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

• Main Author: Akira Shirai
• Format: Article
• Language: English
• Published: AGH Univeristy of Science and Technology Press 2015-01-01
• Series: Opuscula Mathematica
• Subjects: singular partial differential equations; totally characteristic type; nilpotent vector field; formal solution; Gevrey order; Maillet type theorem
• Online Access: http://www.opuscula.agh.edu.pl/vol35/5/art/opuscula_math_3537.pdf

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.