  We analyze the finite-time blow-up of solutions of the heat flow for
$k$-corotational maps $\mathbb R^d\to S^d$. For each dimension
$d>2+k(2+2\sqrt{2})$ we construct a countable family of blow-up solutions via a
method of matched asymptotics by glueing a re-scaled harmonic map to the
singular self-similar solution: the equatorial map. We find that the blow-up
rates of the constructed solutions are closely related to the eigenvalues of
the self-similar solution. In the case of $1$-corotational maps our solutions
are stable and represent the generic blow-up.
