Zeitschrift für Analysis und ihre Anwendungen
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Published online: 2011-01-03
Formal Solutions of Second Order Evolution EquationsGrzegorz Łysik (1) Jan Kochanowski University, Kielce, Poland
We study the initial value problem for a second order evolution equation ∂tu = F(x,u,∇xu,∇2xu, u|t=0 = u0, where F(x, u, p, q) is a polynomial function in variables u ∈ R, p ∈ Rd, q ∈ Rd2 with coefficients analytic on a domain Ω ⊂ Rd, d ≥ 1 and u0 is analytic on Ω. We construct a formal power series solution u(t, x) = Σ∞n=0 φn(x)tn of the equation and prove that it satisfies Gevrey type estimates |φn(x)| ≤ Cn+1n! 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.
Keywords: Nonlinear evolution equations, formal solutions, Gevrey estimates
Łysik Grzegorz: Formal Solutions of Second Order Evolution Equations. Z. Anal. Anwend. 30 (2011), 95-104. doi: 10.4171/ZAA/1426