Non-autonomous Ornstein-Uhlenbeck equations in exterior domains

Abstract

In this paper, we consider non-autonomous Ornstein-Uhlenbeck operators in smooth exterior domains $\Omega \subset {\mathbb{R}}^d$ subject to Dirichlet boundary conditions. Under suitable assumptions on the coefficients, the solution of the corresponding non-autonomous parabolic Cauchy problem is governed by an evolution system $\{P_{\Omega }(t,s){\}}_{0\le s\le t}$ on $L^p(\Omega )$ for $1<p<\infty$. Furthermore, $L^p$-estimates for spatial derivatives and $L^p$-$L^q$ smoothing properties of $P_{\Omega }(t,s)$, $0\le s\le t$, are obtained.

16 pages