  We consider the energy supercritical harmonic heat flow from $\mathbb{R}^d$
into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional
assumption of 1-corotational symmetry, the problem reduces to the one
dimensional semilinear heat equation $$\partial_t u = \partial^2_r u +
\frac{(d-1)}{r}\partial_r u - \frac{(d-1)}{2r^2}\sin(2u).$$ We construct for
this equation a family of $\mathcal{C}^{\infty}$ solutions which blow up in
finite time via concentration of the universal profile $$u(r,t) \sim
Q\left(\frac{r}{\lambda(t)}\right),$$ where $Q$ is the stationary solution of
the equation and the speed is given by the quantized rates $$\lambda(t) \sim
c_u(T-t)^\frac{\ell}{\gamma}, \quad \ell \in \mathbb{N}^*, \;\; 2\ell > \gamma
= \gamma(d) \in (1,2].$$ The construction relies on two arguments: the
reduction of the problem to a finite-dimensional one thanks to a robust
universal energy method and modulation techniques developed by Merle, Rapha\"el
and Rodnianski [Camb. Jour. Math, 3(4):439-617, 2015] for the energy
supercritical nonlinear Schr\"odinger equation and by Rapha\"el and Schweyer
[Anal. PDE, 7(8):1713-1805, 2014] for the energy critical harmonic heat flow,
then we proceed by contradiction to solve the finite-dimensional problem and
conclude using the Brouwer fixed point theorem. Moreover, our constructed
solutions are in fact $(\ell - 1)$ codimension stable under perturbations of
the initial data. As a consequence, the case $\ell = 1$ corresponds to a stable
type II blowup regime.
