## Genuine solutions and formal solutions with Gevrey type estimates of nonlinear partial differential equations

J. Math. Sci. Univ. Tokyo
Vol. 2 (1995), No. 2, Page 375--417.

Åuchi, Sunao
Genuine solutions and 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 $z=(z_{0},z')\in \bm{C} Ã \bm{C}^{n}$. We consider a nonlinear partial differential equation $L(u)=g(z)$, which has {\it a formal solution} $\tilde{u}(z)$ of the form % $$\tilde{u}(z)=z_{0}^{q}(\sum_{n=0}^{+\infty} u_{n}(z') z_{0}^{q_{n}}) u_{0}(z') \not \equiv 0,$$ % where $q \in \bm{R}$ and $0=q_{0}0,$$% which we often call the Gevrey type estimate. It is the main purpose to show under some conditions that there exists {\it a genuine solution}$u_{S_{1}}(z)$with the asymptotic expansion$u_{S_1}(z) \sim \tilde{u}(z)$as$\; z_{0} \rightarrow 0$in some sector$S_{1}\$. We apply the results to formal solutions constructed in Åuchi [7].