On the heat flow for harmonic maps with potential

Let (M, g) and (N, h) be two connected Riemannian manifolds without boundary (M compact, N complete). Let G is an element of C-infinity(N): if u: M --> N is a smooth map, we consider the functional E-G(u) = (1/2) integral (M) [\ du \ (2) - 2G(u)] dV(M) and we study its associated heat equation. In the compact case, we recover a version of the Eells-Sampson theorem, while for noncompact target manifold N, we establish suitable hypotheses and ensure global existence and convergence at infinity. In the second part of the paper, we study phenomena of blowing up solutions.