Lipschitz stability for some coupled degenerate parabolic systems with locally distributed observations of one component

This article presents an inverse source problem for a cascade system of \begin{document}$ n $\end{document} coupled degenerate parabolic equations. In particular, we prove stability and uniqueness results for the inverse problem of determining the source terms by observations in an arbitrary subdomain over a time interval of only one component and data of the \begin{document}$ n $\end{document} components at a fixed positive time \begin{document}$ T' $\end{document} over the whole spatial domain. The proof is based on the application of a Carleman estimate with a single observation acting on a subdomain.

We are interested in answering the following inverse problem: can we retrieve the source terms f 1 , ..., f n in system (1) from incomplete data, that is to say, from a reduced number of measurements of the solution?
The main goal of the present work is to recover all the source terms f k (1 ≤ k ≤ n) using the following observation data: (ay kx ) x (T , ·), ∀k : 1 ≤ k ≤ n, y 1 | ωt 0 and y 1t | ωt 0 .
Our contributions are: • identification of all external forces term for n-coupled degenerate parabolic system (1) from a few interior measurements, • the reduction of the number of observations, and • a global stability estimate (of Lipschitz type).
Inverse problems for non degenerate (a > 0) parabolic systems are well studied over the last decades and there have been a great number of results. The first Lipschitz stability result of an inverse source problem for a parabolic equation (n = 1) was obtained by Imanuvilov and Yamamoto [23]. The essence of their methodology comes from the Bukhgeim-Klibanov method using the Carleman estimate to derive the global uniqueness in inverse problems, see [12]. Inverse problems for parabolic systems of two equations have also been studied, see for example [1,7,16,17,18,19,24,28]. There are, of course, other kinds of models which involve coupled systems of partial differential equations (wave-wave, hyperbolicparabolic coupled systems). For results related to these subjects we refer to e.g. [4] for hyperbolic systems in cascade and [6] in the case of thermoelastic systems. We refer to [5,29] for a more detailed survey concerning the applicability of Carleman estimates to stability of inverse problems.
In the last recent years, an increasing interest has been devoted to the study of degenerate parabolic equations with degeneracy occurring at the boundary or in the interior of the space domain. For such systems, inverse problems of one equation was studied in [9,14,15,25,26]. The main result in these works is the development of adequate Carleman estimates, which are crucial tool to obtain Lipschitz stability for term sources, initial data, potentials and diffusion coefficients. The case of two coupled systems (n = 2), is considered in [11], and in [10].
Recently, in [20] the authors studied the null controllability of n-coupled degenerate parabolic system when m distributed controls are exerted on the system. In this framework, up to now, inverse source problems for coupled systems of n equation with n > 2 were never considered even in the case of non degenerate parabolic coupled systems.
Motivated by this reason, the present paper is devoted to the study of an inverse source problem for such coupled degenerate systems. More precisely, we will follow the approach introduced by Imanuvilov and Yamamoto in [23] 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 given in [20]. As a consequence, we prove a stability estimate of Lipschitz type in determining all the source terms from the knowledge of some measurements of only one component of the solution. To our knowledge, this paper is the first one concerning Lipschitz stability results in inverse problems for degenerate coupled systems such as (1).
For fixed T > T > 0, our first main result is the stability for the inverse source problem.
Then, there exists C = C(T, t 0 , C 0 ) > 0 such that, for all f k ∈ S(C 0 ) and y 0 A brief idea of our strategy is as follows. First, we establish a Carleman estimate with a boundary observation for a single degenerate equation. Then, using a localization argument we deduce a Carleman estimate with a distributed observation for one degenerate equation. Summing up these inequalities we obtain a Carleman estimate for the coupled system with distributed observations of each equation which could be used to show Lipschitz stability estimate in the determination of the source terms from interior measurements of all components of the system. In a second step, by using the equations we try to reduce the number of measurements obtaining a Carleman estimate with a single observation acting on a subdomain. Finally, this estimate is successfully used along with certain energy estimates to obtain the stability result for the inverse source problem of n-coupled degenerate parabolic equations by measurements of one component.

