Determination of singular time-dependent coefficients for wave equations from full and partial data

We study the problem of determining uniquely a time-dependent singular potential $q$, appearing in the wave equation $\partial_t^2u-\Delta_x u+q(t,x)u=0$ in $Q=(0,T)\times\Omega$ with $T>0$ and $\Omega$ a $ \mathcal C^2$ bounded domain of $\mathbb R^n$, $n\geq2$. We start by considering the unique determination of some singular time-dependent coefficients from observations on $\partial Q$. Then, by weakening the singularities of the set of admissible coefficients, we manage to reduce the set of data that still guaranties unique recovery of such a coefficient. To our best knowledge, this paper is the first claiming unique determination of unbounded time-dependent coefficients, which is motivated by the problem of determining general nonlinear terms appearing in nonlinear wave equations.


Statement of the problem.
Let Ω be a C 2 bounded domain of R n , n 2, and fix Σ = (0, T ) × ∂Ω, Q = (0, T ) × Ω with 0 < T < ∞. We consider the wave equation where the potential q is assumed to be an unbounded real valued coefficient. In this paper we seek unique determination of q from observations of solutions of (1.1) on ∂Q.
1.2. Obstruction to uniqueness and set of full data for our problem. Let ν be the outward unit normal vector to ∂Ω, ∂ ν = ν · ∇ x the normal derivative and from now on let be the differential operators := ∂ 2 t − ∆ x . It has been proved by [40], that, for T > Diam(Ω), the data A q = {(u |Σ , ∂ ν u |Σ ) : u ∈ L 2 (Q), u + qu = 0, u |t=0 = ∂ t u |t=0 = 0} (1.2) determines uniquely a time-independent potential q. On the other hand, due to domain of dependence arguments, there is no hope to recover even smooth time-dependent coefficients restricted to the set D = {(t, x) ∈ Q : t ∈ (0, Diam(Ω)/2) ∪ (T − Diam(Ω)/2, T ), dist(x, ∂Ω) > min(t, T − t)} from the data A q (see [32,Subsection 1.1]). Therefore, even when T is large, for the global recovery of general time-dependent coefficients the information on the bottom t = 0 and the top t = T of Q are unavoidable. Thus, for our problem the extra information on {t = 0} and {t = T }, of solutions u of (1.1), can not be completely removed. In this context, we introduce the set of data C q = {(u |Σ , u t=0 , ∂ t u |t=0 , ∂ ν u |Σ , u |t=T , ∂ t u |t=T ) : u ∈ L 2 (Q), u + qu = 0}.
and we recall that [25] proved that, for q ∈ L ∞ (Q), the data C q determines uniquely q. From now on we will refer to C q as the set of full data for our problem and we mention that [31,32,33] proved recovery of bounded time-dependent coefficients q from partial data corresponding to partial knowledge of the set C q . The goal of the present paper is to prove recovery of singular time-dependent coefficients q from full and partial data.
1.3. Physical and mathematical motivations. Physically speaking, our inverse problem consists of determining unstable properties such as some rough time evolving density of an inhomogeneous medium from disturbances generated on the boundary and at initial time, and measurements of the response. The goal is to determine the function q which describes the property of the medium. Moreover, singular time-dependent coefficients can be associated to some unstable time-evolving phenomenon that can not be modeled by bounded time-dependent coefficients or time independent coefficients. Let us also observe that, according to [11,27], for parabolic equations the recovery of nonlinear terms, appearing in some suitable nonlinear equations, can be reduced to the determination of time-dependent coefficients. In this context, the information that allows to recover the nonlinear term is transferred, throw a linearization process, to a time-dependent coefficient depending explicitly on some solutions of the nonlinear problem. In contrast to parabolic equations, due to the weak regularity of solutions, it is not clear that this process allows to transfer the recovery of nonlinear terms, appearing in a nonlinear wave equation, to a bounded time-dependent coefficient. Thus, in order to expect an application of the strategy set by [11,27] to the recovery of nonlinear terms for nonlinear wave equations, it seems important to consider recovery of singular time-dependent coefficients.
