Approximation theorems for parabolic equations and movement of local hot spots

TL;DR

This paper establishes a global approximation theorem for parabolic equations, enabling the construction of solutions with prescribed hot spot trajectories and isothermic hypersurfaces, with applications to heat equations on various domains.

Contribution

It introduces a novel approximation theorem for parabolic equations, allowing global solutions to approximate local solutions and control hot spot movement over time.

Findings

Global solutions can be constructed with hot spots following prescribed curves.

Approximation in Hölder norm between local and global solutions is possible.

Applications include heat equations on flat tori and prescribed isothermic hypersurfaces.

Abstract

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.