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

TL;DR

This paper demonstrates that the cubic nonlinear heat equation exhibits norm inflation and ill-posedness in certain Besov spaces for regularity levels above the critical threshold, highlighting the limits of well-posedness.

Contribution

It proves norm inflation and ill-posedness for the cubic nonlinear heat equation in Besov spaces with regularity above the critical level, extending understanding of ill-posedness.

Findings

Norm inflation occurs in $ ext{C}^s$ for $s \,\leq\, -\frac{2}{3}$.

Ill-posedness is established for $-1 < s \leq -\frac{2}{3}$.

Results are sharp compared to known well-posedness results for $s > -\frac{2}{3}$.

Abstract

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 $\mathcal C^s = B^{s}_{\infty, \infty}$ for $ s \le -\frac 23$. In particular, our result includes the subcritical range $-1< s \le -\frac 23$, 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 $\mathcal C^s$, $s > -\frac 23$, our ill-posedness result is sharp.