1.4. known results. The problem of determining coefficients appearing in hyperbolic equations has attracted many attention over the last decades. This problem has been stated in terms of recovery of a time-independent potential q from the set A q . For instance, [40] proved that A q determines uniquely a time-independent potential q, while [16] proved that partial boundary observations are sufficient for this problem. We recall also that [4,5,30,44] studied the stability issue for this problem.
Several authors considered also the problem of determining time-dependent coefficients appearing in wave equations. In [43], the authors shown that the knowledge of scattering data determines uniquely a smooth time-dependent potential. In [41], the authors studied the recovery of a time-dependent potential q from data on the boundary ∂Ω for all time given by (u |R×∂Ω , ∂ ν u |R×∂Ω ) of forward solutions of (1.1) on the infinite time-space cylindrical domain R t × Ω instead of Q. As for [39], the authors considered this problem at finite time on Q and they proved the recovery of q restricted to some strict subset of Q from A q . Isakov established in [25,Theorem 4.2] unique global determination of general time-dependent potentials on the whole domain Q from the important set of full data C q . By applying a result of unique continuation for wave equation, which is valid only for coefficients analytic with respect to the time variable (see for instance the counterexample of [1]), [17] proved unique recovery of time-dependent coefficients from partial knowledge of the data A q . In [42], the author extended the result of [41]. Moreover, [46] established the stable recovery of X-ray transforms of time-dependent potentials and [2,6] proved log-type stability in the determination of time-dependent coefficients with data similar to [25] and [39]. In [31,32,33], the author proved uniqueness and stability in the recovery of several time-dependent coefficients from partial knowledge of the full set of data C q . It seems that the results of [31,32,33] are stated with the weakest conditions so far that allows to recover general bounded time-dependent coefficients. More recently, [34] proved unique determination of such coefficients on Riemannian manifolds. We mention also the work of [45] who determined some information about time-dependent coefficients from the Dirichlet-to-Neumann map on a cylinder-like Lorentzian manifold related to the wave equation. We refer to the work [10,12,20,21,35] for determination of time-dependent coefficients for fractional diffusion, parabolic and Schrödinger equations have been considered.
In all the above mentioned results, the authors considered time-dependent coefficients that are at least bounded. There have been several works dealing with recovery of non-smooth coefficients appearing in elliptic equations such as [9,15,19,23]. Nevertheless, to our best knowledge, except the present paper, there is no work in the mathematical literature dealing with the recovery of singular time-dependent coefficients q even from the important set of full data C q .
1.5. Main results. The main purpose of this paper is to prove the unique global determination of timedependent and unbounded coefficient q from partial knowledge of the observation of solutions on ∂Q = ({0} × Ω) ∪ Σ ∪ ({T } × Ω). More precisely, we would like to prove unique recovery of unbounded coefficient q ∈ L p1 (0, T ; L p2 (Ω)), p 1 2, p 2 n, from partial knowledge of the full set of data C q . We start by considering the recovery of some general unbounded coefficient q from restriction of C q on the bottom t = 0 and top t = T of the time-space cylindrical domain Q. More precisely, for q ∈ L p1 (0, T ; L p2 (Ω)), p 1 2, p 2 n, we consider the recovery of q from the set of data ). In addition, assuming that T > Diam(Ω), we prove the recovery of q from the set of data Our first main result can be stated as follows Theorem 1.1. Let p 1 ∈ (2, +∞), p 2 ∈ (n, +∞) and let q 1 , q 2 ∈ L p1 (0, T ; L p2 (Ω)). Then, either of the following conditions:

4)
implies that q 1 = q 2 . Moreover, assuming that T > Diam(Ω), the condition We consider also the recovery of a time-dependent and unbounded coefficient q from restriction of the data C q on the lateral boundary Σ. Namely, for all ω ∈ S n−1 = {x ∈ R n : |x| = 1} we introduce the ω-shadowed and ω-illuminated faces of ∂Ω. Here, for all k ∈ N * , · denotes the scalar product in R k given by We define also the parts of the lateral boundary Σ taking the form Σ ±,ω = (0, T ) × ∂Ω ±,ω . We fix ω 0 ∈ S n−1 and we consider V = (0, T ) × V ′ with V ′ a closed neighborhood of ∂Ω −,ω0 in ∂Ω. Then, we study the recovery of q ∈ L p (Q), p > n + 1, from the data and the determination of a time-dependent coefficient q ∈ L ∞ (0, T ; L p (Ω)), p > n, from the data We refer to Section 2 for the definition of this set. Our main result can be stated as follows.
