Inverse Source Problem for a Singular Parabolic Equation with Variable Coefficients

Abstract

We consider a parabolic equation with a singular potential in a bounded domain $\Omega \subset {\mathbb{R}}^n.$ The main result is a Lipschitz stability estimate for an inverse source problem of determining a spatial varying factor $f\left(x\right)$ of the source term $R(x,t)f\left(x\right).$ We obtain a consistent stability result for any $\mu \le p_1{\mu }^{*},$ where $p_1>0$ is the lower bound of $p\left(x\right)$ and ${\mu }^{*}={(n-2)}^2/4,$ and this condition for $\mu$ is also almost a consistently optimal condition for the existence of solutions. The main method we used is the Carleman estimate, and the proof for the inverse source problem relies on the Bukhgeim–Klibanov method.

1. Introduction and Main Result

The purpose of the article is to study an inverse problem to determine the spatially varying parameter $f\left(x\right)$ in a parabolic equation with a singular potential. More precisely, we will study an inverse source result for a parabolic equation with variable coefficients in the principal part and an inverse-squared potential of the form $-\mu /{\left|x\right|}^2$. For the application, we refer to the combustion theory [1,2,3,4] and quantum mechanics [5,6,7].

Let $n\ge 3,n\in N$ be given. $\Omega \subset {\mathbb{R}}^n$ is a bounded domain satisfying $0\in \Omega$ and $\partial \Omega \in C^2$. $T>0.$ Denote

$$\begin{array}{c}{\Omega }_T=\Omega \times (0,T),{\Sigma }_T=\partial \Omega \times (0,T).\end{array}$$

We are interested in determining the spatially varying parameter $f\left(x\right)$ of the source term in the system

$$\left\{\begin{array}{c}{\partial }_{t}u-\mathrm{div}(p\left(x\right)\nabla u)-{\displaystyle \frac{\mu }{{\left|x\right|}^2}}u=f\left(x\right)R(x,t),(x,t)\in {\Omega }_T,\hfill \\ u(x,t)=0,(x,t)\in {\Sigma }_T,\hfill \\ u(x,0)=u_0\left(x\right),x\in \Omega ,\hfill \end{array}\right.$$

(1)

where $u_0\in L^2(\Omega ),$$R(x,t)\in H^1(0,T;L^{\infty }(\Omega ))$, and $f\left(x\right)\in L^2(\Omega )$. $x=(x_1,\dots ,x_n)\in \Omega \subseteq {\mathbb{R}}^n,$${\partial }_t={\displaystyle \frac{\partial }{{\partial }_t}}$, ${\partial }_j={\displaystyle \frac{\partial }{\partial x_j}},j=1,\dots ,n$, $\nabla =({\partial }_1,\dots ,{\partial }_n)$, and $\mu$ is a real constant. Let $p_1,p_2,p_3$ be three positive constants. We assume that $p\left(x\right)\in \mathcal{B}$, where

$$\begin{array}{c}\hfill \mathcal{B}=\left\{p\left(x\right)\in C^3\left(\overline{\Omega }\right)|p_1\le p\left(x\right)\le p_2,|\nabla p\left(x\right)|\le p_3,\forall x\in \overline{\Omega }\right\}.\end{array}$$

(2)

In fact, the inverse-squared potential of the form $-\mu /{\left|x\right|}^2$ appears in some linearized combustion models, such as the following semilinear elliptic equation:

$$\left\{\begin{array}{c}-\Delta u=\mu h\left(u\right),x\in \Omega ,\hfill \\ u\left(x\right)=0,x\in \partial \Omega ,\hfill \end{array}\right.$$

(3)

where the nonhomogeneous term $h:R\to R$ is positive, nonlinear, continuous and increasing. h is a convex function with $h\left(0\right)>0$ and $h\left(x\right)/x\to \infty$ as $x\to +\infty$. Semilinear equations like (3) describe some applications in combustion theory, like the phenomenon of a ball of isothermal gas in gravitational equilibrium, which was proposed by Kelvin (see [8]).

Those inverse-squared potentials also have a number of applications in quantum mechanics. For instance, in [5], the model involving a linear-plus-inversely linear electric field is used to describe the confinement of neutral fermions, which leads to an effective quadratic-plus-inversely quadratic potential in a Sturm–Liouville problem. Readers can also refer to [6] (p. 157) for some other instances on the context of quantum mechanics.

Let $\omega$ be some non-empty open set satisfying $0\notin \omega \subset \Omega$. Let $t_0\in (0,T)$ be fixed; then,

$$\begin{array}{c}\hfill T_0={\displaystyle \frac{t_0+T}{2}}.\end{array}$$

(4)

Denote

$$\begin{array}{c}\hfill {\Omega }_{t_0,T}=\Omega \times (t_0,T),{\Sigma }_{t_0,T}=\partial \Omega \times (t_0,T),{\omega }_T=\omega \times (0,T),{\omega }_{t_0,T}=\omega \times (t_0,T).\end{array}$$

In this article, we consider the following:

Inverse source problem:

Let $T>0,$$R(x,t)$ be given. Determine $f\left(x\right),x\in \Omega$ from $u(·,T_0)$ in Ω and ${\partial }_{t}u$ in $\omega \times (t_0,T)$, where $u(x,t)$ is the solution of (1).

Our main result is the following theorem.

Theorem 1.

