On regularizing-rate estimates for solutions to the heat flow with initial value problem

For a compact Riemannian manifold N ⊂ ℝ K without boundary, we establish the existence of strong solutions to the heat flow for harmonic maps from ℝ n to N , and the regularizing rate estimate of the strong solutions. Moreover, we obtain the analyticity in spatial variables of the solutions. The uniqueness of the mild solutions in C ([0, T ]; W 1, n ) is also considered in this paper.