To our best knowledge the result of Theorem 1.1, 1.2 and 1.3 are the first results claiming unique determination of unbounded time-dependent coefficients for the wave equation. In Theorem 1.1, we prove recovery of coefficients q, that can admit some singularities, by making restriction on the set of full data C q on the bottom t = 0 and the top t = T of Q. While, in Theorem 1.2 and 1.3, we consider less singular time-dependent coefficients, in order to restrict the data on the lateral boundary Σ = (0, T ) × ∂Ω.
We mention also that the uniqueness result of Thorem 1.3 is stated with data close to the one considered by [31,32], where determination of bounded time-dependent potentials is proved with conditions that seems to be one of the weakest so far. More precisely, the only difference between [31,32] and Theorem 1.3 comes from the restriction on the Dirichlet boundary condition ( [31,32] consider Dirichlet boundary condition supported on a neighborhood of the ω 0 -shadowed face, while in Theorem 1.3 we do not restrict the support of the Dirichlet boundary).
In the present paper we consider two different approaches which depend mainly on the restriction that we make on the set of full data C q . For Theorem 1.1, we use geometric optics solutions corresponding to oscillating solutions of the form with λ > 1 a large parameter, R λ a remainder term that admits a decay with respect to the parameter λ and ψ j , j = 1, .., N , real valued. For N = 1, these solutions correspond to a classical tool for proving determination of time independent or time-dependent coefficients (e. g. [2,3,4,6,39,41,40]). In a similar way to [34], we consider in Theorem 1.1 solutions of the form (1.8) with N = 2 in order to be able to restrict the data at t = 0 and t = T while avoiding a "reflection". It seems that in the approach set so far for the construction of solutions of the form (1.8), the decay of the remainder term R λ relies in an important way to the fact that the coefficient q is bounded (or time independent). In this paper, we prove how this construction can be extended to unbounded time-dependent coefficients. The approach used for Theorem 1.1 allows in a quite straightforward way to restrict the data on the bottom t = 0 and on the top t = T of Q. Nevertheless, it is not clear how one can extend this approach to restriction on the lateral boundary Σ without requiring additional smoothness or geometrical assumptions. For this reason, in order to consider restriction on Σ, we use a different approach where the oscillating solutions (1.8) are replaced by exponentially growing and exponentially decaying solutions of the form u(t, x) = e ±λ(t+x·ω) (a(t, x) + w λ (t, x)), (t, x) ∈ Q, (1.9) where ω ∈ S n−1 and w λ admits a decay with respect to the parameter λ. The idea of this approach, which is inspired by [5,31,32,33] (see also [8,29] for elliptic equations), consists of combining results of density of products of solutions with Carleman estimates with linear weight in order to be able to restrict at the same time the data on the bottom t = 0, on the top t = T and on the lateral boundary Σ of Q. For the construction of these solutions, we use Carleman estimates in negative order Sobolev space. To our best knowledge this is the first extension of this approach to singular time-dependent coefficients.
1.6. Outline. This paper is organized as follows. In Section 2, we start with some preliminary results and we define the set of data C q (0), C q (T ), C q (0, T ), C q (T, V ) and C q (0, T, V ). In Section 3, we prove Theorem 1.1 by mean of geometric optics solutions of the form (1.8). Then, Section 4 and Section 5 are respectively devoted to the proof of Theorem 1.2 and Theorem 1.3.