Remark 1.
• Theorem 1.1 provides a global Lipschitz stability estimate that extend the one obtained for a single degenerate heat equation by Cannarsa, Tort and Yamamoto [15] to the case of more general cascade coupled systems. The main difference between our work and [15] is that we consider a coupled system of degenerate parabolic equations, and the additional data are given only for one component of this system. • Although theorem 1.1 provides a useful stability result, we note that it does not ensure that the inverse problem has a unique solution because the class S(C 0 ) is not a vector space.
An important case is when the unknown source terms of (1) take the form where g k are the unknown functions of L 2 (0, 1) while r k ∈ C 1 ([0, T ] × [0, 1]) are the given functions such that for some given constant d k > 0, 1 ≤ k ≤ n. We denote by E the space E := {g(x)r(t, x) for some g ∈ L 2 (0, 1)}.
As an application of Theorem 1.1, we can determine y k (t, 1) = 0, y k (t, 0) = 0, in the weakly degenerate case (WD), (ay kx )(t, 0) = 0, in the strongly degenerate case (SD), x ∈ (0, 1). (10) Here, we prove that we can uniquely recover the spatial components g k of the source terms from the measurement of the solution over the whole spatial domain (0, 1) at any fixed moment T plus additional local observations in space and time of one component of the solution. This is our second main result: ) be a function satisfying (9). Then, there exists C = C(T, t 0 , r k ) > 0 such that, for allf k =g k r k ∈ E andf k =ĝ k r k ∈ E the associated solutions (ỹ 1 , ...,ỹ n ) and (ŷ 1 , ...,ŷ n ) of (10) satisfy In particular, Theorem 1.2 provides the following uniqueness result: if the solutions (ỹ 1 , ...,ỹ n ) and (ŷ 1 , ...,ŷ n ) of (10) associated tog k andĝ k satisfy (aỹ kx ) x (T , ·) = (aŷ kx ) x (T , ·), ∀k : 1 ≤ k ≤ n in (0, 1), a(x) where H 2 a (0, 1) denotes a weighted Hilbert space defined in the next section. The article is organized as follows. In Section 2, we discuss the well-posedness of the problem (1). Then, in Section 3, we establish different Carleman estimates for parabolic equations and parabolic systems. Finally, in Section 4, we apply the Carleman estimates to prove the Lipschitz stability and uniqueness results.

2.
Well-posedness and regularity results. In order to study the well-posedness of system (1), we introduce the following weighted spaces. In the (WD) case: In both cases, the norms are defined as follow We recall from [13,3] that the operator (A, D(A)) defined by Ay := (ay x ) x , y ∈ D(A) = H 2 a (0, 1) is closed negative self-adjoint with dense domain in L 2 (0, 1). Hence, it is infinitesimal generator of an analytic semi-group of contractions in the pivot space L 2 (0, 1).
At this point, as the operator DA with domain D(DA) = H 2 a (0, 1) n is diagonal and since B is a bounded perturbation, the following wellposedness and regularity results hold.
Proof. The proof of statements (i) and (ii) mainly follows from the fact that (DA, D(DA)) generates an analytic semi-group in the pivot space L 2 (0, 1) n . Then it suffices to apply standard semi-groups theory: for example [8,Proposition 3.3] in the case F ∈ H 1 (0, T ; L 2 (0, 1) n ) and [8,Proposition 3.8] in the case F ∈ L 2 (Q) n .
3. Global Carleman estimates. In this section we give a new global Carleman estimate for the system (1). To this end, as in [15], we introduce the following time and space weight functions where t 0 > 0 is a fixed initial time, T > 0 is a final time and where the parameters x 0 y a(y) dy are positive constants that will be chosen later.
In order to state our fundamental result, we need to show first some Carleman estimates in the case of a single parabolic degenerate equation.
3.1. Carleman estimate for one degenerate equation. In this subsection we shall establish a new Carleman estimate for the solution of the following parabolic equation where d > 0 is a positive constant, f ∈ L 2 (Q) and y 0 ∈ L 2 (0, 1).
The following Carleman estimate will be crucial for the aim of this section. Note that the Carleman estimate needed in this work is different from the one showed in [3] where, however, the Carleman inequality was derived to bound just the integrals of sθa(x)y 2 x and s 3 θ 3 x 2 a(x) y 2 (that were sufficient for control purposes). For inverse problems, these estimates are not sufficient to conclude. Hence, as in [23], we need to complete the above result with the estimate of the integrals of 1 sθ y 2 t and sθ 3 2 |ηψ|y 2 .
Theorem 3.1. There exist two positive constants C and s 0 , such that every solution y of (14) satisfies, for all s ≥ s 0 , Proof. The proof is based on the methods developed in [15]. Given a solution (14) and a positive number s > 0, define w = ye sϕ for a.e. (t, x) ∈ Q t0 . We first prove a Carleman-type estimate for w and then we deduce the expected estimate on y. First of all, observe that w satisfies Moreover, w(t 0 , x) = w(T, x) = 0. This property allows us to apply the Carleman estimates established in [3] to w with Q t0 in place of (0, T ) × (0, 1) and d∂ The operators P + s and P − s are not exactly the ones of [3]. However, one can prove that the Carleman estimates do not change.
Using the previous estimate, we will bound the integral Qt 0 Therefore, using the fact that 1 sθ is bounded, it results Since the function x → x 2 a is nondecreasing, then one has Moreover, in what follows we will also need to estimate Qt 0 sθw 2 dx dt. In particular, using Young inequality, we have Qt 0 Let p(x) = x 4/3 a 1/3 , then since the function x → x 2 a is nondecreasing on (0, 1) one has, Thanks to the Hardy-Poincaré inequality (12), we obtain This gives, In a similar way, to bound the integral Qt 0 sθ since |η| ≤ T + t and |ψ| ≤ γd. By using inequality (20), we infer Therefore, since d > 0, we have for s large enough From inequalities (17), (22) and (24), one obtains Consequently, we obtain the estimate (15) which completes the proof.
From the boundary Carleman estimate (15), we will deduce a Carleman estimate for equation (14) on a subregion To this aim, let us set ω = (α , β ) ⊂⊂ ω and consider a smooth cut-off function Our first intermediate Carleman estimate is thus the following.
Proof. Define w := ξy where y is the solution of (14). Then, the function w satisfies the following equation Therefore, applying the Carleman estimate (15) to equation (27) and using the definition of w, we have From the definition of ξ and the Caccioppoli inequality (72), we obtain Moreover, since ξy x = w x − ξ x y, then we get  Proposition 4. Let T > 0. Then, there exist two positive constants C and s 0 such that, for every y 0 ∈ L 2 (0, 1), the solution y of equation (14) satisfies, for all s ≥ s 0 Proof. The function z := ζy is a solution of the uniformly parabolic equation Therefore, using the Caccioppoli inequality (72) and the definitions of z and ζ we deduce From ζy x = z x − ζ x y and supp ζ x ⊂ ω , we obtain for s large enough. Furthermore, using the fact that sθ The estimates (33)-(35) lead to Taking into account the fact that da(x) > 0 and x 2 da(x) > 0 in (α , 1), we deduce This proves Proposition 4.

