Quantitative unique continuation for the linear coupled heat equations

In this paper, we established a quantitative unique continuation results for a coupled heat equations, with the homogeneous Dirichlet boundary condition, on a bounded convex domain Ω of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{d}$\end{document}Rd with smooth boundary ∂Ω. Our result shows that the value of the solutions can be determined uniquely by its value on an arbitrary open subset ω of Ω at any given positive time T.


Introduction
In this paper, we consider an unique continuation of the following linear coupled heat equations: where M is a positive number.
In this paper, we are concerned with the unique continuation for the solution of Eq. (.). The main results obtained in this work can be stated as follows.
The study of unique continuation for the solutions for PDEs began at the early of the last centaury. Besides its own interesting in PDEs theory, it also plays a key role in both inverse problem and control theory. The first quantitative result of strong unique continuation for parabolic equations was derived in  in the literature []. In [], the authors establish the unique continuation for the parabolic equations with time independent coefficients in terms of the eigenfunctions of the corresponding elliptic operator, and their results did not apply to parabolic equations with time dependent coefficients. After , there were more results of unique continuation for parabolic equations, and we refer the reader to [-], and the rich work cited therein. In [], the author discusses the unique continuation for stochastic counterpart. In our paper, we mainly study this property for the coupled heat equations. To the best of our knowledge, this topic has not been studied in past publications. With the aid of frequency function methods, we can establish these quantitative estimates (see [, ]).
The paper is organized as follows. In Section , some preliminary results are presented. The proofs of Theorem . and Theorem . will be given in Section .

Preliminary results
Given a positive number λ, we define Then, for each t ∈ [, T], we write where (y(x, t), z(x, t)) are the solutions of equation (.). The function N λ (t) was first discussed in [], and it was called frequency function (see also [, ], and []). Throughout this section, we always work under the assumption H λ (t) = . Next, we will discuss the properties for the functions G λ (x, t), H λ (t), D λ (t) and N λ (t and, for i = j, Lemma . For each λ > , the following identity holds for t ∈ (, T): Proof By a direct computation, we can obtain Therefore, for each t ∈ (, T), we have This completes the proof of this lemma.

Lemma . For each λ >  and t ∈ (, T), then we have
Proof By a direct computation, we can derive However, and Then, by (.), (.), (.), (.), we obtain In the same way, we get Thus, we can rewrite (.) as This completes the proof of this lemma.
Lemma . For each λ >  and t ∈ (, T), it follows that Proof Firstly, we compute N λ (t) as t ∈ (, T). By (.) and (.), we derive that It follows from the Cauchy-Schwarz inequality that It, together with (.), shows that In what follows, we will discuss the properties of A and B. Since y = z =  on ∂ , we have ∇y = ∂ ν yν, ∇z = ∂ ν zν on ∂ . If the domain is convex, we can derive that ((xx  ) · ν) ≥ .
According to (.) and (.), then In the same way, we get Combining with (.), we get Using equations (.) and (.) can be written as Thus, This completes the proof of this lemma. Let Then we have the following lemma.

Lemma . For each λ > , we have
Proof Integrating (.) over (t, T), we have Integrating the above over (, T  ), we obtain This, alone with (.), shows that We have It, combining with (.), shows that This completes the proof of the lemma.

The proof of Theorem 1.1
Proof We start with proving (.). We take λ >  in the estimate of Lemma . to be such that Therefore, we have Then we derive that Thus, Then we have which is equivalent to the following inequality: Let p = r  r  +C and then the above inequality can be written as Thus, we can obtain (.). This completes the proof of this theorem.

The proof of Theorem 1.2
Proof In order to give (.), we should first prove the following backward uniqueness estimate: For y  , z  ∈ H   ( ), the solutions of equation (.) y(x, t), z(x, t) ∈ L  (, T; H  ( ) ∩ H   ( )). Then we can define a function (t) as follows: Let f = ay + bz, and g = cy + dz. By direct computation, we obtain d dt Integrating by parts and using Cauchy-Schwarz inequality, we have With the aid of Gronwall's inequality, we obtain, for t ∈ (, T), This is the proof of (.), and we have completed the proof of Theorem ..

Conclusions
In this work, we discuss the unique continuation for the linear coupled heat equations.