Non autonomous parabolic problems with unbounded coefficients in unbounded domains

Abstract

Given a class of nonautonomous elliptic operators $\mathcal A(t)$ with unbounded coefficients, defined in $\overline{I \times \Omega}$ (where $I$ is a right-halfline or $I=\mathbb R$ and $\Omega\subset \mathbb R^d$ is possibly unbounded), we prove existence and uniqueness of the evolution operator associated to $\mathcal A(t)$ in the space of bounded and continuous functions, under Dirichlet and first order, non tangential homogeneous boundary conditions. Some qualitative properties of the solutions, the compactness of the evolution operator and some uniform gradient estimates are then proved.