Assume $\mu \le p_1{\mu }^{*}$, where ${\mu }^{*}={(n-2)}^2/4$. $T_0$ is defined as (4). Assume that $p\left(x\right)\in \mathcal{B},$ where $\mathcal{B}$ is given by (2). Assume that, $u_0\in L^2(\Omega ),$ and that u is the solution of (1). Assume that

$$\begin{array}{c}\hfill R(x,t)\in W^{1,\infty }\left({\Omega }_T\right),{\parallel {\partial }_{t}R\parallel }_{L^{\infty }\left({\Omega }_T\right)}\le R_0,\left|R(x,T_0)\right|\ge r_0,x\in \overline{\Omega }\end{array}$$

(5)

for some positive constant $r_0.$ Then, there exists a constant $C=C(n,\Omega ,\omega ,t_0,T,p_1,p_2,p_3,r_0,R_0)>0$ such that

$$\begin{array}{c}{∥f∥}_{L^2(\Omega )}^2\le C\left({∥\mathrm{div}(p\nabla u(·,T_0))+{\displaystyle \frac{\mu }{{\left|x\right|}^2}}u(·,T_0)∥}_{L^2(\Omega )}^2+{∥{\partial }_{t}u∥}_{L^2\left({\omega }_{t_0,T}\right)}^2\right).\end{array}$$

(6)

Remark 1.

In fact, using the same idea, we have a similar result for a general elliptic operator instead of the operator in (1).

Corollary 1.

Assume that ${\left\{a_{ij}\left(x\right)\right\}}_{1\le i,j\le n}$ satisfies $a_{ij}\left(x\right)=a_{ji}\left(x\right)$ and the uniform ellipticity. There exists a constant $a_1,a_2,a_3>0$ such that

$$\begin{array}{c}\hfill \parallel a_{ij}{\parallel }_{C^2(\Omega )}\le a_3,a_1{\left|\xi \right|}^2\le \sum _{i,j=1}^n{a}_{ij}\left(x\right){\xi }_i{\xi }_j\le a_2{\left|\xi \right|}^2,\forall \xi =({\xi }_1,\dots ,{\xi }_n)\in {\mathbb{R}}^n.\end{array}$$

(7)

Assume that $\mu \le a_1{\mu }^{*}$. $T_0$ is defined as (4). Assume that $u_0\in L^2(\Omega ),$ and that u is the solution of (1) with a more general case ${\left\{a_{ij}\left(x\right)\right\}}_{1\le i,j\le n}$. Assume that

$$\begin{array}{c}\hfill R(x,t)\in W^{1,\infty }\left({\Omega }_T\right),{\parallel {\partial }_{t}R\parallel }_{L^{\infty }\left({\Omega }_T\right)}\le R_0,\left|R(x,T_0)\right|\ge r_0,x\in \overline{\Omega }\end{array}$$

for some positive constant $r_0.$ Then, there exists a constant $C=C(n,\Omega ,\omega ,t_0,T,a_1,a_2,a_3,r_0,R_0)>0$ such that

$$\begin{array}{c}{∥f∥}_{L^2(\Omega )}^2\le C\left({∥\sum _{i,j=1}^n{\partial }_i\left(a_{ij}{\partial }_{j}u(·,T_0)\right)+{\displaystyle \frac{\mu }{{\left|x\right|}^2}}u(·,T_0)∥}_{L^2(\Omega )}^2+{∥{\partial }_{t}u∥}_{L^2\left({\omega }_{t_0,T}\right)}^2\right).\end{array}$$

We give the proof of Theorem 1 in Section 4. The idea we followed to determine the source term mainly relies on a global $L^2$—Carleman estimate, which is proposed by Bukhgeim and Klibanov in [9]. Indeed, this is also the fundamental method to prove the uniqueness and the stability in inverse source problems.

The inverse source problem on a parabolic system with a singular potential in the case of $p\left(x\right)\equiv 1$ was firstly solved by Vancostenoble [10]. Later, for $\forall \mu <(p_1^2/p_2){\mu }^{*}$, we solved the null-controllability problem in [11] and the inverse coefficient problem in [12] for the variable coefficient case. Moreover, we consider the inverse source problem for parabolic equations with $m(m\ge 1)$ different singularities when $p\left(x\right)=1$ in [13]. In this article, we extend the inverse source problem in [10] to the case with a general variable coefficient $p\left(x\right)$ in the principal part. As a result, we establish a Lipschitz stability result for the optimal condition $\mu \le p_1{\mu }^{*},$ which is almost a consistently optimal condition for the existence of solutions to the system (1).

For the related controllability issues, many results have been studied by predecessors (e.g., [11,14,15,16,17]). In the case of $p\left(x\right)\equiv 1$, ref. [15] has proven that we can obtain a null-controllability result for $\forall \mu \le {\mu }^{*}$. In [11], we considered the null-controllability problem for a parabolic equation with variable coefficients $p\left(x\right)$ in the principal part and with a singular potential $-\mu /{\left|x\right|}^2$. Ref. [11] indicates that we can obtain the null-controllability problem for a distributed control in an arbitrary open subset of $\Omega$ in the case of $\mu <(p_1^2/p_2){\mu }^{*}$. However, for the case of $\mu >p_2{\mu }^{*}$, we proved that we cannot uniformly stabilize regularized approximations of the system by using a control supported in $\omega$. Furthermore, ref. [16] has proven a null-controllability result in the case $p\left(x\right)\equiv 1$ within the assumption that singular potential $-\mu /{\left|x\right|}^2$ arises at the boundary domain. Ref. [14] analyzed the control properties for a parabolic equation, wherein all the points on the boundary are singular potentials.

This paper is organized as follows: In Section 2, we give the functional setting space. In Section 3, we introduce an improved Carleman estimate which is used to prove our main result. Section 4 is devoted to the proof of our main inverse source result. Section 5 presents some future work. The Appendix A are devoted to the proof of the improved Carleman estimate presented in Section 4.

2. Functional Setting and Well-Posedness

Let w satisfy

$$\left\{\begin{array}{c}{\partial }_{t}w-\mathrm{div}(p\left(x\right)\nabla w)-{\displaystyle \frac{\mu }{{\left|x\right|}^2}}w=g(x,t),(x,t)\in {\Omega }_T,\hfill \\ w(x,t)=0,(x,t)\in {\Sigma }_T,\hfill \\ w(x,0)=w_0\left(x\right),\hfill \end{array}\right.$$

(8)

where $w_0\left(x\right)\in L^2(\Omega ),$$g(x,t)\in L^2\left({\Omega }_T\right),$$p\left(x\right)\in \mathcal{B},$ and where $\mathcal{B}$ is given by (2).

