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Holomorphic and singular solutions of nonlinear singular first order partial differential equations. (English) Zbl 0736.35022
The authors investigate the structure of holomorphic and singular solutions of nonlinear singular first order partial differential equations of the form $$t\partial u/\partial t=F(t,x,u,\partial u/\partial x)$$, where $$t\in\mathbb{C}$$, $$x\in\mathbb{C}^ n$$, $$F(t,x,u,v)$$ is a holomorphic function in a neighborhood of the origin in $$\mathbb{C}\times\mathbb{C}^ n\times\mathbb{C}\times\mathbb{C}^ n$$, and $$F(0,x,0,0)=\partial F/\partial v_ i(0,x,0,0)=0$$ for all $$x$$. The model for this problem is the Briot-Bouquet equation $$t du/dt=f(t,u)$$, $$f(0,0)=0$$.
The nature of the solutions is determined by the characteristic exponent $$\rho:=\partial F/\partial u(0,0,0,0)$$. The authors show that if $$\rho$$ is not a positive integer, then there is a unique holomorphic solution $$u_ 0(t,x)$$ in a neighborhood of the origin in $$\mathbb{C}\times\mathbb{C}^ n$$ satisfying $$u_ 0(0,x)\equiv 0$$. When $$\text{Re }\rho>0$$ (and $$\rho$$ is not a positive integer), there are also solutions that are singular as $$t\to0$$, and the authors characterize all of them within a certain class of functions. When $$\text{Re }\rho\leq0$$, there are no singular solutions within the given class.

MSC:
 35F25 Initial value problems for nonlinear first-order PDEs 32A10 Holomorphic functions of several complex variables 35R05 PDEs with low regular coefficients and/or low regular data 35A20 Analyticity in context of PDEs
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References:
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