Type II Blow-up Mechanism for Supercritical Harmonic Map Heat Flow

The harmonic map heat flow is a geometric flow well known to produce solutions whose gradient blows up in finite time. A popular model for investigating the blow-up is the heat flow for maps , restricted to equivariant maps. This model displays a variety of possible blow-up mechanisms, examples include self-similar solutions for and a so-called Type II blow-up in the critical dimension . Here we present the first constructive example of Type II blow-up in higher dimensions: for each we construct a countable family of Type II solutions, each characterized by a different blow-up rate. We study the mechanism behind the formation of these singular solutions and we relate the blow-up to eigenvalues associated to linearization of the harmonic map heat flow around the equatorial map. Some of the solutions constructed by us were already observed numerically.