Preliminary results
In the present section we define the set of data C q (T, V ), C q (0, T, V ) and we recall some properties of the solutions of (1.1) for any q ∈ L p1 (Q), with p 1 > n + 1, or, for q ∈ L ∞ (0, T ; L p2 (Ω)), with p 2 > n. For this purpose, in a similar way to [32], we will introduce some preliminary tools. We define the space . We consider also the space ) and topologize it as a closed subset of H 1 (Q) (resp L 2 (0, T ; H 1 (Ω))). In view of [32,Proposition 4], the maps can be extended continuously to τ 0 : Here for all w ∈ C ∞ (Q) we set τ 0 w = (τ 0,1 w, τ 0,2 w, τ 0,3 w), τ 1 w = (τ 1,1 w, τ 1,2 w, τ 1,3 w), Therefore, we can introduce . By repeating the arguments used in [32, Proposition 1], one can check that the restriction of τ 0 to S (resp S * ) is one to one and onto. Thus, we can use Let us consider the initial boundary value problem (IBVP in short) (2.1) We have the following well-posedness result for this IBVP when q is unbounded.
Proof. According to the second part of the proof of [ , the proof of this proposition will be completed if we show that for any v ∈ W 2,∞ (0, T ; H 1 0 (Ω)) solving (2.1) the a priori estimate (2.2) holds true. Without lost of generality we assume that v is real valued. From now on we consider this estimate. We define the enery Multiplying (2.1) by ∂ t v and integrating by parts we get On the other hand, we have Applying the Sobolev embedding theorem and the Hölder inequality, for all s ∈ (0, T ) we get with C depending only on Ω. Then, the Poincarré inequality implies where C depends only on Ω. Thus, from (2.4), we get (2.5) In the same way, an application of the Hölder inequality yields Combining this estimate with (2.3)-(2.5), we deduce that where C depends only on T , Ω and any M q L p 1 (0,T ;L p 2 (Ω)) . By taking the power p1 p1−1 on both side of this inequality, we get Then, the Gronwall inequality implies From this last estimate one can easily deduce (2.2).

Let us introduce the IBVP
in Ω, u = g, on Σ. (2.6) We are now in position to state existence and uniqueness of solutions of this IBVP for (g, v 0 , v 1 ) ∈ H and q ∈ L p (Q), p > n + 1.
and the boundary operator (2.9) is the unique solution of (2.6) and estimate (2.9) implies (2.7). Now let us consider the last part of the proposition. For this purpose, let (g, v 0 , v 1 ) ∈ H and let u ∈ H 1 (Q) be the solution of (2.6). Note first that ( Combining this with (2.7), we find that B q is a bounded operator from H to From now on, we define the set C q (T, V ) by In the same way, for q ∈ L p1 (0, T ; L p2 (Ω)), p 1 2, p 2 > n, we consider the set C q (T ), C q (0), C q (0, T ) introduced before Theorem 1.1. Using similar arguments to Proposition 2.2 we can prove the following. Proposition 2.3. Let (g, v 1 ) ∈ H * with v 0 = 0 and let q ∈ L ∞ (0, T ; L p (Ω)), p > n. Then, the IBVP (2.6) admits a unique weak solution u ∈ L 2 (0, T ; H 1 (Ω)) satisfying and the boundary operator B q, * : We define the set C q (0, T, V ) by The goal of this section is to prove Theorem 1.1. For this purpose, we consider special solutions u j of the equation with a large parameter λ > 0 and a remainder term R j,λ that admits some decay with respect to λ. The use of such a solutions, also called oscillating geometric optics solutions, goes back to [40] who have proved unique recovery of time-independent coefficients. Since then, such approach has been used by various authors in different context including recovery of a bounded time-dependent coefficient by [34]. In this section we will prove how one can extend this approach, that has been specifically designed for the recovery of timeindependent coefficients or bounded time-dependent coefficients, to the recovery of singular time-dependent coefficients.
3.1. Oscillating geometric optics solutions. Fixing ω ∈ S n−1 , λ > 1 and a j,k ∈ C ∞ (Q), j = 1, 2, k = 1, 2, we consider solutions of (3.1) taking the form (3.4) Here, the expression a j,k , j, k = 1, 2, are independent of λ and they are respectively solutions of the transport equation The main point in the construction of such solutions, also called oscillating geometric optics (GO in short) solutions, consists of proving the decay of the expression R j,λ with respect to λ → +∞. Actually, we would like to prove the following, lim λ→+∞ R j,λ L ∞ (0,T ;L 2 (Ω)) = 0. . For this reason we can not apply the result of [34] and we need to consider the following.
