## Differential and Integral Equations

- Differential Integral Equations
- Volume 19, Number 6 (2006), 627-668.

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

K.-D. Phung and G. Wang

#### 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.

#### Article information

**Source**

Differential Integral Equations, Volume 19, Number 6 (2006), 627-668.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356050356

**Mathematical Reviews number (MathSciNet)**

MR2234717

**Zentralblatt MATH identifier**

1212.93025

**Subjects**

Primary: 93B05: Controllability

Secondary: 35B10: Periodic solutions 35B60: Continuation and prolongation of solutions [See also 58A15, 58A17, 58Hxx] 35K20: Initial-boundary value problems for second-order parabolic equations 93C20: Systems governed by partial differential equations

#### Citation

Phung, K.-D.; Wang, G. Quantitative uniqueness for time-periodic heat equation with potential and its applications. Differential Integral Equations 19 (2006), no. 6, 627--668. https://projecteuclid.org/euclid.die/1356050356