unis, 78035 Versailles, France
3 Département de mathématiques, Université de Picardie, INSSET, 02109 Saint-Quentin, France
Comptes Rendus. Mathématique, Volume 338 (2004) no. 7, pp. 533-538.

Nous établissons des estimations du gradient pour les solutions bornées d'équations paraboliques semi-linéaires, où la nonlinéarité vérifie seulement des hypothèses unilatérales de croissance quadratique, au lieu des conditions de Bernstein (bilatérales) classiques. Nous étendons ainsi un travail récent de Al. et Ar. Tersenov (Indiana Univ. Math. J. 50 (2001) 1899–1913), où des résultats de ce type ont été obtenus pour les solutions radiales dans une boule, par une technique différente.

We establish gradient estimates for bounded solutions of semilinear parabolic equations, where the nonlinearity only satisfies one-sided quadratic upper growth assumptions, instead of the classical (two-sided) Bernstein's condition. This extends a recent work of Al. and Ar. Tersenov (Indiana Univ. Math. J. 50 (2001) 1899–1913), where results of this kind were obtained for radial solutions in a ball, by a different technique.

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DOI : 10.1016/j.crma.2003.12.030

Jean-Philippe Bartier 1, 2 ; Philippe Souplet 2, 3

1 Ceremade, UMR CNRS 7534, Université Paris IX – Dauphine, place de Lattre de Tassigny, 75775 Paris cedex 16, France
2 Laboratoire de mathématiques appliquées, UMR CNRS 7641, Université de Versailles, 45, avenue des États-Unis, 78035 Versailles, France
3 Département de mathématiques, Université de Picardie, INSSET, 02109 Saint-Quentin, France 
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Jean-Philippe Bartier; Philippe Souplet. Gradient bounds for solutions of semilinear parabolic equations without Bernstein's quadratic condition. Comptes Rendus. Mathématique, Volume 338 (2004) no. 7, pp. 533-538. doi : 10.1016/j.crma.2003.12.030. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.12.030/

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