JournalszaaVol. 23, No. 1pp. 67–93

Non-Analyticity in Time of Solutions to the KdV Equation

  • Grzegorz Łysik

    Jan Kochanowski University, Kielce, Poland
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Abstract

It is proved that formal power series solutions to the initial value problem tu=x3u+x(u2){\partial_tu = \partial_x^3u+\partial_x(u^2)}, u(0,x)=φ(x){u(0,x)=\varphi(x)}, with analytic data φ\varphi belong to the Gevrey class G2{G^2} in time. However, if φ(x)=1/(1+x2){\varphi(x)=1/(1+x^2)}, the formal solution does not belong to the Gevrey class Gs{G^s} in time for 0s<2{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.

Cite this article

Grzegorz Łysik, Non-Analyticity in Time of Solutions to the KdV Equation. Z. Anal. Anwend. 23 (2004), no. 1, pp. 67–93

DOI 10.4171/ZAA/1188