Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations

Luo, Zhuangchu; Chen, Hua; Zhang, Changgui

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 $(t,x)=(0,0)\in {\mathbf{C}}^2$. 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 $(t,x)=(0,0)\in {\mathbf{C}}^2$. 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.

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