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Solutions globales de certaines équations de Fuchs non linéaires dans les espaces de Gevrey

Author: Faiza Derrab
Journal: Proc. Amer. Math. Soc. 135 (2007), 1803-1815
MSC (2000): Primary 35A05; Secondary 35G20, 35A20
Published electronically: December 28, 2006
MathSciNet review: 2286091
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Abstract: We consider nonlinear partial differential equations with several Fuchsian variables of type $ a(t,D_{t}) u(t,x) = f(t,x,Du(t,x))$, where $ a(t,D_{t})$ is a Fuchsian principal part of weight zero. We prove existence and uniqueness of a global solution to this problem in the space of holomorphic functions with respect to the Fuchsian variable $ t$ and in Gevrey spaces with respect to the other variable $ x$. The method of proof is based on the application of the fixed point theorem in some Banach algebras defined by majorant functions that are suitable to this kind of equation.

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  • 1. M. S. Baouendi and C. Goulaouic, Cauchy problems with characteristic initial hypersurface, Comm. Pure Appl. Math. 26 (1973), 455–475. MR 0338532
  • 2. Faiza Derrab, Abdallah Nabaji, Patrice Pongérard, and Claude Wagschal, Problème de Cauchy fuchsien dans les espaces de Gevrey, J. Math. Sci. Univ. Tokyo 11 (2004), no. 4, 401–424 (French, with English summary). MR 2110921
  • 3. Faiza Derrab and Abdallah Nabaji, Solutions holomorphes locale et globale pour un opérateur différentiel linéaire à plusieurs variables fuchsiennes, Osaka J. Math. 42 (2005), no. 3, 653–675 (French, with English summary). MR 2166727
  • 4. F. Derrab, Sur les équations BMG. Thèse de Doctorat d'État, Sidi-Bel-Abbès (2005).
  • 5. Raymond Gérard and Hidetoshi Tahara, Singular nonlinear partial differential equations, Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1996. MR 1757086
  • 6. Maurice Gevrey, Sur la nature analytique des solutions des équations aux dérivées partielles. Premier mémoire, Ann. Sci. École Norm. Sup. (3) 35 (1918), 129–190 (French). MR 1509208
  • 7. Hikosaburo Komatsu, Linear hyperbolic equations with Gevrey coefficients, J. Math. Pures Appl. (9) 59 (1980), no. 2, 145–185. MR 581987
  • 8. Peter D. Lax, Nonlinear hyperbolic equations, Comm. Pure Appl. Math. 6 (1953), 231–258. MR 0056176
  • 9. Nour Saïd Madi and Masafumi Yoshino, Uniqueness and solvability of nonlinear Fuchsian equations, Bull. Sci. Math. 114 (1990), no. 1, 41–60 (English, with French summary). MR 1046700
  • 10. M. Miyake, Singular nonlinear partial differential equations of the first order in a complex domain. Séminaire Adjamagbo, Paris 9, Décember (1997).
  • 11. Patrice Pongérard, Sur une classe d’équations de Fuchs non linéaires, J. Math. Sci. Univ. Tokyo 7 (2000), no. 3, 423–448 (French, with English summary). MR 1792735
  • 12. Patrice Pongérand, Problème de Cauchy caractéristique à solution entière, J. Math. Sci. Univ. Tokyo 8 (2001), no. 1, 89–105 (French, with English summary). MR 1818907
  • 13. Claude Wagschal, Le problème de Goursat non linéaire, J. Math. Pures Appl. (9) 58 (1979), no. 3, 309–337 (French). MR 544256

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Additional Information

Faiza Derrab
Affiliation: 86, Avenue Lieutenant Khelladi, 22000 Sidi-Bel-Abbès, Algérie

Keywords: Nonlinear Fuchsian partial differential equation, several Fuchsian variables, global solution, Gevrey classes, method of majorants, fixed-point theorem.
Received by editor(s): March 4, 2005
Received by editor(s) in revised form: February 6, 2006
Published electronically: December 28, 2006
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.