We study the heat flow for
harmonic maps from a complete noncompact manifold M which satisfies conditions
(a) and (b) in §1. We show that if the target manifold N is complete, the C2 initial
map has bounded image in N and has bounded energy density and bounded tension
field, then the short-time solution of (1.1) in §1 exists and is unique. Additional, if
the sectional curvature of N is bounded from above, either the long-time
solution of (1.1) exists or the energy density of heat flow blows up at a finite
time. Moreover, if N has nonpositive sectional curvature and (1.1) has a
long-time solution u(⋅,t) whose energy density increases logarithmically, and
there is a point p ∈ M and a sequence tν→∞ such that u(⋅,tν) converges
uniformly on compact subsets of M to a harmonic map u∞ by passing to a
subsequence.
For this class of manifolds which satisfy (a) and (b), we also get Lp(p > 0)
mean-value inequalities for subsolutions of heat equations and gradient estimates for
solutions of heat equations.
