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Stable blowup for the supercritical Yang-Mills heat flow. (English) Zbl 1428.58012
Authors’ abstract: The paper considers the heat flow for Yang-Mills connections on \(\mathbb{R}^5\times \mathrm{SO}(5)\). In the \(\mathrm{SO}(5)\)-equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation, for which an explicit self-similar blowup solution was found by B. Weinkove [Calc. Var. Partial Differ. Equ. 19, No. 2, 211–220 (2004; Zbl 1060.53072)]. The authors prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, they show that there exists an open set of initial conditions in a suitable topology, such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar blowup profile on an unbounded domain. Convergence is obtained in suitable Sobolev norms and in \(L^{\infty}\).

MSC:
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
58J35 Heat and other parabolic equation methods for PDEs on manifolds
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
35K05 Heat equation
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