  This paper first proposes a new approximate scheme to construct a harmonic
heat flow $u$ between a parabolic cylinder to a sphere. Y.Chen and M.Struwe
have proved an existence and discussed a partial regularity of harmonic heat
flows by using Ginzburg-Landau heat flow and passing to the limit of a
parameter appeared in the equation. To construct a new harmonic heat flow, we
propose a Ginzburg-Landau type heat flow with a time-dependent parameter. and
we next establish the existence of a harmonic heat flow into spheres with (i) a
global energy inequality, (ii) a monotonicity for the scaled energy, (ii) a
reverse Poincare inequality. These inequalities (i), (ii) and (iii) improves
the Hausdorff dimensional estimates on it's singular set contrast to the former
results. I believe that inequalities (i), (ii) and (iii) allow us to analyze
how it behaves around its singularities.
