Opuscula Math. 35, no. 5 (2015), 689-712
http://dx.doi.org/10.7494/OpMath.2015.35.5.689

Opuscula Mathematica

# Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type. Part II

Akira Shirai

Abstract. In this paper, we study the following nonlinear first order partial differential equation: $f(t,x,u,\partial_t u,\partial_x u)=0\quad\text{with}\quad u(0,x)\equiv 0.$ The purpose of this paper is to determine the estimate of Gevrey order under the condition that the equation is singular of a totally characteristic type. The Gevrey order is indicated by the rate of divergence of a formal power series. This paper is a continuation of the previous papers [Convergence of formal solutions of singular first order nonlinear partial differential equations of totally characteristic type, Funkcial. Ekvac. 45 (2002), 187-208] and [Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type, Surikaiseki Kenkyujo Kokyuroku, Kyoto University 1431 (2005), 94-106]. Especially the last-mentioned paper is regarded as part I of this paper.

Keywords: singular partial differential equations, totally characteristic type, nilpotent vector field, formal solution, Gevrey order, Maillet type theorem.

Mathematics Subject Classification: 35F20, 35A20, 35C10.

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• Akira Shirai
• Sugiyama Jogakuen University, School of Education, Department of Child Development, 17-3 Hoshigaoka Motomachi, Chikusa, Nagoya, 464-8662, Japan
• Communicated by Mirosław Lachowicz.
• Revised: 2014-09-01.
• Accepted: 2014-09-04.
• Published online: 2015-04-27.