JournalszaaVol. 30, No. 1pp. 95–104

Formal Solutions of Second Order Evolution Equations

  • Grzegorz Łysik

    Jan Kochanowski University, Kielce, Poland
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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 xK ⋐ 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