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Abstract and Applied Analysis
Volume 2018, Article ID 2067304, 16 pages
https://doi.org/10.1155/2018/2067304
Research Article

An Inverse Source Problem for Singular Parabolic Equations with Interior Degeneracy

1Faculté des Sciences et Techniques, Université Hassan 1er, Laboratoire MISI, BP 577, Settat 26000, Morocco
2Faculté Polydisciplinaire de Ouarzazate, Université Ibn Zohr, BP 638, Ouarzazate 45000, Morocco

Correspondence should be addressed to Jawad Salhi; moc.liamg@ihlas.js

Received 23 August 2018; Accepted 28 November 2018; Published 9 December 2018

Academic Editor: Changbum Chun

Copyright © 2018 Khalid Atifi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.

1. Introduction

Inverse problems appear in a wide range of scientific applications, such as geophysics, biological and medical imaging, material and structure characterization, electrical, mechanical and civil engineering, and finances. The resolution of inverse problems consists of estimating the parameters of the observed system or structure from available data of solutions. The unknown quantities are diverse, according to the inverse problems and phenomena studied, but typical unknowns are spatially varying coefficients and source terms.

In the present paper, we study the inverse problem of determining the source term in a degenerate heat equation perturbed by a singular potential from the theoretical analysis and numerical computation angles. More precisely, we consider the following problem:where , fixed, and . Moreover, we assume that the constant satisfies suitable assumptions described below and the functions and degenerate at the same interior point of the spatial domain (for the precise assumptions we refer to Section 2). Let us recall that, in inverse source problems, the source term has to satisfy some condition; otherwise uniqueness may be false; see [1]. Let be given and for given, let . In [2, 3], the authors make the assumption that source terms satisfy the conditionTherefore they define the set of admissible source terms as

The Carleman estimate is a class of weighted energy estimates with a large parameter for a solution to a PDE and it is one of the major tools used in the study of unique continuation, observability, and controllability problems for various kinds of PDEs. The idea of using global Carleman estimates to solve inverse problems and prove Lipschitz stability results was first introduced by Puel and Yamamoto [4] in in the context of the wave equation, using a modification of the idea of [5]. Later on, it also has been applied to the standard heat equation by Imanuvilov and Yamamoto [2] in . Their method is based on the use of global Carleman estimates for parabolic problems that were developed by Fursikov and Imanuvilov [6] and used to solve null controllability issues. The novelty of their work is that they not only solve the uniqueness question but they also provide unconditional Lipschitz stability result concerning the reconstruction of the source.

In the last recent years an increasing interest has been devoted to (1) in the case when and the degeneracy can occur at the boundary or in the interior of the space domain. For example, we recall the works [79], where the authors obtain results concerning Carleman estimates and null controllability.

These results are complemented in [3, 1012], where the authors obtain results concerning inverse problems for purely degenerate (i.e., ) parabolic equations and parabolic coupled systems, addressing, in particular, issues such as uniqueness and stability. If , the first results in this direction are obtained in [1] for the nondegenerate heat operator (i.e., ) with a singular potential. But, the study of numerical reconstruction questions are rarely taken into account; see [13, 14].

Furthermore, in both theoretical and numerical aspects, very few results are known regarding the identification of coefficients in degenerate/singular parabolic equations, even though this class of operators occurs in interesting theoretical and applied problems. As far as we know, [15] is the unique published work on this subject; it concerns the reconstruction of the initial heat distribution in a degenerate/singular parabolic equation with degeneracy and singularity at the boundary of the domain.

From the mathematical point of view and in connection with the work of Fragnelli and Mugnai (see [16]), we focus on identifying, on the basis of some observations, the source term, in a parabolic equation presenting both a degenerate diffusion coefficient and a singular potential with degeneracy and singularity inside the spatial domain.

In particular, our results complement the ones of [1, 3] in the purely degenerate case and in the purely singular one, respectively. More precisely, we will follow the approach introduced in [2] for the treatment of uniformly parabolic problems which is based on the use of global Carleman estimates. For this purpose, we use and extend some recent Carleman estimates for degenerate/singular equations obtained by Fragnelli and Mugnai [16]. As a consequence, we prove a stability estimate of Lipschitz type in determining the term source using the following observations data: where the subregion of measurements is a nonempty subinterval of that is assumed to satisfy the following.

Hypothesis 1. The set of observation is such that where , are intervals with , , and .

For fixed , the main result of this paper can be stated as follows.

Theorem 2. Let and suppose that Hypotheses 1 and 14 are satisfied. Then, there exists such that, for all and ,

Remark 3. (i)It is worth noting that the result announced in Theorem 2 is still valid also in the case in which the observation set is an interval containing the degeneracy point. Indeed, if one can always find two subintervals , such that .(ii)If we restrict ourselves to the particular case for some given function , positive at some time and is the unknown function that we want to recover, then uniqueness result can be shown as an immediate consequence of the Lipschitz stability result; see [17, Theorem 2.11].

