Observability estimate for the parabolic equations with inverse square potential

1.   Introduction

Let $\Omega$ be a $C^2$ bounded domain in $\mathbb{R}^d$ $(d\geq3)$ such that $0\in\Omega$. Let $\omega\subset\Omega$ be a non-empty open subset, and $0\notin\omega$. Let $T > 0$, $E\subset(0, T]$ be a Lebesgue measurable subset with positive measure, and denote by $\chi_E$ the characteristic function of $E$.

In this paper, we study the observability estimate for a parabolic equation with a singular potential term, which is described by

where the initial datum $\phi(x)\in L^2(\Omega)$, and the parameter $\mu$ satisfies

It should be emphasized that condition (1.2) is crucial for Eq (1.1). In ^[2], it is proved that if the initial datum $\phi(x)\in L^2(\Omega)$ is non-negative, then Eq (1.1) admits a unique global solution under condition (1.2), and when $\mu > \mu_*$, the local solution may not exist. In recent years, the study on Eq (1.1) has been a hot topic in the theory of PDEs due to its important applications in various branches of applied science and engineering. Extensive related references can be found in ^[4,10,11] and the rich works cited therein. In particular, the well-posedness of Eq (1.1) is further discussed from the point of view of semigroup theory in ^[11]. More specifically, it shows that when $\mu < \mu_*$, for each $\phi\in L^2(\Omega)$, there exists a unique

which is a weak solution of the Eq (1.1).

In this paper, we mainly study the observability estimate for Eq (1.1). The main difficulty in this problem lies in the singularity of Eq (1.1) caused by the potential term. It turns out that Carleman inequalities, frequency functions, and the spectral inequality are the main frameworks in the literature on the observability estimate for the parabolic equations in recent decades. The application of Carleman estimates was described systematically in the works ^[3,14]. About frequency functions, we would like to mention references ^[1,7]. In this article, we apply the spectral inequality for the Schrödinger operator $A = -\triangle-\frac{\mu}{|x|^2}$ to build this estimate.

We denote $\|\cdot\|$ and $\langle\cdot, \cdot\rangle$ to be the usual norm and the inner product in $L^2(\Omega)$, respectively. The main result of the paper is presented as follows:

Theorem 1.1. Let $E\subseteq(0, T]$ be a measurable set with a positive measure, and let $\omega$ be a non-empty open subset of $\Omega$ with $0\notin\omega$. Suppose that

Then, there exists a constant $C = C(\Omega, T, \omega, E)$ such that for each datum $\phi\in L^2(\Omega)$, the solution to Eq (1.1) satisfied

Remark 1.1. Several notes on Theorem 1.1 are given in order.

(a) Inequality (1.4) is called the observability estimate. Observability estimate (1.4) in Theorem 1.1 allows estimating the total energy of the solutions to Eq (1.1) at time $T$ in terms of the partial energy localized in the observation region $\omega\times E$.

(b) With the aid of observability estimate (1.4), we can study the bang-bang property for the time optimal control problem governed by Eq (1.1) with internal control.

(c) From the perspective of control theory, this inequality is equivalent to the null controllability for its adjoint equation. (See ^[5].) We mention ^[3,12,13] in this direction.

(d) In ^[16], a similar problem was investigated for Eq (1.1) when $\Omega$ is a convex domain of $R^d$ and $0\in\omega$, where $\omega$ is a non-empty open subset of $\Omega$. It obtains in ^[16] that there exist two positive numbers $\alpha = \alpha(\Omega, \omega)\in(0, 1)$, $C = C(\Omega, \omega)$ such that for each $T > 0$, the solution of (1.1) satisfies

where $\phi(x)\in L^2(\Omega)$ is the initial datum. Clearly, (1.5) is a Hölder type interpolation inequality for Eq (1.1), from which and the method in Theorem 4 of ^[8], we can also get (1.4) in this case.

We organize the paper as follows. In Section 2, we give some preliminary results. Section 3 is devoted to the proof of Theorem 1.1.

2.   Preliminary results

We first introduce certain notations. Let $\{\lambda_i\}_{i = 1}^\infty$, with $0 < \lambda_1 < \lambda_2\leq\cdots\leq \lambda_i\rightarrow +\infty$ as $i\rightarrow \infty$, be the set of eigenvalues of the Schrödinger operator $A: = -\triangle-\frac{\mu}{|x|^2}$ with homogeneous Dirichlet boundary conditions, and let $\{e_i(x)\}_{i = 1}^\infty$ be the set of corresponding eigenfunctions with $\|e_i(x)\|_{L^2(\Omega)} = 1$, $(i = 1, 2, 3, \cdots)$ which constitutes an orthonormal basis of $L^2(\Omega)$ (e.g., see Theorem 3.1 in ^[12]). Now, for each $r > 0$, we define two subspaces of $L^2(\Omega)$ as follows

Clearly, we have the following orthogonal direct sum decomposition

Let $S(t)$ be the compact semigroup generated by the operator $-A$ on $L^{2}(\Omega)$. Then, the unique solution to Eq (1.1) corresponding to the initial datum $\phi(x)\in L^2(\Omega)$ can be written as (e.g., see section 4.1 in ^[9])

In addition,

It follows from (2.1) that

Remark 2.1. In fact, it follows from (2.2) that the solution to Eq (1.1) possesses the energy decay property.

