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

Niveau: Supérieur, Licence, Bac+2
Di?erential and Integral Equations Volume 19, Number 6 (2006), 627–668 QUANTITATIVE UNIQUENESS FOR TIME-PERIODIC HEAT EQUATION WITH POTENTIAL AND ITS APPLICATIONS K-D. Phung Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China G. Wang Department of Mathematics, Wuhan University, Wuhan 430072, China The Center for Optimal Control and Discrete Mathematics Huazhong Normal University, Wuhan 730079, China (Submitted by: Viorel Barbu) Abstract. In this paper, we establish a quantitative unique contin- uation property for some time-periodic linear parabolic equations in a bounded domain Ω. We prove that for a time-periodic heat equa- tion with particular time-periodic potential, its solution u(x, t) satis- fies ?u(·, 0)?L2(Ω) ≤ C ?u(·, 0)?L2(?) where ? ? Ω. Also we deduce the asymptotic controllability for the heat equation with an even, time- periodic potential. Moreover, the controller belongs to a finite dimen- sional subspace and is explicitly computed. 1. Introduction and main results Throughout this paper, Ω is a connected bounded domain in Rd, d ≥ 1, with a boundary ∂Ω of class C2, ? is a non-empty open subset of Ω. Let 1 > 0 be the first eigenvalue of the operator ?? with the Dirichlet boundary condition (i.

• periodic potential


• stable eigenvalues


• strong unique


• unique solution


• heat equation


• boun- ded domains


• continuation property


• linear time-periodic