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.