Approximation theorems for parabolic equations and movement of local hot spots
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 $Ω\subset \mathbb{R}^{n+1}$ satisfying certain necessary topological condition, then it can be approximated in a Hölder norm by a global solution $u$ to the equation. If $Ω$ 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.