In this paper, we consider the heat flow for Yang-Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)-$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 Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we 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 $L^{\infty}$.

中文翻译：

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超临界杨-米尔斯热流的稳定爆破

在本文中，我们考虑了 $\mathbb{R}^5 \times SO(5)$ 上 Yang-Mills 连接的热流。在 $SO(5)-$equivariant 设置中，Yang-Mills 热方程简化为单个半线性反应扩散方程，Weinkove \cite{Wei04} 找到了该方程的显式自相似膨胀解。我们证明了该解在小扰动下的非线性渐近稳定性。特别是，我们表明在合适的拓扑中存在一组开放的初始条件，使得相应的解决方案在有限时间内爆炸并收敛到无界域上的非平凡自相似爆炸剖面。在合适的 Sobolev 范数和 $L^{\infty}$ 中获得收敛。