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

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.

There are new references and other minor changes suggested by a referee. The paper will appear in J. Math. Anal. Appl