3.2.
Carleman estimate for n-coupled degenerate equations. Now, we show the main result of this section, which is the ω-Carleman estimate for the coupled system (1). For this purpose, the parameters γ, ρ and d will be chosen such that where n is the size of the system (1).

Remark 2. By (38) and proceeding as in [20, Lemma 3.1], it can be shown that
is nonempty. We can then choose γ in this interval.

Using Propositions 3 and 4 and the Hardy-Poincaré inequality we deduce an intermediate important result which could be used to show a Lipschitz stability estimate for parabolic systems of determining n term sources from measurements of all components of the solution.
Theorem 3.3. There exist two positive constants C > 0 and s 0 > 0 such that for all (y 0 1 , ..., y 0 n ) ∈ (L 2 (0, 1)) n , the solution (y 1 , ..., y n ) of (1) satisfies, for all s ≥ s 0 where J (y) := Qt 0 1 sθ y 2 t + sθ 3 2 |ηψ|y 2 + sθa(x)y 2 x + s 3 θ 3 x 2 a(x) y 2 e 2sϕ dx dt.
Proof. Since y k is the solution of the system t ∈ (0, T ), y k (t, 0) = 0, for (WD), (ay kx )(t, 0) = 0, for (SD), t ∈ (0, T ), applying Proposition 3, for s big enough, we have On the other hand, since d k > 0, we have Hence, Moreover, since x → x 2 a is nondecreasing, we have Applying Hardy-Poincaé inequality to v := ξy j e sϕ , one has and by v x = ξy jx e sϕ + ξsθ x a(x) y j e sϕ + ξ x y j e sϕ , we obtain Qt 0 Now, using the fact that θ is bounded from below and since supp(ξ x ) ⊂ ω , one has Therefore, by taking the Carleman parameter s large enough, we obtain Thus, combining (44) with (45), we deduce n k=1 Qt 0 and by this, it results that n k=1 Qt 0 Similarly, applying Proposition 4 to y k , the solution of (43), and using the Hardy-Poincaré inequality, we obtain the estimate n k=1 Qt 0 Since e 2sϕ ≤ e 2sΦ , 1 2 ≤ ξ 2 + ζ 2 ≤ 1 and θ is bounded from below, then by adding (46) and (47) we obtain n k=1 Qt 0 for s large enough. This ends the proof.
Let us recall that our goal is to determine the term sources f k , k ∈ {1, ..., n} from measurements of one component of the solution using data on a prescribed subregion ω of (0, 1). To this aim, the key point is given by the next lemma which play a crucial role to absorb the observation on the components y k , k ∈ {2, ..., n}.
Proof. The proof follows the one of [20, Lemma 3.7], but it is different due to the fact that we have to deal here with a non-homogeneous system. For this reason we only point out the difference that appear when we consider the nonhomogeneous system (1) in place of homogeneous one. Let us consider a nonnegative smooth cut-off function χ ∈ C ∞ (0, 1), such that and Observe that, the k − 1th equation of the system (1) can be written as Multiplying the above equation by s l θ l χe 2sΦ k y k and integrating on Q t0 , since O 1 ⊂ ω 0 by (6), it follows that Moreover, proceeding as in [20,Lemma 3.7], one can get where l = max(3, 2l − 3) and J = max(3, 2l + 1, 4l − 5). On the other hand, for K 4 we have But by Lemma 3.2 we know that Consequently, by (52) we have Now, at this level, using the fact that supp χ ⊂ O 0 and thus a(x) x 2 is bounded in O 0 , then for s large enough, Putting together inequalities (50), (51) and (53), we finally obtain As a consequence of (41) and Lemma 3.4, we deduce the following fundamental Carleman estimate for the system (1) with one observed component. Qt 0 Proof. To