Optimal global well-posedness for semilinear heat equations

We address the question of global in time existence for the semilinear parabolic equation \begin{equation*} \left{ \begin{array}{rcl} \partial_t u-Δu&=&|u|^{b-1}u \quad {\rm ~in~} $\mathbb{R}$^n\times (0,\infty),\ u(\cdot,0)&=& u_0 \quad {\rm ~in~} $\mathbb{R}$^n, \end{array} \right. \end{equation*} where $b>1+2\mathrm{/}n,n\ge 1$. We found a space of initial data $Z$ that guarantees the existence of a global in time solution whenever the norm of the initial datum $u_0$ in $Z$ is small. The space $Z$ is optimal in the sense that if $u_0$ is a nonnegative distribution and the equation admits a nonnegative solution $u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^b({\mathbb{R}}^n\times (0,\mathrm{\infty }))$, then $u_0\in Z$. Moreover, our result contains all of the known results in the literature. In particular, it implies that global in time existence does hold for end-point Besov and Triebel-Lizorkin spaces, which resolves an open problem posed by L. Brandolese and F. Cortez (J. Funct. Anal. $\mathbf{\text{276}}$ (2019), 2589-2604).