  We study the heat flow for Yang-Mills connections on $\mathbb R^d \times
SO(d)$. It is well-known that in dimensions $5 \leq d \leq 9$ this model admits
homothetically shrinking solitons, i.e., self-similar blowup solutions, with an
explicit example given by Weinkove \cite{Wei04}. We prove the nonlinear
asymptotic stability of the Weinkove solution under small equivariant
perturbations and thus extend a result by the second author and Donninger for
$d=5$ to higher dimensions. At the same time, we provide a general framework
for proving stability of self-similar blowup solutions to a large class of
semilinear heat equations in arbitrary space dimension $d \geq 3$, including a
robust and simple method for solving the underlying spectral problems.