As a direct consequence of (2.1) and (2.2), we have

Lemma 2.1. For each positive number $r > 0$ and each $\eta\in X_r^{\perp}$, the following conclusion is true:

Next, the following result is brought from ^[6] (see Theorem 1.2 in ^[6]).

Lemma 2.2. Let $\omega$ be a non-empty open subset of $\Omega$ with $0\notin\omega$. Suppose (1.3) holds. Then, there exist two positive constants $C_1 > 1$ and $C_2 > 0$, which only depend on $\Omega, \omega$, such that

for each positive number $r > 0$ and each choice of the coefficients $\{a_i\}_{\lambda_i\leq r}$ with $a_i\in\mathbb{R}$.

Remark 2.2. $(i)$ Inequality (2.4) is called the spectral inequality for the Schrödinger operator $A = -\triangle-\frac{\mu}{|x|^2}$.

$(ii)$ By Lemma 2.2, we have that for each positive number $r > 0$ and each $\psi\in X_r$,

Throughout the rest of this paper, the following notation will be used. For each measurable set $A\subseteq \mathbb{R}$, we denote by $|A|$ its Lebesgue measure in $\mathbb{R}$. The following lemma is quoted from ^[7,15] (e.g., see Proposition 2.1 in ^[7]).

Lemma 2.3. Let $E\subseteq[0, T]$ be a measurable set with a positive measure, and let $l$ be a density point of $E$. Then, for each $z > 1$, there exists $l_1\in(l, T)$ such that the sequence $\{l_i\}_{i = 1}^\infty$ given by

satisfies

3.   The proof of the main result

Proof. The proof will be organized in three steps.

Step 1. We first construct a subset sequence of $(0, T]$.

Let $l$ be a density point for $E$. We arbitrarily fix a positive number $z > 1$. By Lemma 2.3, there exists $l_1\in(l, T]$ and a sequence $\{l_i\}_{i = 1}^\infty$ satisfying (2.6) and (2.7). We now define a subset sequence of $(0, T]$ as follows:

It is obvious that

Then, it follow from (2.7) that

It follows from (2.6) that

Thus,

Let $b > 1$ be a positive number satisfying

where $C_1$ and $C_2$ are the positive numbers given in Lemma 2.2. Taking $r_i = b^i$ $(i = 2, 3, 4\cdots)$, and using (3.3) and (3.5), we obtain that

It, together with (3.4), indicates

Step 2. For each $i\in\{2, 3, 4, \cdots\}$, we estimate the term $\frac{l_i-l_{i+1}}{4}\|S(l_{i})\phi\|_{L^2(\Omega)}$, where $\phi\in L^2(\Omega)$.

From (3.2) and (2.2), it follows that for each $\varphi\in X_{r_i}$ $(i\in\{2, 3, 4\cdots\})$,

For each initial datum $\phi\in L^2(\Omega)$, there exists a unique decomposition

where $\phi_1\in X_{r_i}$ and $\phi_2\in X^{\bot}_{r_i}$. Taking $\varphi = \phi_1$ in (3.7), and then using (2.5), we obtain

By the triangle inequality and (3.1), we deduce

Together with (2.2), we have

This, along with Lemma 2.1, indicates that

By Lemma 2.2 and $\phi_2\in X^{\bot}_{r_i}$, we have

Applying (3.8), and the triangle inequality, we obtain

Then, it, together with (3.9), and (3.10), indicates

Thus,

which leads to

Step 3. We prove (1.4).

Summing (3.11) from $2$ to $\infty$, it follows that

where

By (3.6), we obtain

This, together with (3.12) and (2.2), indicates that

That is

from which and Cauchy's inequality, it follows (1.4). This completes the proof.

4.   Conclusions

Using the Lebeau-Robbiano strategy, we obtain the observation estimate (1.4) for Eq (1.1) in this paper. From (1.4), we have that if the solution $y = 0$ over $\omega\times E$, then $y(\cdot, T) = 0$ over $\Omega$. It, together with (2.1), shows that $y\equiv0$ over $\Omega\times[0, T]$. From the application point of view, we can recover the initial state and the evolution history, according to the observation of $y$ in $\omega\times E$.

Acknowledgments

The authors would like to express their sincere thanks to the referees for their valuable suggestions. This work was partially supported by the Natural Science Foundation of Henan Province (No. 202300410248), and by the Natural Science Foundation of Henan Normal University (No. 2019PL15).

Conflict of interest

The authors declare that they have no competing interests in this paper.