Non-autonomous Ornstein-Uhlenbeck equations in exterior domains

In this paper, we consider non-autonomous Ornstein-Uhlenbeck operators in smooth exterior domains $\mathrm{\Omega }\subset {\mathbb{R}}^d$ 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 $\{P_{\mathrm{\Omega }}(t,s){\}}_{0\le s\le t}$ on $L^p(\mathrm{\Omega })$ for $1< p < \infty$\mathrm{.}Furthermore,$L^p$-estimatesforspatialderivativesand$L^p$-$L^q$smoothingpropertiesof$P_\Omega(t,s)$,$0\leq s\leq t$, are obtained.