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 Lam & eacute; 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 Lam & eacute; operator to another system that resembles the extension problem for the fractional heat operator. Finally in the case when s >= 1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \ge 1/2$$\end{document}, 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 Hsu=Vu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}<^>s \textbf{u}=V\textbf{u}$$\end{document}.