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In this paper, we consider the fractional heat equation with criticalexponent in $\mathbb{R}^n$ for $n>6s,s\in(0,1),$ \begin{equation*}u_t=-(-\Delta)^su+|u|^{\frac{4s}{n-2s}}u,\quad (x,t)\in\mathbb{R}^n\times\mathbb{R}. \end{equation*} We construct a bubble tower typesolution both for the forward and backward problem by establishing theexistence of the sign-changing solution with multiple blow-up at a single pointwith the form \begin{equation*}u(x,t)=(1+o(1))\sum_{j=1}^{k}(-1)^{j-1}\mu_j(t)^{-\frac{n-2s}{2}}U\left(\frac{x}{\mu_j(t)}\right)\quad\mbox{as}\quad t\to+\infty, \end{equation*} and the positive solution withmultiple blow-up at a single point with the form \begin{equation*}u(x,t)=(1+o(1))\sum_{j=1}^{k}\mu_j(t)^{-\frac{n-2s}{2}}U\left(\frac{x}{\mu_j(t)}\right)\quad\mbox{as}\quad t\to-\infty, \end{equation*} respectively. Here $k\ge2$ isa positive integer,$$U(y)=\alpha_{n,s}\left(\frac{1}{1+|y|^2}\right)^{\frac{n-2s}{2}},$$ and\begin{equation*} \mu_j(t)=\beta_j|t|^{-\alpha_j}(1+o(1))~\mathrm{as}~t\to\pm\infty, \quad\alpha_j=\frac{1}{2s}\left(\frac{n-2s}{n-6s}\right)^{j-1}-\frac{1}{2s},\end{equation*} for some certain positive numbers $\beta_j,j=1,\cdots,k.$

Infinite time bubble towers in the fractional heat equation with critical exponent