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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).

35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35A10 Cauchy-Kovalevskaya theorems
35A20 Analyticity in context of PDEs
Full Text: DOI
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