## Formal solutions with Gevrey type estimates of nonlinear partial differential equations

J. Math. Sci. Univ. Tokyo
Vol. 1 (1994), No. 1, Page 205--237.

Åuchi, Sunao
Formal solutions with Gevrey type estimates of nonlinear partial differential equations
Let $L(u)=L(z, \d ^{Î±}u; |Î± | \leq m )$ be a nonlinear partial differential operator defined in a neighbourhood $Î©$ of $z=0$ in $\bm{C}^{n+1},$ where $z=(z_{0},z')\in \bm{C} Ã \bm{C}^{n}$. $L(u)$ is a polynomial of the unknown and its derivatives $\{ \d ^{Î±}u ; |Î±| \leq m \}$ with degree $M.$ In this paper we consider a nonlinear partial differential equation $L(u)=g(z)$. The main purpose of this paper is to find a formal solution $u(z)$ of $L(u)=g(z)$ with the form % $$u(z)=z_{0}^{q}(\sum_{n=0}^{+\infty} u_{n}(z') z_{0}^{q_{n}}) u_{0}(z') \not \equiv 0,$$ % where $q \in \bm{R}$ and \$0=q_{0}