Lemma 3.1. Let q j ∈ L p1 (0, T ; L p2 (Ω)), j = 1, 2, p 1 > 2, p 2 > n. Then, we can find u j ∈ K(Q) solving (3.1), of the form (3.3)-(3.4), with R j,λ , j = 1, 2, satisfying (3.8) and the following estimate Proof. We will consider this result only for j = 2, the proof for j = 1 being similar by symmetry. Note first that, (3.5) implies that Thus, in light Proposition 2.1, we have R 2,λ ∈ K(Q) with In particular, this proves (3.9). The only point that we need to check is the decay with respect to λ given by (3.8). For this purpose, we consider v(t, x) := t 0 R 2,λ (s, x)ds and we easily check In view of [38, Theorem 2.1, Chapter 5], since G λ ∈ H 1 (0, T ; L 2 (Ω)) we have v ∈ H 2 (Q). We define the energy E(t) at time t associated with v and given by Multiplying (3.12) by ∂ t v and taking the real part, we find Applying Fubini's theorem, we obtain (3.13) On the other hand, applying the Hölder inequality, we get .
Then, combining the Sobolev embedding theorem with the Poincarré inequality, we deduce that with C depending only on Ω. Applying again the Hölder inequality, we get (3.14) In the same way, we obtain .

(3.15)
Finally, fixing Combining (3.13)-(3.16), we deduce that and we get with C depending only on Ω and T . Now taking the power p1 p1−2 on both side of this inequality, we get and applying the Gronwall inequality, we obtain where C 1 depends only on p 1 and C 2 on q 2 L p 1 (0,T ;L p 2 (Ω)) , Ω and T . According to this estimate, the proof of the lemma will be completed if we prove that lim λ→+∞ β λ L ∞ (0,T ;L 2 (Ω)) = 0. Therefore, for all t ∈ [0, T ] and almost every x ∈ Ω, we obtain Moreover, from the definition of H λ , we get Thus, we deduce from Lebesgue's dominated convergence theorem that Combining this with the estimate we deduce (3.17). This completes the proof the lemma.

Now, in view of [28, Lemma 2.2] we can multiply u 1 to the equation in (3.19) and apply Green formula to get
with n the outward unite normal vector to ∂Q. Since C q1 (0) = C q2 (0) and v| t=0 = u 2 | t=0 = 0, we see ∂ ν u| Σ = u| t=T = ∂ t u| t=T = 0, in addition to the boundary conditions of u in (3.19). Consequently, it follows from (3.20) that Inserting the expressions of u j (j = 1, 2) given by (3.18) to the previous identity gives the relation for all λ > 1. Using the fact that q ∈ L 2 (Q) and applying the Riemann-Lebesgue lemma and (3.8), we deduce that On the other hand, by Cauchy-Schwarz inequality it holds that C ||q|| L p 1 (0,T ;L p 2 (Ω)) ||R 1,λ || L ∞ (0,T ;H 1 (Ω)) ||R 2,λ || L ∞ (0,T ;L 2 (Ω)) , which tends to zero as λ → ∞ due to the decaying behavior of R j,λ (see (3.8)) and estimate (3.9). Therefore, |R λ | → 0 as λ → ∞. It then follows that Since ω ∈ S n−1 is arbitrary chosen, we deduce that for any ω ∈ S n−1 and any ξ lying in the hyperplane {ζ ∈ R 1+n : ζ · (1, −ω) = 0} of R 1+n , the Fourier transform F q is null at ξ. On the other hand, since q ∈ L 1 (R 1+n ) is compactly supported in Q, we know that F q is a complex valued real-analytic function and it follows that F q = 0. By inverse Fourier transform this yields the vanishing of q, which implies that q 1 = q 2 in Q.
We have proved so far that either of the conditions (1.3) and (1.4) implies q 1 = q 2 . It remains to prove that for T > Diam(Ω), the condition (1.5) implies q 1 = q 2 .