prove Theorem 3.5 we will follow the same argument given in [22] and which is used to obtain the null controllability property for nondegenerate cascade parabolic systems with one control force. Given ω 0 ⊂ ω, we choose ω ⊂⊂ ω 0 and let Y = (y 1 , ..., y n ) be the solution to (1) associated to Y 0 ∈ L 2 (0, 1) n . From the definition of L B (l, φ, y) and recalling that Φ n = Φ, by (41) we have For l 1 := max(3, l, J), using the fact that Φ n ≤ Φ n−1 and L ω (l 1 , Φ n−1 , y k ) ≤ L On (l 1 , Φ n−1 , y k ), we deduce that where C n is a new positive constant and R 1 = 2l − 2. Observe that in (55) we have eliminated from the right hand side the local term involving y n . We can go on applying (48) for k = n − 1, l = l 1 , O 1 = O n , O 0 = O n−1 and ε = 1 2 Cn and eliminate in (56) the local term L On (J 1 , Φ n−1 , y n−1 ). By (a finite) iteration of this argument we obtain Qt 0 for some positive constants l n−1 and R n−1 . Now, since Φ n = Φ ≤ Φ k and sup Finally, by setting R = R n−1 in the previous estimate, we end the proof. 4. Stability estimate and uniqueness for the inverse source problem. In this section, we establish a stability and a uniqueness result using certain ideas from [23] and [15]. More precisely, we obtain an inequality which estimates the term sources f k , k ∈ {1, ..., n} over the entire domain (0, 1) with an upper bound given by some Sobolev norm of the solution Y at some fixed time T ∈ (0, T ) and the partial knowledge of y 1 and y 1t over the subdomain ω ⊂⊂ (0, 1). In proving these kinds of stability estimates, the global Carleman estimate obtained in Theorem 3.5 will play a crucial part along with certain energy estimates.
Applying the Carleman estimate of Theorem 3.5 to problem (57), we get: Let us note that, if we replace Qt 0 s R θ R f 2 kt e 2sΦ k dx dt by Qt 0 f 2 kt e 2sΦ k dx dt, the inequality (58) would be the kind of estimate that one would obtain when dealing with the more standard inverse problem that consists in retrieving the source term f in the scalar equation y t − (ay x ) x = f . Hence the next step mainly consists in adapting the reasoning of [15] to the present case, taking into account this extra term and the coupling of the equations. We shall first prove the following lemma.
Proof of Lemma 4.1. Since e 2sϕ(t,x) dx = 0, for a.a. x ∈ (0, 1), we can write Using Young's inequality and taking into account the fact that b kj ∈ L ∞ (Q), we estimate Moreover, applying once more Young's inequality, one has Arguing as in (20), by the Hardy-Poincaré inequality applied to y j e sϕ , we obtain Similarly, we have On the other hand, since |ϕ t | ≤ Cθ 3 2 |ηψ|, it follows that Thus, (60)-(64) yield the estimate (59).
Going back to the proof of Theorem 1.1, we note that the k equation of the system (4) can be written as Therefore, In particular, since sup the previous estimate yields Using the boundedness of e −2sϕ(T ,x) , we can prove that there exists a positive constant C such that Next, using the assumption that f 1 , ..., f n ∈ S(C 0 ), one has |f kt (t, x)| ≤ C 0 |f k (T , x)|, and |f k (t, x)| ≤ |f k (T , x)| + Then we observe that, for a fixed ε > 0, such that t 0 < T − ε < T < T + ε < T, we may write  s R+1 θ R+1 e 2sΦ k > 0.
Thus, in view of (67), we conclude that for some constant C = C(T, t 0 , C 0 ) > 0. This gives (7) and completes the proof of Theorem 1.1.