  Let $(M,g)$ be a four dimensional compact Riemannian manifold with boundary
and $(N,h)$ be a compact Riemannian manifold without boundary. We show the
existence of a unique, global weak solution of the heat flow of extrinsic
biharmonic maps from $M$ to $N$ under the Dirichlet boundary condition, which
is regular with the exception of at most finitely many time slices. We also
discuss the behavior of solution near the singular times. As an immediate
application, we prove the existence of a smooth extrinsic biharmonic map from
$M$ to $N$ under any Dirichlet boundary condition.
