Formal Solutions of Second Order Evolution Equations

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 u(t, x) = Σ∞_n=0 φ_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.