The heat semigroup on weighted Sobolev and asymptotic spaces.

We prove that the heat equation on $\R^d$ is well-posed in weighted Sobolev spaces and 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 $t\to\infty$. We apply these results to nonlinear heat equations on $\R^d$, including global existence in time.