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On the holomorphic solution of nonlinear totally characteristic equations. (English) Zbl 1017.35006
The authors consider the following equation $t{\partial u\over \partial t}=F\left(t,x,u, {\partial u\over\partial x}\right) \tag{1}$ for the unknown $$u(t,x)$$ with function $$F(t,x,u, {\partial u\over \partial x})$$ of the variable $$(t,x,u,v) \in\mathbb{C}_t \times\mathbb{C}_x \times\mathbb{C}_u \times\mathbb{C}_v$$ such that $$F(t,x,u,v)$$ is a holomorphic function defined in a neighborhood of the origin $$(0,0,0,0)\in \mathbb{C}_t\times \mathbb{C}_x\times \mathbb{C}_n\times \mathbb{C}_u$$ and $$F(0,x,0,0) \equiv 0$$ near $$x=0$$. Let $$\beta(x): ={\partial F\over\partial u} (0,x,0,0)$$, $$\gamma(x):= {\partial F\over\partial v}(0,x,0,0)$$, the authors consider the simplest case $$\gamma(x)=xc(x)$$, $$c(0)\neq 0$$ and prove the following Theorem:
If $$\text{Re} c(0)<0$$ and $${\beta(0)- k\over c(0)} \notin\mathbb{Z}_-=\{0,-1,-2,\dots\}$$ for any $$k\geq 1$$, then equation (1) has a unique holomorphic solution $$u(t,x)$$ near $$(0,0)\in \mathbb{C}_t\times \mathbb{C}_x$$ with $$u(0,x) =0$$ near $$x=0$$.
The result is extended also to a higher-order equation of type (1).

##### MSC:
 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 35A10 Cauchy-Kovalevskaya theorems 35A20 Analyticity in context of PDEs
##### Keywords:
holomorphic singular differential equation
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##### References:
  Gérard, Proc. Japan Acad. 66 pp 72– (1990)  Gérard, Publ. RIMS, Kyoto Univ. 26 pp 979– (1990)  Gérard, Publ. RIMS, Kyoto Univ. 29 pp 121– (1993)  and : Singular Nonlinear Partial Differential Equations, Aspects of Mathematics, E 28, Vieweg, 1996  : Ordinary Differential Equations in the Complex Domain, John Wiley and Sons, 1976  : Linear Partial Differential Operators, Springer, 1963 · Zbl 0108.09301  Yamane, Kyoto Univ., 33 pp 801– (1997)
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