  In this paper, we study time optimal control problems for heat equations on
$\Omega\times \mathbb{R}^+$. Two properties under consideration are the
existence and the bang-bang properties of time optimal controls. It is proved
that those two properties hold when controls are imposed on some proper subsets
of $\Omega$; while they do not stand when controls are active on the whole
$\Omega$. Besides, a new property for eigenfunctions of $-\Delta$ with
Dirichlet boundary condition is revealed.
