Norm inflation for the cubic nonlinear heat equation above the scaling critical regularity

Abstract

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 ${\mathcal{C}}^s=B_{\infty ,\infty }^s$ for $s\le -\frac{2}{3}$. In particular, our result includes the subcritical range $-1<s\le -\frac{2}{3}$, which is above the scaling critical regularity $s=-1$ with respect to the Hölder-Besov scale. In view of the well-posedness result in ${\mathcal{C}}^s$, $s>-\frac{2}{3}$, our ill-posedness result is sharp.

15 pages. Minor updates. To appear in Funkcial. Ekvac