  The paper pursues two connected goals. Firstly, we establish the
Li-Yau-Hamilton estimate for the heat equation on a manifold $M$ with nonempty
boundary. Results of this kind are typically used to prove monotonicity
formulas related to geometric flows. Secondly, we establish bounds for a
solution $\nabla(t)$ of the Yang-Mills heat equation in a vector bundle over
$M$. The Li-Yau-Hamilton estimate is utilized in the proofs. Our results imply
that the curvature of $\nabla(t)$ does not blow up if the dimension of $M$ is
less than 4 or if the initial energy of $\nabla(t)$ is sufficiently small.
