Spatial asymptotics and equilibria of heat flow on [formula omitted

Abstract

We prove that the heat equation on Rd is well-posed in certain spaces of functions allowing spatial asymptotic expansions as |x|→∞ of any a priori given order. In fact, we show that the Laplacian on such function spaces generates an analytic semigroup of angle π/2 with polynomial growth as t→∞. 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.