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We shall study the first order singular nonlinear partial differential equation of the form f(x, u(x), ∂xu(x)) = 0 with u(0) = 0, where x ∈ C and f(x, u, ξ) is holomorphic in a neighborhood of the origin. This equation is said to be singular if f(0, 0, ξ) ≡ 0 for all ξ ∈ C. The purpose of this paper is to study the Maillet type theorem which means to determine the Gevrey order of a formal power series solution which may diverge. §

A Maillet type theorem for first order singular nonlinear partial differential equations