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Maillet type theorem for nonlinear partial differential equations and Newton polygons. (English) Zbl 0995.35002
The author considers the following Cauchy problem for non-Kowalevskian nonlinear partial differential equations: $t^nD^m_t u(t,x)=a(x) t^{k-m+n}+ f(t,x,D_t^j D_x^\alpha u),$
$u(t,x)= 0(t^k),$ where in the nonlinearity $$0\leq j\leq m_0$$, $$0\leq j+|\alpha|\leq N$$, with $$n,m,m_0, N,k$$ given nonnegative integers, $$m\leq m_0\leq N$$, $$m_0<k$$ and $$a(x)\neq 0$$ in a neighborhood of the origin. The functions $$a$$ and $$f$$ are holomorphic. Under an additional assumption on the Taylor expansion of $$f$$, the author proves existence and uniqueness of a formal solution $$u(t,x)= \sum^\infty_{j=k} u_j(x) t^j$$ in a neighborhood of the origin. Moreover, this solution belongs to the formal Gevrey class $$G^s$$, where $$s$$ is charcterized in terms of the Newton polygon associated to the equation.
Reviewer: L.Rodino (Torino)

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