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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
##### Keywords:
formal solution; formal Gevrey class; Gevrey index
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##### References:
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