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

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.