3.3. Proof of Theorem 1.1 with restriction at t = 0 and t = T . In this section, we assume that T > Diam(Ω) is fulfilled and we will show that (1.5) implies q 1 = q 2 . For this purpose, we fix λ > 1, ω ∈ S n−1 and ε = T −Diam(Ω)
Therefore, we have and by the same way that supp(a 2,2 ) ∩ supp(a 1,2 ) = ∅. This implies (3.24) and by the same way that q 1 = q 2 . Thus, the proof of Theorem 1.1 is completed.

Proof of Theorem 1.2
In the previous section we have seen that the oscillating geometric optics solutions (3.2) can be used for the recovery of some general singular time-dependent potential. We have even proved that, by adding a second term, we can restrict the data on the bottom t = 0 and top t = T of Q while avoiding a "reflection". Nevertheless, as mentioned in the introduction, it is not clear how one can adapt this approach to restrict data on the lateral boundary Σ without requiring additional smoothness or geometrical assumptions. In this section, we use a different strategy for restricting the data at Σ. Namely, we replace the oscillating GO solutions (3.2) by exponentially growing and decaying solutions, of the form (1.9), in order to restrict the data on Σ by mean of a Carleman estimate. In this section, we assume that q 1 , q 2 ∈ L p (Q), with p > n + 1, and we will prove that (1.6) implies q 1 = q 2 . For this purpose, we will start with the construction of solutions of (1.1) taking the form (1.9). Then we will show Carleman estimates for unbounded potentials and we will complete the proof of Theorem 1.2.

Geometric optics solutions for Theorem 1.2.
Let ω ∈ S n−1 and let ξ ∈ R 1+n be such that ξ · (1, −ω) = 0. In this section we consider exponentially decaying solutions u 1 ∈ H 1 (Q) of the equation and exponentially growing solution where λ > 1 and the term w j ∈ H 1 (Q), j = 1, 2, satisfies with C independent of λ. We summarize these results in the following way.  We start by considering Proposition 4.1. To build solutions u 1 ∈ H 1 (Q) of the form (4.1), we first recall some preliminary tools and a suitable Carleman estimate in Sobolev space of negative order borrowed from [33]. For all m ∈ R, we introduce the space H m λ (R 1+n ) defined by Note that here we consider these spaces with λ > 1 and, for λ = 1, one can check that H m λ (R 1+n ) = H m (R 1+n ). Here for all tempered distribution u ∈ S ′ (R 1+n ), we denote byû the Fourier transform of u. We fix the weighted operator P ω,±λ := e ∓λ(t+x·ω) e ±λ(t+x·ω) = ± 2λ(∂ t − ω · ∇ x ) and we recall the following Carleman estimate with C > 0 independent of v and λ.
Proof. We start by considering the case q ∈ L p1 (Q). Note first that On the other hand, fixing by the Sobolev embedding theorem we deduce that Combining this with the fact that we deduce from the Hölder inequality that Thus, applying (4.4) and (4.6), we deduce (4.5) for λ > 1 sufficiently large. Now let us consider the case q ∈ L ∞ (0, T ; L p2 (Ω)). Note first that qv L 2 (0,T ;H −1 λ (R n )) . Therefore, by repeating the above arguments, we obtain which implies (4.5) for λ > 1 sufficiently large. Combining these two results, one can find λ ′′ 1 > λ ′ 1 such that (4.5) is fulfilled.
Thus, by the Hahn Banach theorem we can extend L to a continuous linear form on H −1 λ (R 1+n ) still denoted by L and satisfying L C F L 2 (Q) . Therefore, there exists . This proves that w 1 fufills (4.3) which completes the proof of the proposition. Now let us consider the construction of the exponentially growing solutions given by Proposition 4.2. Combining [33,Lemma 5.4] with the arguments used in Lemma 4.2 we obtain the Carleman estimate.
with C > 0 independent of v and λ.
In a similar way to Proposition 4.1, we can complete the proof of Proposition 4.2 by applying estimate (4.8).

4.2.
Carleman estimates for unbounded potential. This subsection is devoted to the proof of a Carleman estimate similar to [33,Theorem 3.1]. More precisely, we consider the following estimate.

