Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations
Annales de l'Institut Fourier, Volume 62 (2012) no. 2, pp. 571-618.

In this paper, we study a class of first order nonlinear degenerate partial differential equations with singularity at (t,x)=(0,0)C 2 . Using exponential-type Nagumo norm approach, the Gevrey asymptotic analysis is extended to case of holomorphic parameters in a natural way. A sharp condition is then established to deduce the k-summability of the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.

Dans cet article, nous étudions une classe d’équations aux dérivées partielles du premier ordre, non linéaires, dégénérées et ayant une singularité en (t,x)=(0,0)C 2 . Au moyen d’une famille de normes de Nagumo de type exponentiel, l’analyse asymptotique Gevrey s’étend naturellement au cas de paramètres holomorphes. Une condition optimale est ainsi établie pour déduire la k-sommabilité des solutions formelles. En outre, des solutions analytiques dans des domaines coniques sont obtenues pour chaque type de ces PDE singulières non linéaires.

DOI: 10.5802/aif.2688
Classification: 30E15,  32D15,  35C10,  35C20
Keywords: Nagumo norm, singular differential equations, Fuchsian singularity, Borel summability, Stokes phenomenon, k-summability, holomorphic parameters.
Luo, Zhuangchu 1; Chen, Hua 1; Zhang, Changgui 2

1 Wuhan University, School of Mathematics and Statistics, Wuhan 430072, China
2 Université de Lille 1, Laboratoire P. Painlevé (UMR–CNRS 8524), UFR Math., Cité scientifique, 59655 Villeneuve d’Ascq cedex, France
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Luo, Zhuangchu; Chen, Hua; Zhang, Changgui. Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations. Annales de l'Institut Fourier, Volume 62 (2012) no. 2, pp. 571-618. doi : 10.5802/aif.2688. http://archive.numdam.org/articles/10.5802/aif.2688/

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