Extension problem for the fractional parabolic Lamé operator and unique continuation

In this paper, we introduce and analyse an explicit formulation of fractional powers of the parabolic Lamé operator and we then study the extension problem associated to such non-local operators. We also study the various regularity properties of solutions to such an extension problem via a transformation as in Ang et al. (Commun Partial Differ Equ 23:371–385, 1998), Alessandrini and Morassi (Commun Partial Differ Equ 26(9–10):1787–1810, 2001), Eller et al. (Nonlinear partial differential equations andtheir applications, North-Holland, Amsterdam, 2002), and Gurtin (in: Truesdell, C. (ed.) Handbuch der Physik, Springer, Berlin, 1972), which reduces the extension problem for the parabolic Lamé operator to another system that resembles the extension problem for the fractional heat operator. Finally in the case when \(s \ge 1/2\) $s\ge 1/2$ , by proving a conditional doubling property for solutions to the corresponding reduced system followed by a blowup argument, we establish a space-like strong unique continuation result for \(\mathbb {H}^s \textbf{u}=V\textbf{u}\) ${\mathbb{H}}^s\mathbf{u}=V\mathbf{u}$ .