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Nonlinear singular first order partial differential equations of Briot- Bouquet type. (English) Zbl 0711.35034
Die gewöhnliche Differentialgleichung vom Briot-Bouquet Typ [see E. Hille, Ordinary differential equations in the complex domain (1976; Zbl 0343.34007)] wird zu einer partiellen Differentialgleichung derart verallgemeinert, daß analoge Sätze wie im gewöhnlichen Fall gelten. Betrachtet wird
(*) t \(\partial u/\partial t=F(t,x,u,\partial u/\partial x)\), wobei \(t\in {\mathbb{C}}\), \(x\in {\mathbb{C}}^ n\), \(\partial u/\partial x=(\partial u/\partial x_ 1,...,\partial u/\partial x_ n)\) und u komplexwertig sind. \(\Delta\) sei Nullumgebung in \({\mathbb{C}}\times {\mathbb{C}}^ n\times {\mathbb{C}}\times {\mathbb{C}}^ n\) und \(\Delta_ 0=\Delta \cap \{t=0\), \(u=0\), \(v=0\}\). Es seien \(F=F(t,x,u,v)\) holomorph in \(\Delta\), \(F(0,x,0,0)=0\) in \(\Delta_ 0\), \((\partial F/\partial \nu_ i)(0,x,0,0)=0\) in \(\Delta_ 0\) \((i=1,...,n)\) und \(\rho (x)=(\partial F/\partial u)(0,x,0,0)\). Es werden Bedingungen für Existenz und Eindeutigkeit holomorpher und singulärer Lösungen in einer Nullumgebung von \({\mathbb{C}}\times {\mathbb{C}}^ n\) formuliert, die u(0,x)\(\equiv 0\) erfüllen.

MSC:
35F25 Initial value problems for nonlinear first-order PDEs
35B65 Smoothness and regularity of solutions to PDEs
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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[1] Ch. Briot and J. CI. Bouquet: Recherches sur les proprietes des f onctions definies par des equations differentielles. J. Ecole Polytech., 21, 133-197 (1856).
[2] R. Gerard: Etude locale des equations differentielles de la forme xyf=f(xfy) au voisinage de x-0. J. Fac. Sci. Univ. Tokyo, 36 (1989). · Zbl 0709.34029
[3] E. Hille: Ordinary Differential Equations in the Complex Domain. John Wiley and Sons (1976). · Zbl 0343.34007
[4] M. Hukuhara, T. Kimura, and T. Matuda: Equations differentielles ordinaires du premier order dans le champ complexe. Publ. of the Math. Soc. of Japan (1961). · Zbl 0101.30002
[5] T. Kimura: Ordinary Differential Equations. Iwanami Shoten (1977) (in Japanese). · Zbl 0676.34001
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