  We study $O(d)$-equivariant biharmonic maps in the critical dimension. A
major consequence of our study concerns the corresponding heat flow. More
precisely, we prove that blowup occurs in the biharmonic map heat flow from
$B^4(0, 1)$ into $S^4$. To our knowledge, this was the first example of blowup
for the biharmonic map heat flow. Such results have been hard to prove, due to
the inapplicability of the maximum principle in the biharmonic case.
Furthermore, we classify the possible $O(4)$-equivariant biharmonic maps from
$\mathbf{R}^4$ into $S^4$, and we show that there exists, in contrast to the
harmonic map analogue, equivariant biharmonic maps from $B^4(0,1)$ into $S^4$
that wind around $S^4$ as many times as we wish. We believe that the ideas
developed herein could be useful in the study of other higher-order parabolic
equations.
