Non-self-similar blow-up in the heat flow for harmonic maps in higher dimensions

We analyse the finite-time blow-up of solutions of the heat flow for k-corotational maps $\mathbb{R}^{d}\rightarrow S^{d}$. For each dimension $d> 2+k\left ( 2+2\sqrt{2} \right )$ 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