Quantitative uniqueness for time-periodic heat equation with potential and its applications

Abstract

In this paper, we establish a quantitative unique continuation property for some time-periodic linear parabolic equations in a bounded domain $\Omega$. We prove that for a time-periodic heat equation with particular time-periodic potential, its solution $u(x,t)$ satisfies $\left\| u(\cdot,0) \right\| _{L^{2}(\Omega) }\leq C\left\| u(\cdot,0) \right\| _{L^{2}(\omega) }$ where $\omega\subset\Omega$. Also we deduce the asymptotic controllability for the heat equation with an even, time-periodic potential. Moreover, the controller belongs to a finite dimensional subspace and is explicitly computed.