Spatial asymptotics and equilibria of heat flow on $\mathbb{R}^d$

TL;DR

This paper establishes well-posedness and asymptotic behavior of the heat equation on ^d in function spaces with prescribed spatial asymptotics, and explores equilibrium solutions for nonlinear heat flows.

Contribution

It introduces new function spaces allowing detailed asymptotic analysis and demonstrates the generation of analytic semigroups for the Laplacian in these spaces, extending understanding of nonlinear heat flow equilibria.

Findings

Heat equation is well-posed in spaces with prescribed asymptotics.

Laplacian generates an analytic semigroup with polynomial growth.

Existence of equilibrium solutions with specified spatial asymptotics.

Abstract

We prove that the heat equation on $\mathbb{R}^d$ is well-posed in certain spaces of functions allowing spatial asymptotic expansions as $|x|\to\infty$ of any a priori given order. In fact, we show that the Laplacian on such function spaces generates an analytic semigroup of angle $\pi/2$ with polynomial growth as $t\to\infty$. Generically, a large class of nonlinear heat flows have equilibrium solutions with spatial asymptotics of the considered type. We provide a simple nonlinear model that features global in time existence with such asymptotics at spatial infinity.