  We prove a global approximation theorem for a general parabolic operator $L$,
which asserts that if $v$ satisfies the equation $Lv=0$ in a spacetime region
$\Omega \subset \mathbb{R}^{n+1}$ satisfying certain necessary topological
condition, then it can be approximated in a H\"older norm by a global solution
$u$ to the equation. If $\Omega$ is compact and $L$ is the usual heat operator,
one can instead approximate the local solution $v$ by the unique solution that
falls off at infinity to the Cauchy problem with a suitably chosen smooth,
compactly supported initial datum. These results are next applied to prove the
existence of global solutions to the equation $Lu=0$ with a local hot spot that
moves along a prescribed curve for all time, up to a uniformly small error.
Global solutions that exhibit isothermic hypersurfaces of prescribed topologies
for all times and applications to the heat equation on the flat torus are
discussed too.
