On a Backward Estimate for Solutions of Parabolic Differential Equations and its Application to Unique Continuation

Abstract

We prove a new backward estimate and a new strong unique continuation property for solutions $u \in \mathcal{C} = C^o ((0, T); H^2 ({\mathbf{R}}^n; e^{-\alpha|x|^2} dx)) \cap C^1 ((0, T); L^2 ({\mathbf{R}}^n; e^{-\alpha|x|^2} dx))$ of parabolic differential equations $\frac{\partial u}{\partial t} = \Delta u + V(x, t)u$ under certain conditions on $V$, where $\alpha > 0$ is a fixed number.