The heat flow of harmonic maps from noncompact manifolds

We consider the existence, uniqueness and convergence for the long time solution to the harmonic map heat equation between two complete noncompact Riemannian manifolds, where the target manifold is assumed to have nonpositive curvature. As a corollary, we show that if the tension field is in LpLp, the heat flow exists globally and uniquely which converges at infinity under an additional condition.