Abstract
Using mixed analytical and numerical methods we investigate the development of singularities in the heat flow for corotational harmonic maps from the d-dimensional sphere to itself for 3 ⩽ d ⩽ 6. By gluing together shrinking and expanding asymptotically self-similar solutions we construct global weak solutions which are smooth everywhere except for a sequence of times at which there occurs the type I blow-up at one of the poles of the sphere. We give evidence that in the generic case the continuation beyond blow-up is unique, the topological degree of the map changes by one at each blow-up time Ti, and eventually the solution comes to rest at the zero energy constant map.
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