Heat flow for extrinsic biharmonic maps with small initial energy

Let M-m and N-n hooked right arrow R-k be two compact Riemannian manifolds without boundary. We consider the L-2 gradient flow for the energy F(u) := 1/2 integral(M) \Deltau\(2). If m less than or equal to 3 or if m = 4 and F(u(0)) is small, we show that the heat flow for extrinsic biharmonic maps exists for all time, and that the solution subconverges to a smooth extrinsic biharmonic map as time goes to infinity.