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Zeitschrift für Analysis und ihre Anwendungen


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Volume 23, Issue 1, 2004, pp. 67–93
DOI: 10.4171/ZAA/1188

Published online: 2004-03-31

Non-Analyticity in Time of Solutions to the KdV Equation

Grzegorz Łysik[1]

(1) Jan Kochanowski University, Kielce, Poland

It is proved that formal power series solutions to the initial value problem ${\partial_tu = \partial_x^3u+\partial_x(u^2)}$, ${u(0,x)=\varphi(x)}$, with analytic data $\varphi$ belong to the Gevrey class ${G^2}$ in time. However, if ${\varphi(x)=1/(1+x^2)}$, the formal solution does not belong to the Gevrey class ${G^s}$ in time for ${0\le s<2}$, so it is not analytic in time. The proof is based on the estimation of a double sum of products of binomial coefficients.

Keywords: KdV equation, non-analyticity, Gevrey spaces, binomial coefficients

Łysik Grzegorz: Non-Analyticity in Time of Solutions to the KdV Equation. Z. Anal. Anwend. 23 (2004), 67-93. doi: 10.4171/ZAA/1188