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arXiv · 1404.2209

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

Paweł Biernat
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Abstract

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.

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Paweł Biernat. 2014-10-20. Non-self-similar blow-up in the heat flow for harmonic maps in higher dimensions. https://doi.org/10.1088/0951-7715%2F28%2F1%2F167

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