A Carleman estimate for the fractional heat equation and its application in final state observability

Abstract

In the paper, we show a global Carleman estimate for the non-local heat equation. To be more precise, let $\Omega\subset\RR^d$ be a bounded domain and $\CO\subset\Omega$ an open subdomain, $s\in(0,1)$. We show that there exist constants $C_1,C_2,r_0, T_0>0$ and a weight function $\alpha:\Omega\to(0,\infty)$ such that any solution $u$ of %consider the following system % \begin{eqnarray}\label{oben1} \left\{ \begin{array}{rcl} \timed u(x,t)+(-\De)^s u (x,t) &=&f(x,t) \quad\mbox{for}\quad (x,t)\in \Om \times (0,\infty), \\ u(x,t) &=& 0 \quad\mbox{for}\quad(x,t)\in \partial \Om \times (0,\infty), \end{array}\right. \end{eqnarray} satisfies for all $r\ge r_0$ and $T>0$ \begin{eqnarray}\label{Carle} % \lefteqn{ \int_0^T\Big[ \int_\Om e^{-2r\frac {\alpha(x)}{t(T-t)}} |f(x,t)|^2\,dx+C_1\int_\CO e^{-2r\frac {\alpha(x)}{t(T-t)}} \frac {r^2}{t^4(T-t)^4}|u(x,t)|^2dx\,\Big] dt \vspace{2cm} }&& \\ \nonumber &\ge & C_2 \Bigg[\int_0^T \int_\Om e^{-2r\frac {\alpha(x)}{t(T-t)}}\Big\{ \big|(-\Delta)^s u(x,t)\big|^2 + \frac 12 \Big|\timed u(x,t)\Big|^2+ \frac r{t^4(T-t)^4}\,|u(t,x)|^2\Big\} dx\, dt. %\del{\\ %&&\vspace{2cm}}+ r^3\int_0^T \int_\mathcal{O} \frac {r^3}{t^3(T-t)^3} \Phi^2(x,t) |u(x,t)|^2 \, dx\, dt\Bigg]. \end{eqnarray} % In order to prove this result, we use the Caffarelli-Silvestre extension procedure. To illustrate the applicability of the result, we prove as a second main result the final state observability of the non-local heat equation.