An Inverse Source Problem for Singular Parabolic Equations with Interior Degeneracy

and Applied Analysis 3 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, C denotes a generic positive constant, which may vary from line to line. 2. Well-Posedness The ways in which a and b degenerate at x0 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 existsx0 ∈ (0, 1) such thata(x0) = b(x0) = 0, a, b > 0 in [0, 1]\{x0}, a, b ∈ C1([0, 1]\{x0}) and there existsK, L ∈ (0, 1) such that (x − x0)a󸀠 ≤ Ka and (x − x0)b󸀠 ≤ Lb a.e. in [0, 1]. Hypothesis 5 (weakly strongly degenerate case (WSD)). There exists x0 ∈ (0, 1) such that a(x0) = b(x0) = 0, a, b > 0 in [0, 1] \ {x0}, a ∈ C1([0, 1] \ {x0}), b ∈ C1([0, 1] \ {x0}) ∩ W1,∞(0, 1), ∃K ∈ (0, 1), L ∈ [1, 2) such that (x − x0)a󸀠 ≤ Ka and (x − x0)b󸀠 ≤ Lb a.e. in [0, 1]. Hypothesis 6 (strongly weakly degenerate case (SWD)). There exists x0 ∈ (0, 1) such that a(x0) = b(x0) = 0, a, b > 0 in [0, 1] \ {x0}, a ∈ C1([0, 1] \ {x0}) ∩ W1,∞(0, 1), b ∈ C1([0, 1] \ {x0}), ∃K ∈ [1, 2), L ∈ (0, 1) such that (x − x0)a󸀠 ≤ Ka and (x − x0)b󸀠 ≤ Lb a.e. in [0, 1]. Hypothesis 7 (double strongly degenerate case (SSD)). There exists x0 ∈ (0, 1) such that a(x0) = b(x0) = 0, a, b > 0 in [0, 1] \ {x0}, a, b ∈ C1([0, 1] \ {x0}) ∩ W1,∞(0, 1), there exists K, L ∈ [1, 2) such that (x − x0)a󸀠 ≤ Ka and (x − x0)b󸀠 ≤ Lb a.e. in [0, 1]. 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 H1 a (0, 1) fl {u ∈ W1,1 0 (0, 1) : √aux ∈ L2 (0, 1)} (7) and H1 a,b (0, 1) fl {u ∈ H1 a (0, 1) : u √b ∈ L2 (0, 1)} (8) endowed with the inner products ⟨u, V⟩H1 a fl ∫ 1 0 au󸀠V󸀠dx + ∫1 0 uV dx, (9)


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.
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 1996 in the context of the wave equation, using a modification of the idea of [5].Later on, it also has been 2 Abstract and Applied Analysis applied to the standard heat equation by Imanuvilov and Yamamoto [2] in 1998.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  = 0 and the degeneracy can occur at the boundary or in the interior of the space domain.For example, we recall the works [7][8][9], where the authors obtain results concerning Carleman estimates and null controllability.
These results are complemented in [3,[10][11][12], where the authors obtain results concerning inverse problems for purely degenerate (i.e.,  = 0) parabolic equations and parabolic coupled systems, addressing, in particular, issues such as uniqueness and stability.If  ̸ = 0, the first results in this direction are obtained in [1] for the nondegenerate heat operator (i.e.,  > 0) 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: and where the subregion of measurements  is a nonempty subinterval of (0, 1) that is assumed to satisfy the following.
For fixed  >   > 0, the main result of this paper can be stated as follows.
(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  0 ∈  one can always find two subintervals  1 ⊂ (0,  0 ),  2 =⊂ ( 0 , 1) such that ( 1 ∪  2 ) ⊂⊂  \ { 0 }.
(ii) If we restrict ourselves to the particular case ℎ ∈ {,  ∈  2 (0, 1)} for some given function  ∈  1 ([0, ] × [0, 1]), 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 ℎ(, ).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.

Well-Posedness
The ways in which  and  degenerate at  0 can be quite different, and for this reason, following [16], to establish our results, we give the following definitions and assumptions.
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 [ 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 ‖ ⋅ ‖ 2 H is equivalent to From now on, we make the following assumptions on , , and .Hypothesis 9. (1) One among the Hypotheses 4, 5, or 6 holds true with  +  ≤ 2 and we assume that (2) Hypotheses 4, 5, 6, or 7 hold with  < 0.
Using Lemma 8, the next inequality is proved in [16, Proposition 2.18], which is crucial not only to obtain the wellposedness 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 Λ ∈ (0, 1] such that, for all  ∈ H, there holds Now, let us go back to problem (1), recalling the following definition.
Hence, the next result holds thanks to the theory of semigroups.
Proposition 13.The following assertions hold.