Proof of Theorem 1.3
Let us first remark that, in contrast to Theorem 1.1, in Theorem 1.2 we do not restrict the data to solutions of (1.1) satisfying u |t=0 = 0. In this section we will show Theorem 1.3 by combining the restriction on the bottom t = 0, the top t = T of Q stated in Theorem 1.1 with the restriction on the lateral boundary Σ stated in Theorem 1.2. From now on, we fix q 1 , q 2 ∈ L ∞ (0, T ; L p (Ω)), p > n, and we will show that condition (1.7) implies q 1 = q 2 . For this purpose we still consider exponentially growing and decaying GO solutions close to those of the previous subsection, but this time we need to take into account the constraint u 2 (0, x) = 0 required in Theorem 1.3. For this purpose, we will consider a different construction comparing to the one of the previous section which will follow from a Carleman estimate in negative order Sobolev space only with respect to the space variable. 5.1. Carleman estimate in negative Sobolev space for Theorem 1.3. In this subsection we will derive a Carleman estimate in negative order Sobolev space which will be one of the main tool for the construction of exponentially growing solutions u 2 of (3.1) taking the form with the restriction τ 0,2 u 2 = 0 (recall that for v ∈ C ∞ (Q), τ 0,2 v = v |t=0 ). In a similar way to the previous section, for all m ∈ R, we introduce the space H m λ (R n ) defined by In order to construct solutions u 2 of the form (4.2) and satisfying τ 0,2 u 2 = 0, instead of the Carleman estimate (4.8), we consider the following.
with C > 0 independent of v and λ.
In order to prove this theorem, we start by recalling the following intermediate tools. From now on, for m ∈ R and ξ ∈ R n , we set ξ, λ = (|ξ| 2 + λ 2 ) 1 2 and D x , λ m u defined by . For m ∈ R we define also the class of symbols Following [24,Theorem 18.1.6], for any m ∈ R and c λ ∈ S m λ , we define c λ (x, D x ), with D x = −i∇ x , by with C > 0 independent of v and λ.
Proof. Consider w(t, x) = v(T − t, x) and note that according to (5.2), we have w ∈ C 2 ([0, T ]; C ∞ 0 (Ω)) and Therefore, in a similar way to the proof of [32, Lemma 4.1], one can check that with c > 0 independent of w and λ. Now, recalling that w solves where C depends only on T and Ω. Combining this with (5.5), we obtain Using the fact that P −ω,λ w(t, x) = P ω,−λ v(T − t, x), we deduce (5.4).
Applying the Carleman estimate (5.3), we can now build solutions u 2 of the form (5.1) and satisfying τ 0,2 u 2 = 0 and complete the proof of Theorem 1.3.
Using this proposition, we are now in position to complete the proof of Theorem 1.3. Proof of Theorem 1.3. Let us remark that since Lemma 4.2 and Theorem 4.1 are valid when q ∈ L ∞ (0, T ; L p (Ω)) one can easily extend Proposition 4.1 to the case q 1 ∈ L ∞ (0, T ; L p (Ω)). Therefore, in the context of this section, Proposition 4.1 holds true. Combining Proposition 4.1 with Proposition 5.1, we deduce existence of a solution u 1 ∈ H 1 (Q) of u 1 + q 1 u 1 = 0 in Q taking the form (4.1), with w 1 ∈ H 1 (Q) satisfying (4.3) for j = 1, as well as the existence of a solution u 2 ∈ L 2 (0, T ; H 1 (Ω)) of u 2 + q 2 u 2 = 0 in Q, τ 0,2 u 2 = 0, taking the form (5.1) with the term w 2 ∈ L 2 (0, T ; H 1 (Ω)) satisfying (5.9). Repeating the arguments of the end of the proof of Theorem 1.2 (see Subsection 4.4), we can deduce the following orthogonality identity lim λ→+∞ Q qu 1 u 2 dxdt = 0. (5.12) Moreover, one can check that with Y λ (t, x) = q[e −2λt e −i(t,x)·ξ + e −i(t,x)·ξ w 2 + w 1 + w 1 w 2 ]. Combining Combining this asymptotic property with (5.12), we can conclude in a similar way to Theorem 1.2 that q 1 = q 2 .