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Non-Analyticity in Time of Solutions to the KdV Equation

It is proved that formal power series solutions to the initial value problem ∂tu = ∂ 3 xu + ∂x(u ), u(0, x) = φ(x) with analytic data φ belong to the Gevrey class G in time. However, if φ(x) = 1 1+x2 , the formal solution does not belong to the Gevrey class G in time for 0 ≤ 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.