# Formal Solutions of Second Order Evolution Equations

### Grzegorz Łysik

Jan Kochanowski University, Kielce, Poland

## Abstract

We study the initial value problem for a second order evolution equation $∂_{t}u=F(x,u,∇_{x}u,∇_{x}u),u∣_{t=0}=u_{0}$, where $F(x,u,p,q)$ is a polynomial function in variables $u∈R$, $p∈R_{d}$, $q∈R_{d_{2}}$ with coefficients analytic on a domain $Ω⊂R_{d}$, $d≥1$ and $u_{0}$ is analytic on $Ω$. We construct a formal power series solution $u^(t,x)=∑_{n=0}φ_{n}(x)t_{n}$ of the equation and prove that it satisfies Gevrey type estimates $∣φ_{n}(x)∣≤C_{n+1}n!$ for $x∈K⋐Ω$ and $n∈N_{0}$, where $C$ does not depend on $n$. The proof is based on some combinatorial identities and estimates which may be of independent interest.

## Cite this article

Grzegorz Łysik, Formal Solutions of Second Order Evolution Equations. Z. Anal. Anwend. 30 (2011), no. 1, pp. 95–104

DOI 10.4171/ZAA/1426