Firstly, there are many related results about the well-posedness of the solution of system (8). Baras and Goldsteint [7,18] discovered that the existence and non-existence of positive solutions to (8) in the case of $p\equiv 1$ and $g\equiv 0$ is strongly related to the value of $\mu$ and Hardy’s inequality [19]:

$${\mu }^{*}\left(n\right){\int }_{\Omega }{\displaystyle \frac{{\left|u\right|}^2}{{\left|x\right|}^2}}\mathrm{d}x\le {\int }_{\Omega }{|\nabla u|}^2\mathrm{d}x,\forall u\in H_0^1(\Omega ),$$

(9)

where ${\mu }^{*}={\mu }^{*}\left(n\right)={(n-2)}^2/4$ is the optimal constant. The following discoveries were presented by Goidstein and Zhang [20]:

(i)

For $\forall \mu \le p_1{\mu }^{*}$, (8) has a unique non-negative solution $u(·,t)\in H_0^1(\Omega )$ for any positive initial value $u_0\in L^2(\Omega ),$ and $f(·,t)\in L^2(\Omega )$ for all $t\in (0,T)$;

(ii)

For $\forall \mu \ge p_2{\mu }^{*},$ (8) has no non-negative solutions except $u\equiv 0.$

It should be noted that the more general singular potential case $V\in L_{loc}^1(\Omega )$ when $p\left(x\right)\equiv 1$ was settled by Cabré and Martel [21].

Secondly, since we assume $\mu \le p_1{\mu }^{*}$ and $p\left(x\right)\in \mathcal{B},$ using (9), we have

$$\mu {\int }_{\Omega }{\displaystyle \frac{{\left|u\right|}^2}{{\left|x\right|}^2}}\mathrm{d}x\le p_1{\mu }^{*}{\int }_{\Omega }{\displaystyle \frac{{\left|u\right|}^2}{{\left|x\right|}^2}}\mathrm{d}x\le {\int }_{\Omega }p\left(x\right){|\nabla u|}^2\mathrm{d}x,\forall u\in H_0^1(\Omega ).$$

(10)

For any p satisfying (2), we introduce a new Hilbert space $H_p(\Omega )$ with the following norm:

$$\begin{array}{c}\forall w\in H_0^1(\Omega ),{∥w∥}_{H_p(\Omega )}={\left({\int }_{\Omega }\left({p\left(x\right)|\nabla w|}^2-{\displaystyle \frac{\mu }{{\left|x\right|}^2}}{\left|w\right|}^2\right)\mathrm{d}x\right)}^{1/2}.\end{array}$$

In the sub-critical case, $\mu <p_1{\mu }^{*}$, and when $n\ne 2,$ (9) implies that ${∥w∥}_{H_p(\Omega )}$ is equivalent to the usual norm in $H_0^1(\Omega ).$ However, in the critical case $p\left(x\right)\equiv p_1$ on $\overline{\Omega }$ and $\mu =p_1{\mu }^{*},$ the space ${∥w∥}_{H_p(\Omega )}$ is strictly larger than $H_0^1(\Omega ).$

For $\forall \mu \le p_1{\mu }^{*}$ and $\forall p\in \mathcal{B}$, we introduce an unbounded operator:

$$\begin{array}{c}D\left(A_p\right)=\left\{w\in H_p(\Omega )|\left(\nabla ·(p\left(x\right)\nabla w)+{\displaystyle \frac{\mu }{{\left|x\right|}^2}}w\right)\in L^2(\Omega )\right\},\\ \forall w\in D\left(A_p\right),A_{p}w=-\nabla ·(p\left(x\right)\nabla w)-{\displaystyle \frac{\mu }{{\left|x\right|}^2}}w.\end{array}$$

For $\forall w\in D\left(A_p\right),$ the norm is defined by the following:

$$\begin{array}{c}{∥w∥}_{D\left(A_p\right)}={\parallel w\parallel }_{L^2(\Omega )}+{∥\nabla ·(p\left(x\right)\nabla w)+{\displaystyle \frac{\mu }{{\left|x\right|}^2}}w∥}_{L^2(\Omega )}.\end{array}$$

Using the standard theory of semi-groups, we can obtain:

Theorem 2.

μ is a constant satisfying $\mu \le p_1{\mu }^{*}$ and $p\in \mathcal{B}.$ Then, the following assertions hold:

(i) 

For all $w_0\in D\left(A_p\right)$, and  $g\in H^1\left(0,T;L^2(\Omega )\right),$ system (8) has a unique solution satisfying

$$w\in C^0\left([0,T];D\left(A_p\right)\right)\cap C^1\left([0,T];L^2(\Omega )\right).$$

(ii) 

For all $w_0\in L^2(\Omega )$ and $g\in L^2\left(0,T;L^2(\Omega )\right),$ system (8) has a unique solution satisfying

$$\forall t_0>0,w\in L^2\left([t_0,T];D\left(A_p\right)\right)\cap H^1\left([t_0,T];L^2(\Omega )\right).$$

(iii) 

For all $w_0\in L^2(\Omega )$ and $g\in H^1\left(0,T;L^2(\Omega )\right),$ system (8) has a unique solution satisfying

$$\forall t_0>0,w\in C^0\left([t_0,T];D\left(A_p\right)\right)\cap C^1\left([t_0,T];L^2(\Omega )\right).$$

The proof of Theorem 2 mainly relies on the fact that the operator $A_p$ generates an analytic semi-group of contractions in the pivot space $L^2(\Omega )$, whereby applying the standard semi-group theory is sufficient. For more details, readers can refer to [10,20,22].

3. Carleman Estimate

In this paper, we aim to solve the inverse source problem of (1). Indeed, the inverse source problem is one of the most important subjects in partial differential equations. The normal method to prove inverse problems is via the Carleman estimate, which was proposed by Carleman [23] in 1939 to obtain a unique continuation for a two-dimensional elliptic equation. Later, the method of Carleman estimates was applied to more problems, such as the controllability and inverse problems.