In fact, we will not only investigate the theoretical aspect of the inverse source problem due to our interest in mathematics, but also consider the numerical reconstruction of the source term . To this end, we adopt the classical Tikhonov regularization to reformulate the inverse problem into a related optimization problem, for which we develop an iterative thresholding algorithm by using the corresponding adjoint system. In particular, we will focus on the determination of the unknown source term from the measured data at the final time. The resolution of this problem is standard and it is based on the gradient of the cost functional. More precisely, the most important issue in numerical solutions of inverse problems is the Lipschitz continuity of the Fréchet gradient. Indeed, in order to construct an effective minimization algorithm for an inverse problem, one needs to analyze the gradient of the considered cost functional. There is a vast literature on inverse problems for linear parabolic equations with final overdetermination. For example, we mention the pioneering work [18]. Compared to a standard parabolic equation, the main challenge here is the nonstandard degeneracy of the diffusion coefficient as well as the singularity of the potential of the partial differential equation (1).

The rest of this article is organized as follows. In Section 2, we recall the well-posedness of the problem (1). Then Section 3 is devoted to the proof of the main stability result of Lipschitz type. In Section 4, we reformulate our inverse source problem as a minimization problem with the Tikhonov regularization and provide a monotone iteration scheme based on a gradient method.

Throughout the paper, denotes a generic positive constant, which may vary from line to line.

2. Well-Posedness

The ways in which and degenerate at can be quite different, and for this reason, following [16], to establish our results, we give the following definitions and assumptions.

Hypothesis 4 (double weakly degenerate case (WWD)). There exists such that , in , and there exists such that and a.e. in

Hypothesis 5 (weakly strongly degenerate case (WSD)). There exists such that , in , , such that and a.e. in

Hypothesis 6 (strongly weakly degenerate case (SWD)). There exists such that , in , , , , such that and a.e. in .

Hypothesis 7 (double strongly degenerate case (SSD)). There exists such that , in , , there exists such that and a.e. in

For the well-posedness of the problem (1), as in [16], we consider different classes of weighted Hilbert spaces, which are suitable to study the four different situations given above, namely, the (WWD), (WSD), (SWD), and (SSD) cases. Thus, we consider the Hilbert spaces and endowed with the inner products and respectively.

In order to deal with the singularity of we need the following inequality proved in [16, Proposition 2.14].

Lemma 8. If one among Hypotheses 46 holds with , then there exists a constant such that for all we have

In order to study well-posedness of problem (1) and in view of Lemma 8, we consider the space where the Hardy-Poincaré inequality (11) holds.

We underline that, from Lemma 8, the standard norm is equivalent to

From now on, we make the following assumptions on , , and .

Hypothesis 9. One among the Hypotheses 4, 5, or 6 holds true with and we assume that Hypotheses 4, 5, 6, or 7 hold with .

Using Lemma 8, the next inequality is proved in [16, Proposition 2.18], which is crucial not only to obtain the well-posedness of problem (1), but also to prove that the inverse problem posed as weak solution minimization problem has a solution.

Proposition 10. Assume Hypothesis 9. Then there exists a positive constant such that, for all , there holds

Now, let us go back to problem (1), recalling the following definition.

Definition 11. Let and . A function is said to be a (weak) solution of (1) if and satisfies the following differential equation: for all .

Finally, we introduce the Hilbert space where with domain

Remark 12. Observe that if , then , so that and inequality (11) holds.

Hence, the next result holds thanks to the theory of semigroups.

Proposition 13. The following assertions hold.
(i) The operator is the infinitesimal generator of a strongly continuous semigroup of contractions on . Moreover, the semigroup is analytic.
(ii) For all and , problem (1) admits a unique strict solution belonging to the class Moreover, if , then for all there holds (iii) For all and for all , problem (1) has a unique weak solution such that for all there holds Moreover, if and , we have

Proof. The proof of statement can be found in [16], whereas statements and are a consequence of and [19, Proposition 3.3 and Proposition 3.8].

3. Lipschitz Stability Result

In this section, we aim at obtaining a Lipschitz stability result on determining the source term in problem (1) in the spirit of the result by Imanuvilov and Yamamoto [2]. The key ingredient to obtain such a result is Carleman estimates. Here we use specific Carleman estimates for degenerate/singular parabolic equations (inspired by [16]). Thus, we first recall this fundamental tool in the following section before proving Theorem 2 in Section 3.2.

3.1. Carleman Estimate

The aim of this subsection is to prove a Carleman type inequality for solutions of problem (1). First of all, let us make precise the assumptions under which we consider problem (1).

Hypothesis 14. Hypothesis 9 holds. Moreover, if , then there exists that the following condition is satisfied.
The function is nonincreasing on the left of and nondecreasing on the right of .
Moreover, if , we require that

As usual, the derivation of global Carleman estimates relies on the introduction of some suitable weight function of the form wherewith , , and where is defined by the following way:Observe that , and clearly Eventually, we define as in [3] the second time weight function:Let us now turn to the following linear initial-boundary value problem:where . In the following, we denote

Now we are ready to state global Carleman estimates with boundary observation for system (31).

