Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains

We consider the oblique derivative problem for linear nonautonomous second order parabolic equations with bounded measurable coefficients on bounded Lipschitz cylinders. We derive an optimal elliptic-type Harnack inequality for positive solutions of this problem and use it to show that each positive solution exponentially dominates any solution which changes sign for all times. We show several nontrivial applications of both the exponential estimate and the derived Harnack inequality.