# Blowup for the nonlinear heat equation with small initial data in scale-invariant Besov norms

## TL;DR

This paper demonstrates that solutions to a nonlinear heat equation can blow up in finite time even with arbitrarily small initial data in certain scale-invariant Besov norms, answering a longstanding open question.

## Contribution

It proves finite-time blowup for small initial data in scale-invariant Besov norms for the nonlinear heat equation, resolving a question posed by Yves Meyer.

## Key findings

- Solutions blow up in finite time despite small initial data.
- The result confirms the optimality of previous smallness conditions for global existence.
- Addresses an open problem for the case b=3.

## Abstract

We consider the Cauchy problem of the nonlinear heat equation $u_t -\Delta u= u^{b},\ u(0,x)=u_0$, with $b\geq 2$ and $b\in \mathbb{N}$. We prove that initial data $u_0\in \mathcal{S}(\mathbb{R}^{n})$ (the Schwartz class)arbitrarily small in the scale invariant Besov-norm$\dot B^{-2/b}_{n(b-1) b/2,q}(\mathbb{R}^{n})$, can produce solutions that blow up in finite time. The case $b=3$ answers a question raised by Yves Meyer.Our result also proves that the smallness assumption put in an earlier work by C. Miao, B.~Yuan and B. Zhang, for the global-in-time solvability, is essentially optimal.