The object of this work is to study the trichotomy dynamics of fractional heat equation with critical exponent in \({\mathbb {R}}^{n}\) \(\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}u+(-\Delta )^{s} u=|u|^{\frac{4s}{n-2s}}u, & \text {in}\quad {\mathbb {R}}^{n} \times (t_{0},+\infty ), \\ u(x,t_{0})=u_{0}(x), & \text {in}\quad {\mathbb {R}}^{n}, \end{array}\right. } \end{aligned}\) where \(4s<n<6s,\) \(0<s<1.\) For \(t_{0}\) sufficiently large, we construct the positive solution, which is smooth and globally defined in time, provided that the initial value satisfies \(u_{0}(x)\sim |x|^{-\gamma }\) with \(\gamma > \frac{n-2s}{2}.\) The global solution has the approximate form: (i) for \(\frac{n-2s}{2}<\gamma < 2s,\) the solution exhibits a algebraic decay; (ii) for \(\gamma =2s,\) the solution exhibits a slow logarithmic decay; (iii) for \(2s< \gamma <n-2s,\) the solution converges to a constant. Our strategy of main proof is based on the inner–outer gluing method in the fractional parabolic equations.
