  In this paper, we will study the existence of finite time singularity to
harmonic heat flow and their formation patterns. After works of
Coron-Ghidaglia, Ding and Chen-Ding, one knows blow-up solutions under
smallness of initial energy for m>=3. soon later, 2 dimensional blowup
solutions were found by Chang-Ding-Ye. The first part of this paper is devoted
to construction of new examples of finite time blow-up solutions without
smallness conditions for 3<=m<7. In fact, when considering rotational symmetric
harmonic heat flow from B_1\subset R^m to S^m\subset R^{m+1}, we will prove
that the maximal solution blows up in finite time if b>\vartheta_m, and exists
for all time if 0<b<\pi/2. This result can be regarded as a generalization of
results of Chang-Ding-Ye nad Chang-Ding to higher dimensional case, which
relies on a completely different argument. The second part of the paper study
the rate of blow-up solutions. When M is a bounded domain in R^2 and consider
Dirichlet boundary condition on \partial M, Hamilton has obtained that the
blowup rate must be faster than (T-t)^{-1}. Under a similar setting, it was
later improved a litttle by Topping to (T-t)^{-1}|log(T-t)|. In this paper, we
will extend the results to all Riemmanian surfaces M and improve the rate of
Topping to (T-t)^{-1}a(|log(T-t)|) for any positive nondecreasing function
a(\tau) satisfying $\int^\infty_1\frac{d\tau}{a(\tau)}=+\infty$, which is
comparable to a recent result of Raphael-Schweyer for rotational symmetric
solutions. Turning to the higher dimensional case 3<=m<7, we will demonstrate a
completely different phenomenon by showing that all rotational symmetric
blow-up solutions can not be type II, which is different to the case m>=7 by
Bizon-Wasserman. Finally, we also present result of finite time type I blowup
for heat flow from S^m to S^m\subset R^{m+1}, when 3<=m<7 and degree is no less
than 2.