Lipschitz Stability Result
In this section, we aim at obtaining a Lipschitz stability result on determining the source term ℎ(, ) 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.

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).
(2) Moreover, if  < 0, we require that As usual, the derivation of global Carleman estimates relies on the introduction of some suitable weight function of the form where and with and where ã is defined by the following way: Observe that () → +∞ as  →  + 0 ,  − , 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 ℎ ∈  2 ( 0 , ,  2 (0, 1)).In the following, we denote and    0 fl ( 0 , ) × . (32) 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  0 such that the solution  of (31) in Some part of estimate ( 33) is already proved in [16] and, even if we refer to [20] a few times, our proof is quite selfcontained.In [16], the authors prove a Carleman inequality that estimates the integrals of () 2   and  3  3 (( −  0 ) 2 /()) 2 (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  > 0.Then, from (31), we obtain Moreover, ( 0 , ) = (, ) = 0.This property allows us to apply the Carleman estimates established in [16,Lemma 3.8] to  with    0 in place of (0, ) × (0, 1) 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 Replacing  by   , we immediately get from (35) Moreover,   =     +     .Therefore, In conclusion, thanks to (35)-(37), we get Step 2. Estimate ∬    0 To 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 where   is the Hardy-Poincaré constant and  1 = max( 4/3 0 /(0), (1 −  0 ) 4/3 /(1)).In the previous inequality, we have used the property that the map   → | −  0 |  /() is nonincreasing on the left of  0 and nondecreasing on the right of  0 for all  ≥ ; see [20,Lemma 2.1].If  > 4/3, we can consider the function () = (()| −  0 | 4 ) 1/3 .Then we have where Moreover, using the condition (C) given in Hypothesis 14, one has that the function ()/| −  0 |  , with  fl (4 + )/3 ∈ (1, 2), is nonincreasing on the left of  0 and nondecreasing on the right of  0 .The Hardy-Poincaré inequality given in [20, Proposition 1.1] implies Thus, in every case, for  large enough there holds Hence, Finally, coming back to , we get Step 3. Estimate of ∬    0 (1/) We then estimate ∬    0 2   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 has Now, 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 that Conclusion.We immediately deduce the expected Carleman estimate (33) from ( 38), (47), and (54).

Proof of Lipschitz Stability.
The object of this subsection is to prove Theorem 2 which recovers a source term ℎ from the measured final data and the partial knowledge of   over the subdomain  ⊂ (0,1).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  ∈ [, ], Φ (, ) =  () Ψ () , where  > 0 and  a  2 ([, ]) function such that () > 0 in (, ), () = () = 0 and   () ̸ = 0 in [, ] \ ω, ω is an open subset of .Now, choose the constant  1 in (27) so that Thus, one can show that weight functions satisfy the following properties which are needed in the sequel.(i)  ≤ Φ.For all  ∈ [0, 1], we have Thus, to show that  ≤ Ψ, it suffices to have This means ) and the conclusion follows immediately.
To complete the proof, it is sufficient to prove a similar inequality on the interval [0,  2 ].To this aim, we follow a reflection procedure.Consider the function  ( Observe that   (, − 1 ) =   (, 1) = 0 and, by the assumption on  and the fact that ,  ∈  2 ((0, ) × ), where  fl (− 1 , 1).Thus, we can apply the analogue of Theorem 15 (which still holds true, since ã belongs to  1,1 (−1, 1) in the weakly degenerate case and to  1,∞ (−1, 1) in the strongly degenerate one) on (− 1 , 1) in place of (0, 1), obtaining that there exist two positive constants  and  0 ( 0 sufficiently large), such that  satisfies, for all  ≥  0 , Using again the fact that   is supported in   2 ] and the boundedness of ã and ã (recall that, using the assumption on , ã is  1 far away from  0 and 0 in the weakly degenerate case and it is  1,∞ (−1, 1) in the strongly degenerate one), it follows that Now, by the definitions of , we note that Using a change of variable, by (57), one has Consequently, Going back to (78), by (81), and using the fact that  ≤ Φ, we obtain Thus, applying the Caccioppoli inequality and recalling that  is defined through ℎ  , one can find  > 0 such that Hence, using the definitions of  and , there exists a constant  > 0 such that Finally adding up (71) and (84), we conclude that for all  ≥  0 and for a positive constant .
Step 2 (estimate from above of  1 ).In this step, our purpose is to show that there exists some constant  > 0 such that Let us recall that, by the definition of Ψ, there exists some constant  < 0 such that Ψ() ≤  for all  ∈ [0, 1].Therefore we have As a consequence, setting  = sup R {  →  3  2 } > 0, we have In order to complete the proof of (86), it remains to prove the following lemma.
Lemma 17.There exists a constant  > 0 such that We omit the proof of Lemma 17 which is classical and we refer the reader to [2].Using (88) and (89), we obtain (86).

