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For the Novikov equation, on both the line and the circle, we construct a 2-peakon solution with an asymmetric antipeakon-peakon initial profile whose $H^s$-norm for $s 3/2$, these results together establish $3/2$ as the critical index of well-posedness for this equation. The case $s=3/2$ remains an open question.

Construction of 2-Peakon Solutions and Ill-Posedness for the Novikov Equation