Optimal existence classes and nonlinear--like dynamics in the linear heat equation in ${\mathbb R}^d$

TL;DR

This paper investigates the linear heat equation in or rom initial data in Radon measure classes, revealing optimal conditions for solution existence, and demonstrating nonlinear-like phenomena such as blowup, oscillations, and prescribed boundary behaviors.

Contribution

It characterizes optimal initial data classes for solution existence and uncovers nonlinear-like behaviors in solutions of the linear heat equation.

Findings

Initial data classes are optimal for solution existence.

Solutions can exhibit finite-time blowup and complex spatial patterns.

Generic solutions show wild oscillations and customizable boundary behaviors.

Abstract

We analyse the behaviour of solutions of the linear heat equation in ${\mathbb R}^d$ for initial data in the classes $M_\varepsilon({\mathbb R}^d)$ of Radon measures with $\int_{{\mathbb R}^d}{\rm e}^{-\varepsilon|x|^2}\,{\rm d}|u_0|<\infty$. We show that these classes are in some sense optimal for local and global existence of non-negative solutions: in particular $M_0({\mathbb R}^d)=\cap_{\varepsilon>0}M_\varepsilon({\mathbb R}^d)$ consists precisely of those initial data for which the a solution of the heat equation can be given for all time using the heat kernel representation formula. After considering properties of existence, uniqueness, and regularity for such initial data, which can grow rapidly at infinity, we go on to show that they give rise to properties associated more often with nonlinear models. We demonstrate the finite-time blowup of solutions, showing that the set of blowup points is the complement of a convex set, and that given any closed convex set there is an initial condition whose solutions remain bounded precisely on this set at the `blowup time'. We also show that wild oscillations are possible from non-negative initial data as $t\to\infty$ (in fact we show that this behaviour is generic), and that one can prescribe the behaviour of $u(0,t)$ to be any real-analytic function $\gamma(t)$ on $[0,\infty)$.