Heat Flow of Extrinsic Biharmonic Maps from a four Dimensional Manifold with Boundary

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.