Norm Inflation for the Cubic Nonlinear Heat Equation above the Scaling Critical Regularity

We consider the ill-posedness issue for the cubic nonlinear heat equation and prove norm inflation with infinite loss of regularity in the Hölder-Besov space = Bs∞,∞ for s ≤ -2/3. In particular, our result includes the subcritical range -1 < s ≤ -2/3, which is above the scaling critical regularity s = -1 with respect to the Hölder-Besov scale. In view of the well-posedness result in , s > -2/3, our ill-posedness result is sharp.