Formal Solutions of Second Order Evolution Equations

Abstract

We study the initial value problem for a second order evolution equation ${\partial }_{t}u=F(x,u,{\nabla }_{x}u,{\nabla }_x^{2}u),u{|}_{t=0}=u_0$, where $F(x,u,p,q)$ is a polynomial function in variables $u\in \mathbb{R}$, $p\in {\mathbb{R}}^d$, $q\in {\mathbb{R}}^{d^2}$ with coefficients analytic on a domain $\Omega \subset {\mathbb{R}}^d$, $d\ge 1$ and $u_0$ is analytic on $\Omega$. We construct a formal power series solution $\hat{u}(t,x)={\sum }_{n=0}^{\infty }{\phi }_n(x)t^n$ of the equation and prove that it satisfies Gevrey type estimates $|{\phi }_n(x)|\le C^{n+1}n!$ for $x\in K⋐\Omega$ and $n\in {\mathbb{N}}_0$, where $C$ does not depend on $n$. The proof is based on some combinatorial identities and estimates which may be of independent interest.