The idea of applying Carleman estimates to inverse problems was first proposed by Bukhgeim and Klibanov [9]. So far, there have been many studies published about inverse problems related to normal parabolic equations, for example, Bukhgeim [24], Huang et al. [25], Isakov [26], Klibanov [27,28,29,30], Klibanov and Li [31], Imanuvilov and Yamamoto [32], Yamamoto [33], Yuan and Yamamoto [34], and related references therein. For inverse problems in relation to parabolic equations with singular potentials, we refer to the works of Qin and Li [12], Qin [13], Vancostenoble [10].

In this section, we will give an improved Carleman estimate result compared with [11] for system (8). The most prominent feature of Carleman estimates in this section is the optimization of conditions on $\mu$. Compared with the condition $\mu <{\displaystyle \frac{p_1^2}{p_2}}{\mu }^{*}$ in [11], Carleman estimates in this section holds for all $\mu \le p_1{\mu }^{*}$. According to Section 2, this condition is consistently optimal for the well-posedness of (1). Meanwhile, all constants C in the Carleman estimates are independent of the value of $\mu$.

In fact, this optimization of conditions on $\mu$ is not obvious. Since the coexistence of the singular potential $\mu /{\left|x\right|}^2$ and the variable coefficient $p\left(x\right)$ in the principal term, the method of calculation of the Carleman estimate in [10,11,15] cannot be directly applied to our problems. The existence of the variable coefficient $p\left(x\right)$ leads to the deduction of certain terms like $\left(\nabla p·\nabla \sigma \right){|\nabla z|}^2$ in calculations, which then leads to the fact that $\mu z/\left|x\right|$ cannot be completely controlled by $|\nabla z|$ for all $\mu \le {\mu }^{*}$ in the norm of $L^2(\Omega )$. In [11], combining the assumption $p\in \mathcal{B}$, $\mu <{\displaystyle \frac{p_1^2}{p_2}}{\mu }^{*}$ and Hardy’s inequality (9), we used the following estimate inequality:

$$\begin{array}{cc}\hfill {\int }_{\Omega }{p}^2\left(x\right){|\nabla z|}^2\mathrm{d}x-\mu {\int }_{\Omega }p\left(x\right){\displaystyle \frac{{\left|z\right|}^2}{{\left|x\right|}^2}}\mathrm{d}x& \ge p_1^2{\int }_{\Omega }{|\nabla z|}^2\mathrm{d}x-p_2\mu {\int }_{\Omega }{\displaystyle \frac{{\left|z\right|}^2}{{\left|x\right|}^2}}\mathrm{d}x\hfill \\ \hfill & \ge \left(p_1^2{\mu }^{*}-p_2\mu \right){\int }_{\Omega }{\displaystyle \frac{{\left|z\right|}^2}{{\left|x\right|}^2}}\mathrm{d}x>0,\forall z\in H_0^1(\Omega ).\hfill \end{array}$$

In this article, to offset negative terms and obtain an optimal condition on $\mu$, we introduce some new terms (see (A5)) in the grouping step of the proof of Carleman estimates, and we propose a weighted Hardy’s inequality:

Lemma 1.

Assume that $p\left(x\right)\in \mathcal{B}$, where $\mathcal{B}$ is given by (2). Then, for any $\mu \le p_1{\mu }^{*}$, there exists a positive constant C such that the following inequality holds:

$$\begin{array}{c}\hfill {\int }_{\Omega }{p}^2{\left(x\right)|\nabla z|}^2\mathrm{d}x+C{\int }_{\Omega }{\left|z\right|}^2\mathrm{d}x\ge \mu {\int }_{\Omega }p\left(x\right){\displaystyle \frac{{\left|z\right|}^2}{{\left|x\right|}^2}}\mathrm{d}x,\forall z\in H_0^1(\Omega ).\end{array}$$

(11)

For the proof of (11), see (A33).

In this section, we assume that $p\left(x\right)\in \mathcal{B}$, where $\mathcal{B}$ is given by (2). Let r be a fixed small positive constant that satisfies

$$\left\{\begin{array}{c}0<r<1,p_1>5p_{3}r,\hfill \\ \overline{B}(0,r)\subseteq \Omega ,\overline{B}(0,r)\cap \overline{\omega }=⌀.\hfill \end{array}\right.$$

(12)

Here, $B(0,r)$ is a ball, and henceforth, we denote $B^r\triangleq B(0,r),$$\overline{B^r}\triangleq \overline{B(0,r)}.$

Unlike [11,12], r, determined by (12), does not depend on the value of $\mu$. In fact, this is the key to deduce the improved result that all constants C in the Carleman estimates are independent of the value of $\mu$.

As in [11], for some $\delta >0$, we introduce a smooth function satisfying

$$\left\{\begin{array}{c}\psi \left(x\right)=ln\left({\displaystyle \frac{\left|x\right|}{r}}\right),x\in B^r,\hfill \\ \psi \left(x\right)=0,x\in \partial \Omega ,\hfill \\ \psi \left(x\right)>0,x\in \Omega \setminus \overline{B^r}\hfill \end{array}\right.$$

(13)

and

$$\begin{array}{c}{\overline{\omega }}_0\subset \subset \omega ,|\nabla \psi \left(x\right)|\ge \delta ,x\in \overline{\Omega }\setminus {\omega }_0.\end{array}$$

(14)

We introduce $\theta \left(t\right)$ as

$$\begin{array}{c}\theta \left(t\right)={\left({\displaystyle \frac{1}{t(T-t)}}\right)}^3,\end{array}$$

(15)

which is different from $\theta \left(t\right)$ in [10,11,12,13,32] but is similar to $\theta \left(t\right)$ in [14,15].