Theorem 15. Assume Hypothesis 14. Then, there exist two positive constants and such that the solution of (31) in satisfies, for all ,

Some part of estimate (33) is already proved in [16] and, even if we refer to [20] a few times, our proof is quite self-contained. In [16], the authors prove a Carleman inequality that estimates the integrals of and (that were sufficient for control purposes). For inverse problems, these estimates are not sufficient and one also needs some additional estimate of with a special weight and some estimate of the derivative term that we added here in the statement of Theorem 15. The proof is based on the methods developed in [3].

Proof. The proof of Theorem 15 relies on the change of variables with . Then, from (31), we obtain Moreover, . This property allows us to apply the Carleman estimates established in [16, Lemma 3.8] to with in place of The operators and are not exactly the ones of [16]. However, one can prove that the Carleman estimates do not change. Using the previous estimate, we aim at proving estimate (33) that concerns the variable .
Step 1. Estimate of and . Replacing by , we immediately get from (35)Moreover, . Therefore,In conclusion, thanks to (35)-(37), we getStep 2. Estimate . Observe that, since and , one hasTo estimate the integral on the right-hand side of (39), we follow the technique of [20, Lemma 4.1]. Using the Young inequality, we find Now, if , we consider the function . Obviously, there exists such that the function is nonincreasing on the left of and nondecreasing on the right of . Then, we can apply the Hardy-Poincaré inequality given in [20, Proposition 1.1], obtaining where is the Hardy-Poincaré constant and
In the previous inequality, we have used the property that the map is nonincreasing on the left of and nondecreasing on the right of for all ; see [20, Lemma 2.1]. If , we can consider the function . Then we have where Moreover, using the condition given in Hypothesis 14, one has that the function , with , is nonincreasing on the left of and nondecreasing on the right of . The Hardy-Poincaré inequality given in [20, Proposition 1.1] implies Thus, in every case, for large enough there holdsHence, Finally, coming back to , we getStep 3. Estimate of . First of all, coming back to the definition of , we have Note that is bounded on and (see [20, Lemma 2.1]). Therefore, there exists such thatWe then estimate thanks to Hardy inequality, as we have done in the previous step in (45). In this way, we find Finally, using (49) and (35), one hasNow, in order to obtain the estimate of we have to use the estimate of (47). From the definition of , we have . Hence The second term in the above right-hand side is estimated as follows: Hence using (47) and (51) we conclude thatConclusion. We immediately deduce the expected Carleman estimate (33) from (38), (47), and (54).

3.2. Proof of Lipschitz Stability

The object of this subsection is to prove Theorem 2 which recovers a source term from the measured final data and the partial knowledge of over the subdomain . In proving these kinds of stability estimates, the global Carleman estimate obtained in Theorem 15 will play a crucial part along with certain energy estimates.

In order to obtain our main result, we need to define the following weights function associated with nondegenerate Carleman estimates in a general interval which are suited to our purpose. For ,where and a function such that in , and in , is an open subset of .

Now, choose the constant in (27) so thatThus, one can show that weight functions satisfy the following properties which are needed in the sequel.

Lemma 16. For , we have

Proof. First, let us set .
(i). For all , we have Thus, to show that , it suffices to have This means and the conclusion follows immediately.
(ii). In a similar manner, using (56), one has and the thesis follows.

Proof of Theorem 2. Let , , and . Observe that, according to Proposition 13, the solution of the problem (1) belongs to It follows that the solution of (1) satisfies sufficient regularity properties to proceed to the following computations. Let where satisfies (1). Then and satisfiesLet us recall that our goal is to provide an estimate of . For this purpose, we divide the proof into four steps and we recall that .
Step 1 (Carleman estimates with locally distributed observation). Such estimates are obtained by studying some auxiliary problems, introduced by means of a suitable cut-off argument. First, by the assumption on the observation set, we can fix two subintervals , and four points , , with . Then, fix and consider a smooth function such that Let us now introduce where is the solution of (61). Hence, fixed ; is a solution of the nondegenerate nonsingular parabolic equation with . Observe that, by the assumption on and the fact that is supported in , we have .
Thus, we can apply the analogue of [6, Lemma 1.2] for in , obtaining that there exist two positive constants and ( sufficiently large), such that satisfies, for all , Using again the fact that is supported in and the boundedness of (far away from in the weakly degenerate case and since in the strongly degenerate case), we have, by the Caccioppoli inequality for the nondegenerate case, Hence, we can choose so large that for all and for a positive constant , the following estimate holds:Furthermore, using the fact that , by (66) one hasThe estimates (66)-(67) lead toNow, since , by Lemma 16 we can prove that there exists a positive constant such that for every Then, using (69), (68) becomes Hence, using the definition of , it resultsfor a positive constant and for large enough.
To complete the proof, it is sufficient to prove a similar inequality on the interval . To this aim, we follow a reflection procedure. Consider the function Therefore, solves the problembeing Now, consider a smooth function such that