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 H\"older-Besov space C^s = B^s_{\infty, \infty} for s \le -2/3. In particular, our result includes the subcritical range -1< s \le -2/3, which is above the scaling critical regularity s = -1 with respect to the H\"older-Besov scale. In view of the well-posedness result in C^s, s > -2/3, our ill-posedness result is sharp.