Let $\psi \left(x\right)$ be a smooth function satisfying (13) and (14). $s>0$ and $\lambda >0$ are two large parameters. Let

$$\begin{array}{c}\varphi \left(x\right)={\mathrm{e}}^{\lambda \psi \left(x\right)},\Psi =\underset{x\in \Omega }{sup}\psi \left(x\right),\end{array}$$

(16)

$$\begin{array}{c}\sigma (x,t)=s\theta \left(t\right)\left({\mathrm{e}}^{2\lambda \Psi }-{\displaystyle \frac{1}{2}}{\left|x\right|}^2-{\mathrm{e}}^{\lambda \psi \left(x\right)}\right).\end{array}$$

(17)

Denote

$$\begin{array}{c}\mathsf{\Theta }\triangleq \Omega \setminus \left(\overline{B^r}\cup {\overline{\omega }}_0\right),\tilde{\mathsf{\Theta }}\triangleq \Omega \setminus \overline{B^r},B_T^r\triangleq B^r\times (0,T),\\ {\mathsf{\Theta }}_T\triangleq \mathsf{\Theta }\times (0,T),{\tilde{\mathsf{\Theta }}}_T\triangleq \tilde{\mathsf{\Theta }}\times (0,T),{\omega }_{0,T}\triangleq {\omega }_0\times (0,T).\end{array}$$

Now, we are in a position to state the Carleman estimate.

Theorem 3.

Let $p\left(x\right)\in \mathcal{B}$, r satisfy (12), and $\mu \le p_1{\mu }^{*}$. Then, there exists a positive constant ${\lambda }_0={\lambda }_0(\Omega ,\omega ,n,T,p_1,p_2,p_3,r,{\lambda }_{*}),$ such that, for all $\lambda \ge {\lambda }_0$, there exist constants $s_0=s_0\left(\lambda \right)>0$ and $C=C(\Omega ,\omega ,n,T,p_1,p_2,p_3,r,s_0,{\lambda }_0)>0$ such that for all $s\ge s_0\left(\lambda \right)$, the solution of (8) satisfies

$$\begin{array}{cc}\hfill & s{\lambda }^2{\int }_{{\tilde{\mathsf{\Theta }}}_T}\theta \varphi {\mathrm{e}}^{-2\sigma }{|\nabla w|}^2\mathrm{d}x\mathrm{d}t+{\displaystyle \frac{s}{{\lambda }^2}}{\int }_{{\Omega }_T}\theta {\mathrm{e}}^{-2\sigma }{\left|x\right||\nabla w|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +s{\int }_{{\Omega }_T}\theta {\mathrm{e}}^{-2\sigma }{\displaystyle \frac{{\left|w\right|}^2}{\left|x\right|}}\mathrm{d}x\mathrm{d}t+s^3{\int }_{{\Omega }_T}{\theta }^3{\mathrm{e}}^{-2\sigma }{\left|x\right|}^2{\left|w\right|}^2\mathrm{d}x\mathrm{d}t+s^3{\lambda }^4{\int }_{{\tilde{\mathsf{\Theta }}}_T}{\theta }^3{\varphi }^3{\mathrm{e}}^{-2\sigma }{\left|w\right|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\displaystyle \frac{1}{s}}{\mathrm{e}}^{-4\lambda \Psi }{\int }_{B_T^r}{\displaystyle \frac{1}{\theta }}{\mathrm{e}}^{-2\sigma }|{\partial }_t{w|}^2\mathrm{d}x\mathrm{d}t+{\displaystyle \frac{1}{s}}{\mathrm{e}}^{-4\lambda \Psi }{\int }_{{\tilde{\mathsf{\Theta }}}_T}{\displaystyle \frac{1}{\theta \varphi }}{\mathrm{e}}^{-2\sigma }{\left|{\partial }_{t}w\right|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{s}^3{\lambda }^4{\mathrm{e}}^{2\lambda \Psi }{\int }_{{\omega }_T}{\theta }^3{\varphi }^3{\mathrm{e}}^{-2\sigma }{\left|w\right|}^2\mathrm{d}x\mathrm{d}t+C{\int }_{{\Omega }_T}{\mathrm{e}}^{-2\sigma }{\left|g\right|}^2\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

(18)

We present the proof in Appendix A.

Remark 2.

In fact, using the same idea, we can also obtain a similar estimate (18) for a general elliptic operator instead of the operator in (8).

Corollary 2.

Assume that ${\left\{a_{ij}\left(x\right)\right\}}_{1\le i,j\le n}$ satisfies (7). Assume $\mu \le a_1{\mu }^{*}$. Let r be small enough. Then, there exists a positive constant ${\lambda }_0={\lambda }_0(\Omega ,\omega ,n,T,a_1,a_2,a_3,r,{\lambda }_{*}),$ such that, for all $\lambda \ge {\lambda }_0$, there exist constants $s_0=s_0\left(\lambda \right)>0$ and $C=C(\Omega ,\omega ,n,T,a_1,a_2,a_3,r,s_0,{\lambda }_0)>0$, such that for all $s\ge s_0\left(\lambda \right)$, the solution of (8) with a more general case ${\left\{a_{ij}\left(x\right)\right\}}_{1\le i,j\le n}$ satisfies