Numerical Approach
In this section, we develop an algorithm for numerically reconstructing the unknown source term from the measured final data.

Solvability of the Inverse Problem and Gradient Formula.
As the theoretical stability is guaranteed by Theorem 2, in this subsection we study the inverse source problem from the numerical viewpoint.To this end, let us define our inverse problem which we use in computations.

Inverse Source Problem (ISP).
Let  be the solution to (1).Determine the source term ℎ(, ) from the measured data at the final time (, .).Problem (ISP) will be defined as the inverse source problem associated with the singular parabolic equation (1).Accordingly, for a given function ℎ ∈ U, from some class of admissible source functions U, which will be defined below, the singular parabolic problem (1) will be referred to as a direct problem.A solution of the direct problem (1), corresponding to the function ℎ ∈ U, will be defined as (, ; ℎ).Remark 18.It should be mentioned that we do not need the supplement distributed measurements to obtain the numerical solution of the inverse problem.
An immediate consequence of Lemma 19 is the following result.
Proposition 20.Assume Hypothesis 9.Then, the functional  is continuous on U and there exists a minimizer ℎ ⋆ ∈ U of (ℎ), i.e.,

𝐽 (ℎ
Here we will propose an iterative method to solve the nonlinear optimization problem (104), and thus the information of the derivatives of the objective functional plays a significant and important role.Our approach for computing the Fréchet derivatives is based on adjoint state method.This used method is also called the variational adjoint method or the adjoint problem approach [18,21,22].A distinct advantage of using such a method is due to the fact that it permits to reduce the computational costs as well as its relatively simple numerical implementation.
The following proposition characterizes the derivative of the cost functional (104).
Proposition 21.The Tikhonov functional  is Fréchet differentiable and its derivative at each ℎ ∈ U is given by where V ∈ ([0,]; 2 (0, 1)) is the mild solution of the following adjoint equation: For the proof of the above result, we shall use the following lemma which derives an integral relationship relating the change ℎ in the term source to the change of the output () through the solution of an adjoint problem.
Choosing the arbitrary input  ∈  2 (0, 1) in the adjoint problem (119) as () fl (() − ũ), we deduce from the integral relationship (118) that Using the integral equality (121) in the increment formula (120), we deduce that To obtain the Fréchet differential of (ℎ), by definition, we need to show that To this aim, observe that if we substitute  by  in (111) it follows that i.e., the second right-hand side integral of ( 122) is of the order O(‖ℎ‖ 2  2 () ).So, since the last two integrals in (122) are of the order O(‖ℎ‖ 2  2 () ), we deduce that the cost functional (ℎ) is Frćhet-differentiable, with Fréchet differential: Multiply both sides of the parabolic equation ( 126) by V, integrate on , and then apply the integration by parts formula on both sides multiple times.Then, taking into account the initial and boundary conditions in problem (126), we obtain the following identity: where ℎ 0 ∈ U is a given initial iteration.The choice of   defines different gradient methods, although in many situation the estimation of this parameter is a difficult problem.However, in the case of Lipschitz continuity of the gradient   (ℎ) the parameter   can be estimated via the Lipschitz constant as follows (see [18]): where  0 ,  1 > 0 are arbitrary parameters.Now we will prove the Lipschitz continuity of the Fréchet gradient.
An important application of this theorem is the following lemma (see [23,Lemmas 3.4.4 and 3.4.5]).
Lemma 24.Let the Fréchet gradient of the Tikhonov functional (104) defined on the set of admissible source terms U be Lipschitz continuous with the Lipschitz constant  > 0. Denote by {ℎ  } ⊂ U the sequence of iterations obtained by the Landweber iteration algorithm (129).If the relaxation parameter   ∈ (0, 2/), then the following statements hold: where  ⋆ fl inf ℎ∈U (ℎ).

Conclusion
In this paper, we have considered an inverse source problem for a class of degenerate and singular parabolic equations.Based on Carleman estimates, global Lipschitz stability result is proved.Then, the identification of the source term is formulated as a minimization problem combining the output least squares and the Tikhonov regularization.It is proved that the Fréchet derivative of the cost functional can be formulated via the solution of the adjoint parabolic problem.Lipschitz continuity of the gradient functional was also proved, which implies the monotonicity of the numerical sequence of iterations obtained by the Landweber iteration algorithm.Some applications with numerical implementations are in progress.