Non-analyticity in time of solutions to the KdV equation (Q1885056)

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