$$\begin{array}{cc}\hfill & s{\lambda }^2{\int }_{{\tilde{\mathsf{\Theta }}}_T}\theta \varphi {\mathrm{e}}^{-2\sigma }{|\nabla w|}^2\mathrm{d}x\mathrm{d}t+{\displaystyle \frac{s}{{\lambda }^2}}{\int }_{{\Omega }_T}\theta {\mathrm{e}}^{-2\sigma }{\left|x\right||\nabla w|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +s{\int }_{{\Omega }_T}\theta {\mathrm{e}}^{-2\sigma }{\displaystyle \frac{{\left|w\right|}^2}{\left|x\right|}}\mathrm{d}x\mathrm{d}t+s^3{\int }_{{\Omega }_T}{\theta }^3{\mathrm{e}}^{-2\sigma }{\left|x\right|}^2{\left|w\right|}^2\mathrm{d}x\mathrm{d}t+s^3{\lambda }^4{\int }_{{\tilde{\mathsf{\Theta }}}_T}{\theta }^3{\varphi }^3{\mathrm{e}}^{-2\sigma }{\left|w\right|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\displaystyle \frac{1}{s}}{\mathrm{e}}^{-4\lambda \Psi }{\int }_{B_T^r}{\displaystyle \frac{1}{\theta }}{\mathrm{e}}^{-2\sigma }|{\partial }_t{w|}^2\mathrm{d}x\mathrm{d}t+{\displaystyle \frac{1}{s}}{\mathrm{e}}^{-4\lambda \Psi }{\int }_{{\tilde{\mathsf{\Theta }}}_T}{\displaystyle \frac{1}{\theta \varphi }}{\mathrm{e}}^{-2\sigma }{\left|{\partial }_{t}w\right|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{s}^3{\lambda }^4{\mathrm{e}}^{2\lambda \Psi }{\int }_{{\omega }_T}{\theta }^3{\varphi }^3{\mathrm{e}}^{-2\sigma }{\left|w\right|}^2\mathrm{d}x\mathrm{d}t+C{\int }_{{\Omega }_T}{\mathrm{e}}^{-2\sigma }{\left|g\right|}^2\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

Remark 3.

Using (18), we can improve the null-controllability result in [11] to the case of $\mu \le p_1{\mu }^{*}$ for problem (8).

4. Proof of Theorem 1

Since $u_0\left(x\right)\in L^2(\Omega )$ and $R\in H^1(0,T;L^{\infty }(\Omega ))$, the solution of (1) satisfies sufficient regularity properties (see Theorem 2 (iii)) to proceed to the following computations:

Let $y={\partial }_{t}u.$ By (1), y satisfies

$$\left\{\begin{array}{c}{\partial }_{t}y-\mathrm{div}(p\nabla y)-{\displaystyle \frac{\mu }{{\left|x\right|}^2}}y=f\left(x\right){\partial }_{t}R,(x,t)\in {\Omega }_T,\hfill \\ y(x,t)=0,(x,t)\in {\Sigma }_T.\hfill \end{array}\right.$$

(19)

To apply Theorem 3 on ${\Omega }_{t_0,T}$ instead of ${\Omega }_T$, we rewrite the definition of $\theta$ as the following formula:

$$\begin{array}{c}\hfill \theta \left(t\right)={\left({\displaystyle \frac{1}{(t-t_0)(T-t)}}\right)}^3.\end{array}$$

(20)

The other weight functions are defined as in (13), (16) and (17). Denote

$$\begin{array}{c}{\Omega }_{t_0,T}\triangleq \Omega \times (t_0,T),B_{t_0,T}^r\triangleq B^r\times (t_0,T),{\tilde{\mathsf{\Theta }}}_{t_0,T}\triangleq \tilde{\mathsf{\Theta }}\times (t_0,T),{\omega }_{t_0,T}\triangleq \omega \times (t_0,T).\end{array}$$

Fixing $\lambda \ge {\lambda }_0$ and applying the Carleman estimate (18) with $w=y$ and $g=f\left(x\right){\partial }_{t}R$ on ${\Omega }_{t_0,T},$ there exist $s_0>0$ and $C=C(\Omega ,\omega ,t_0,T,n,p_1,p_2,p_3)>0$ such that, for all $s\ge s_0,$ the following inequality holds:

$$\begin{array}{cc}\hfill & s{\int }_{{\Omega }_{t_0,T}}\theta {\mathrm{e}}^{-2\sigma }{\displaystyle \frac{{\left|y\right|}^2}{\left|x\right|}}\mathrm{d}x\mathrm{d}t+s^3{\int }_{{\Omega }_{t_0,T}}{\theta }^3{\mathrm{e}}^{-2\sigma }{\left|x\right|}^2{\left|y\right|}^2\mathrm{d}x\mathrm{d}t+s^3{\int }_{{\tilde{\mathsf{\Theta }}}_{t_0,T}}{\theta }^3{\varphi }^3{\mathrm{e}}^{-2\sigma }{\left|y\right|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\displaystyle \frac{1}{s}}{\int }_{B_{t_0,T}^r}{\displaystyle \frac{1}{\theta }}{\mathrm{e}}^{-2\sigma }|{\partial }_t{y|}^2\mathrm{d}x\mathrm{d}t+{\displaystyle \frac{1}{s}}{\int }_{{\tilde{\mathsf{\Theta }}}_{t_0,T}}{\displaystyle \frac{1}{\theta \varphi }}{\mathrm{e}}^{-2\sigma }{\left|{\partial }_{t}y\right|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill \le & C{s}^3{\int }_{{\omega }_{t_0,T}}{\theta }^3{\varphi }^3{\mathrm{e}}^{-2\sigma }{\left|y\right|}^2\mathrm{d}x\mathrm{d}t+C{\int }_{{\Omega }_{t_0,T}}{\mathrm{e}}^{-2\sigma }{\left|f\left(x\right)\right|}^2{\left|{\partial }_{t}R\right|}^2\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

(21)

