Formal solutions with Gevrey type estimates of nonlinear partial differential equations

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

Ōuchi, Sunao
Formal solutions with Gevrey type estimates of nonlinear partial differential equations
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Abstract:
Let $L(u)=L(z, \d ^{α}u; |α | \leq m ) $ be a nonlinear partial differential operator defined in a neighbourhood $Ω$ of $z=0$ in $\bm{C}^{n+1},$ where $z=(z_{0},z')\in \bm{C} × \bm{C}^{n}$. $L(u)$ is a polynomial of the unknown and its derivatives $\{ \d ^{α}u ; |α| \leq m \}$ 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}
Mathematics Subject Classification (1991): Primary 35A99; Secondary 35C10, 35C20
Mathematical Reviews Number: MR1298544

Received: 1993-07-07