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Harmonic maps are critical points of the energy functional for maps between Riemannian manifolds. In this paper we study the heat equation for harmonic maps from a non-compact manifold M into N. We show that if the target manifold N is compact and has non-positive sectional curvature, and if the initial map has finite total energy, then there exists a solution u(x,t) : M×[0,∞)→N and a sequence t j →∞, such that u(.,t j ) converges on compact subsets of M to a harmonic from M into N. We also obtain some basic properties of the solution u(x,t). In particular, we prove a uniqueness theorem for the solution and a monotonicity theorem for the energy functional

On the heat equation for harmonic maps from noncompact manifolds