By (5), we substitute $y={\partial }_{t}u$ to (21) to obtain

$$\begin{array}{cc}\hfill & s{\int }_{{\Omega }_{t_0,T}}\theta {\mathrm{e}}^{-2\sigma }{\displaystyle \frac{|{\partial }_t{u|}^2}{\left|x\right|}}\mathrm{d}x\mathrm{d}t+s^3{\int }_{{\Omega }_{t_0,T}}{\theta }^3{\mathrm{e}}^{-2\sigma }{\left|x\right|}^2|{\partial }_t{u|}^2\mathrm{d}x\mathrm{d}t+s^3{\int }_{{\tilde{\mathsf{\Theta }}}_{t_0,T}}{\theta }^3{\varphi }^3{\mathrm{e}}^{-2\sigma }{\left|{\partial }_{t}u\right|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\displaystyle \frac{1}{s}}{\int }_{B_{t_0,T}^r}{\displaystyle \frac{1}{\theta }}{\mathrm{e}}^{-2\sigma }|{\partial }_t^2{u|}^2\mathrm{d}x\mathrm{d}t+{\displaystyle \frac{1}{s}}{\int }_{{\tilde{\mathsf{\Theta }}}_{t_0,T}}{\displaystyle \frac{1}{\theta \varphi }}{\mathrm{e}}^{-2\sigma }{\left|{\partial }_t^{2}u\right|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{s}^3{\int }_{{\omega }_{t_0,T}}{\theta }^3{\varphi }^3{\mathrm{e}}^{-2\sigma }|{\partial }_t{u|}^2\mathrm{d}x\mathrm{d}t+C{\int }_{{\Omega }_{t_0,T}}{\mathrm{e}}^{-2\sigma }{\left|f\left(x\right)\right|}^2\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

(22)

On the other hand, since $R(x,T_0)\ge r_0>0$ on $\overline{\Omega }$ and

$$f\left(x\right)R(x,T_0)={\partial }_{t}u(x,T_0)-\mathrm{div}(p\nabla u(x,T_0))-{\displaystyle \frac{\mu }{{\left|x\right|}^2}}u(x,T_0),$$

we have

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\mathrm{e}}^{-2\sigma (x,T_0)}{\left|f\left(x\right)\right|}^2\mathrm{d}x\le C{\int }_{\Omega }{\mathrm{e}}^{-2\sigma (x,T_0)}{\left|f\left(x\right)R(x,T_0)\right|}^2\mathrm{d}x\hfill \\ \hfill & \le C{\int }_{\Omega }{\mathrm{e}}^{-2\sigma (x,T_0)}{\left|{\partial }_{t}u(x,T_0)\right|}^2\mathrm{d}x+C{\int }_{\Omega }{\mathrm{e}}^{-2\sigma (x,T_0)}{\left|\mathrm{div}\left(p\nabla u(x,T_0)\right)+{\displaystyle \frac{\mu }{{\left|x\right|}^2}}u(x,T_0)\right|}^2\mathrm{d}x.\hfill \end{array}$$

(23)

Using $\underset{t\to t_0^{+}}{lim}\sigma =+\infty ,x\in \Omega ,$ we have

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\mathrm{e}}^{-2\sigma (x,T_0)}{\left|{\partial }_{t}u(x,T_0)\right|}^2\mathrm{d}x={\int }_{t_0}^{T_0}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\int }_{\Omega }{\mathrm{e}}^{-2\sigma (x,t)}{\left|{\partial }_{t}u(x,t)\right|}^2\mathrm{d}x\right)\mathrm{d}t\hfill \\ \hfill & \le C{\int }_{{\Omega }_{t_0,T}}{\mathrm{e}}^{-2\sigma }\left(\left|{\partial }_{t}u\right|\left|{\partial }_t^{2}u\right|+s\left|{\theta }^{\prime }\right|{\left|{\partial }_{t}u\right|}^2\right)\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

(24)

Using Young’s inequality and $\left|x\right|\le r<1$ in $B^r$ and (22), we have

$$\begin{array}{cc}\hfill & 2{\int }_{{\Omega }_{t_0,T}}{\mathrm{e}}^{-2\sigma }\left|{\partial }_{t}u\right|\left|{\partial }_t^{2}u\right|\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le {\int }_{B_{t_0,T}^r}{\mathrm{e}}^{-2\sigma }\left(s\theta {\left|{\partial }_{t}u\right|}^2+{\displaystyle \frac{1}{s\theta }}{\left|{\partial }_t^{2}u\right|}^2\right)\mathrm{d}x\mathrm{d}t+{\int }_{{\tilde{\mathsf{\Theta }}}_{t_0,T}}{\mathrm{e}}^{-2\sigma }\left(s\theta \varphi |{\partial }_t{u|}^2+{\displaystyle \frac{1}{s\theta \varphi }}{\left|{\partial }_t^{2}u\right|}^2\right)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le Cs{\int }_{{\Omega }_{t_0,T}}\theta {\mathrm{e}}^{-2\sigma }{\displaystyle \frac{|{\partial }_t{u|}^2}{\left|x\right|}}\mathrm{d}x\mathrm{d}t+{\displaystyle \frac{1}{s}}{\int }_{B_{t_0,T}^r}{\displaystyle \frac{1}{\theta }}{\mathrm{e}}^{-2\sigma }{\left|{\partial }_t^{2}u\right|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +s^3{\int }_{{\tilde{\mathsf{\Theta }}}_{t_0,T}}{\theta }^3{\varphi }^3{\mathrm{e}}^{-2\sigma }|{\partial }_t{u|}^2\mathrm{d}x\mathrm{d}t+{\displaystyle \frac{1}{s}}{\int }_{{\tilde{\mathsf{\Theta }}}_{t_0,T}}{\displaystyle \frac{1}{\theta \varphi }}{\mathrm{e}}^{-2\sigma }{\left|{\partial }_t^{2}u\right|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{s}^3{\int }_{{\omega }_{t_0,T}}{\theta }^3{\varphi }^3{\mathrm{e}}^{-2\sigma }|{\partial }_t{u|}^2\mathrm{d}x\mathrm{d}t+C{\int }_{{\Omega }_{t_0,T}}{\mathrm{e}}^{-2\sigma }{\left|f\left(x\right)\right|}^2\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

