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Formal solutions of nonlinear first order totally characteristic type PDE with irregular singularity. (English) Zbl 0993.35003
Consider a holomorphic function \(F(t,x,u,v)\) defined on an open polydisk \( \Delta \) centered at the origin of \(\mathbb{C}_{t}\times \mathbb{C}_{x}\times \mathbb{C}_{u}\times \mathbb{C}_{v}\) and which satisfies \(F(0,x,0,0) \equiv 0\) on \( \Delta _{0}= \Delta \cap \{t=0,u=0,v=0\}\). The authors study the first order scalar nonlinear singular partial differential equation \[ t( \partial u / \partial t)=F(t,x,u, \partial u/\partial x). \] From the assumptions it follows that \(F(t,x,u,v)= a(x)t +b(x)u + \gamma (x) v + \sum _{i+j+\alpha } a _{i,j,\alpha }(x) t ^{i} u ^{j} v ^{\alpha }\) where \(a(x),b(x)\), \(\gamma (x)\) and the \( a _{i,j,\alpha }(x)\) are holomorphic in \(x\) on \( \Delta _{0}\). The case when \( \gamma (x) \equiv 0\) has been studied in a series of papers by R. Gerard and H. Tahara and when \( \gamma (0)\neq 0\), one can solve for \( (\partial u/\partial x)\) and reduce oneself to the Cauchy-Kowalewsky theorem. Accordingly, the authors assume in this paper that \( \gamma (x)= x ^{p } c(x)\), with \(p\) some natural number and \(c(0)\neq 0\). The main result of the authors now refers to the case \( p \geq 2\), the case \(p=1\) having been already studied in a paper by Chen and Tahara. It states that when \( b(0) \neq 0\), then the above equation admits a formal solution \( u \in \mathbb{C}[[t,x]]\) with \(u(0,x) \equiv 0\), which belongs to some formal Gevrey class. For the discussion of the relation of formal solutions to true solutions, the authors refer to a forthcoming paper.

MSC:
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35C10 Series solutions to PDEs
35A20 Analyticity in context of PDEs
35A10 Cauchy-Kovalevskaya theorems
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