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

Bartier, Jean-Philippe1, 2 ; Souplet, Philippe2, 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, Tome 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2003.12.030
Bartier, Jean-Philippe 1, 2 ; Souplet, Philippe 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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Bartier, Jean-Philippe; Souplet, Philippe. Gradient bounds for solutions of semilinear parabolic equations without Bernstein's quadratic condition. Comptes Rendus. Mathématique, Tome 338 (2004) no. 7, pp. 533-538. doi : 10.1016/j.crma.2003.12.030. http://www.numdam.org/articles/10.1016/j.crma.2003.12.030/

[1] Angenent, S.; Fila, M. Interior gradient blow-up in a semilinear parabolic equation, Differential Integral Equations, Volume 9 (1996), pp. 865-877

[2] J.-Ph. Bartier, Ph. Souplet, Gradient bounds for solutions of quasilinear parabolic equations without Bernstein's quadratic condition, in preparation

[3] Bernstein, S.N. Sur les équations du calcul des variations, Ann. Sci. École Norm. Sup., Volume 29 (1912) no. 3, pp. 431-485

[4] Boccardo, L.; Murat, F.; Puel, J.-P. Existence results for some quasilinear parabolic equations, Nonlinear Anal., Volume 13 (1989), pp. 373-392

[5] Edmunds, D.E.; Peletier, L.A. Quasilinear parabolic equations, Ann. Scuola Norm. Sup. Pisa, Volume 25 (1971) no. 3, pp. 397-421

[6] Ivanov, A.V. The first boundary value problem for quasilinear second order parabolic equation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), Volume 38 (1973), pp. 10-32 (in Russian). English translation J. Soviet Math., 8, 1977, pp. 354-372

[7] Ladyzenskaya, O.; Ladyzenskaya, O. Solution of the first boundary problem in the large for quasi-linear parabolic equations, Trudy Moskov. Mat. Obshch., Volume 107 (1956), pp. 636-639 (in Russian) (in Russian)

[8] Ladyzenskaya, O.; Solonnikov, V.A.; Uralceva, N.N. Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. Transl., American Mathematical Society, Providence, RI, 1967

[9] Lieberman, G. Interior gradient bounds for nonuniformly parabolic equations, Indiana Univ. Math. J., Volume 32 (1983), pp. 579-601

[10] Quittner, P. On global existence and stationary solutions for two classes of semilinear parabolic equations, Comment. Math. Univ. Carolin., Volume 34 (1993), pp. 105-124

[11] Serrin, J. Gradient estimates for solutions of nonlinear elliptic and parabolic equations, Contributions to Nonlinear Functional Analysis, Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, WI, 1971, Academic Press, New York, 1971, pp. 565-601

[12] Souplet, Ph. Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differential Integral Equations, Volume 15 (2002), pp. 237-256

[13] Souplet, Ph.; Weissler, F.B. Poincaré's inequality and global solutions of a nonlinear parabolic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 16 (1999), pp. 337-373

[14] Tersenov, Al.; Tersenov, Ar. Global solvability for a class of quasilinear parabolic problems, Indiana Univ. Math. J., Volume 50 (2001), pp. 1899-1913

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