(25)

and

$$\begin{array}{cc}\hfill & {\int }_{{\Omega }_{t_0,T}}s|{\theta }^{\prime }|{\mathrm{e}}^{-2\sigma }{\left|{\partial }_{t}u\right|}^2\mathrm{d}x\mathrm{d}t\le C{\int }_{{\Omega }_{t_0,T}}s{\left|\theta \right|}^{1+{\displaystyle \frac{1}{3}}}{\mathrm{e}}^{-2\sigma }{\left|{\partial }_{t}u\right|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & =C{\int }_{{\Omega }_{t_0,T}}s{\mathrm{e}}^{-2\sigma }\left({\left|\theta \right|}^{{\displaystyle \frac{2}{3}}}{\left|x\right|}^{{\displaystyle \frac{2}{3}}}{\left|{\partial }_{t}u\right|}^{{\displaystyle \frac{2}{3}}}\right)\left({\left|\theta \right|}^{{\displaystyle \frac{2}{3}}}{\left|x\right|}^{-{\displaystyle \frac{2}{3}}}{\left|{\partial }_{t}u\right|}^{{\displaystyle \frac{4}{3}}}\right)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{\int }_{{\Omega }_{t_0,T}}s{\mathrm{e}}^{-2\sigma }\left\{{\left({\left|\theta \right|}^{{\displaystyle \frac{2}{3}}}{\left|x\right|}^{{\displaystyle \frac{2}{3}}}{\left|{\partial }_{t}u\right|}^{{\displaystyle \frac{2}{3}}}\right)}^3+{\left({\left|\theta \right|}^{{\displaystyle \frac{2}{3}}}{\left|x\right|}^{-{\displaystyle \frac{2}{3}}}{\left|{\partial }_{t}u\right|}^{{\displaystyle \frac{4}{3}}}\right)}^{{\displaystyle \frac{3}{2}}}\right\}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & =C{\int }_{{\Omega }_{t_0,T}}s{\mathrm{e}}^{-2\sigma }\left\{{\theta }^2{\left|x\right|}^2{\left|{\partial }_{t}u\right|}^2+\theta {\displaystyle \frac{{\left|{\partial }_{t}u\right|}^2}{\left|x\right|}}\right\}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{s}^3{\int }_{{\Omega }_{t_0,T}}{\theta }^3{\mathrm{e}}^{-2\sigma }{\left|x\right|}^2{\left|{\partial }_{t}u\right|}^2\mathrm{d}x\mathrm{d}t+Cs{\int }_{{\Omega }_{t_0,T}}\theta {\mathrm{e}}^{-2\sigma }{\displaystyle \frac{|{\partial }_t{u|}^2}{\left|x\right|}}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & \le C{s}^3{\int }_{{\omega }_{t_0,T}}{\theta }^3{\varphi }^3{\mathrm{e}}^{-2\sigma }|{\partial }_t{u|}^2\mathrm{d}x\mathrm{d}t+C{\int }_{{\Omega }_{t_0,T}}{\mathrm{e}}^{-2\sigma }{\left|f\left(x\right)\right|}^2\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

(26)

Using (23)–(26), we obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\mathrm{e}}^{-2\sigma (x,T_0)}{\left|f\left(x\right)\right|}^2\mathrm{d}x& \le C{s}^3{\int }_{{\omega }_{t_0,T}}{\theta }^3{\varphi }^3{\mathrm{e}}^{-2\sigma }|{\partial }_t{u|}^2\mathrm{d}x\mathrm{d}t+C{\int }_{{\Omega }_{t_0,T}}{\mathrm{e}}^{-2\sigma }{\left|f\left(x\right)\right|}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +C{\int }_{\Omega }{\mathrm{e}}^{-2\sigma (x,T_0)}{\left|\mathrm{div}\left(p\nabla u(x,T_0)\right)+{\displaystyle \frac{\mu }{{\left|x\right|}^2}}u(x,T_0)\right|}^2\mathrm{d}x.\hfill \end{array}$$

Here, we introduce a lemma obtained from [10].

Lemma 2.

There exists some $C>0$ such that, for all $s>0,$

$$\begin{array}{c}\hfill {\int }_{{\Omega }_{t_0,T}}{\left|f\left(x\right)\right|}^2{\mathrm{e}}^{-2\sigma (x,t)}\mathrm{d}x\mathrm{d}t\le {\displaystyle \frac{C}{\sqrt{s}}}{\int }_{\Omega }{\left|f\left(x\right)\right|}^2{\mathrm{e}}^{-2\sigma (x,T_0)}\mathrm{d}x.\end{array}$$

(27)

For the proof of Lemma 2, we refer to section VI.2 in [10].

Combining Lemma 2 and the fact that ${\theta }^3{\varphi }^3{\mathrm{e}}^{-2\sigma }$ is bounded on ${\omega }_{t_0,T},$ the following equality holds for some $s\ge s_0$, which is fixed and sufficiently large:

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\mathrm{e}}^{-2\sigma (x,T_0)}{\left|f\left(x\right)\right|}^2\mathrm{d}x& \le C{\int }_{{\omega }_{t_0,T}}{\left|{\partial }_{t}u\right|}^2\mathrm{d}x\mathrm{d}t+C{\int }_{\Omega }{\left|\mathrm{div}\left(p\nabla u(x,T_0)\right)+{\displaystyle \frac{\mu }{{\left|x\right|}^2}}u(x,T_0)\right|}^2\mathrm{d}x\hfill \end{array}$$

We complete the proof of Theorem 1.

5. Future Works

In this article, we consider the inverse source problem for the parabolic equation with a single singularity. It would also be interesting to study the case of multipolar inverse-squared singular potentials. This situation has been considered in [13] for the case of $p\left(x\right)=1$. However, there are no results for the case of variable coefficients in the principal part.

Moreover, in this article, we consider the case when the singularity is in the interior of the domain. However, there are also many applications in combustion theory and quantum mechanics when the singularity is placed on a point of the boundary. In fact, there have not been any inverse problem results in this case.

Finally, we are aiming to solve the inverse problem for the hyperbolic equation with singularities. According to the Bukhgeim–Klibanov method, we need to derive Carleman estimates for a hyperbolic operator with singular potentials. This would be difficult because normal weight functions are hard to compensate the singularity of the potential term. This will be the object of a forthcoming work.