Slowly Oscillating Solution of the Cubic Heat Equation

In this paper, we are considering the Cauchy problem of the nonlinear heat equation $u\mathrm{\_}t-\mathrm{\Delta }u=u^3,u(0,x)=u\mathrm{\_}0$. After extending Y. Meyer’s result establishing the existence of global solutions, under a smallness condition of the initial data in the homogeneous Besov spaces $\dot{B}\mathrm{\_}{p}^{-\sigma ,\mathrm{\infty }}({\mathbb{R}}^3)$, where $3 \textless{} p \textless{} 9$and$\sigma=1-3/p$,weprovethatinitialdata$u_0\in $\mathcal{S}$($\mathbb{R}$^{3})$,arbitrarilysmallin${\dot B^{-2/3,\infty}_{9}}($\mathbb{R}$^{3})$, can produce solutions that explode in finite time. In addition, the blowup may occur after an arbitrarily short time.