Maillet type theorem for nonlinear partial differential equations and Newton polygons. (English) Zbl 0995.35002

Shirai, Akira

J. Math. Soc. Japan 53, No. 3, 565-587 (2001).

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

Keywords:

non-Kowalevskian nonlinear partial differential equations; formal Gevrey class