# 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 ∂_tu_ = *F*(*x*,*u*,∇_xu_,∇2_xu_, *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*=0 φ_n_(

*u*(*t, x*) = Σ∞_n*x*)

*tn*of the equation and prove that it satisfies Gevrey type estimates |φ_n_(

*x*)| ≤

*C__n*+1_n_! for

*x*∈

*K*⋐ Ω and

*n*∈ N0, 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