UNIQUENESS AND STABILITY FOR THE RECOVERY OF A TIME-DEPENDENT SOURCE IN ELASTODYNAMICS

. This paper is concerned with inverse source problems for the time-dependent Lam´e system in an unbounded domain corresponding to either the exterior of a bounded cavity or the full space R 3 . If the time and spatial variables of the source term can be separated with compact support, we prove that the vector valued spatial source term can be uniquely determined by boundary Dirichlet data in the exterior of a given cavity. Uniqueness and stability for recovering some class of time-dependent source terms are also obtained by using, respectively, partial and full boundary data.


Introduction.
1.1. Statement of the problem. Consider the radiation of an elastic source F outside a cavity D described by the system where ρ denotes the density, µ and λ are the Lamé coefficients, U = (u 1 , u 2 , u 3 ) is the displacement vector, D ⊂ R 3 is the region of the cavity and L λ,µ U is the Lamé operator defined by L λ,µ U := −µ(x)∇ × ∇ × U + (λ(x) + 2µ(x))∇∇ · U +(∇ · U )∇λ(x) + ((∇U ) + (∇U ) T )∇µ(x). (2) Throughout the paper, it is supposed that ρ > 0 is a constant and µ, λ ∈ C 3 (R 3 ) satisfy µ > 0, λ ≥ 0. Further, the density function and the Lamé coefficients are supposed to be constants in |x| > R for some sufficiently large R > 0 such that D ⊂ B R := {x ∈ R 3 : |x| < R}. Together with the governing equation, we impose the initial conditions and the traction-free boundary condition on ∂D: where T U is the stress boundary condition defined by (11) (see Section 2). In this paper we consider the inverse problem of determining the source term F from knowledge of U on the surface ∂B R = {x ∈ R 3 : |x| = R} with R > 0 sufficiently large. According to [6,Remark 4.5] there is an obstruction for the recovery of general time-dependent source terms F . Facing this obstruction we consider this problem for some specific type of source terms.

1.2.
Motivations. We recall that the Lamé system (1)- (2) is frequently used for the study of linear elasticity and imaging problems. In this context our inverse problems can be seen as the recovery of an external force provided by the source term F . Elastodynamics source terms corresponding to the product of a spatial function g and a temporal function f can be regarded as an approximation of the elastic pulse and are commonly used in modeling vibration phenomena in seismology and teleseismic inversion [1,47]. This type of sources has been also considered in numerous applications in biomedical imaging (see the references [2,3] and the references therein) where our inverse problem can be seen as the recovery of the information provided by the parameter under consideration.
Let us also observe that the identification of time-dependent sources (see Theorems 1.3 and 1.4) is associated to the recovery of a moving source which can be thought as an approximation of a pulsed signal transmitted by a moving antenna (see [23] and the references therein for more details).
1.3. Known results. Inverse source problems are a class of inverse problems which have received many interest. These problems take different forms and have many applications (environment, imaging, seismology · · · ). For an overview of these problems we refer to [28]. Among the different arguments considered for solving these problems we can mention the approach based on applications of Carleman estimates arising from the work of [15] (see also [38,39]). This approach has been applied successfully to hyperbolic equations by [50] in order to extend his previous work [49] to a wider class of source terms. More precisely, in [50] the author considered the recovery of source terms of the form f (x)G(x, t), where G is known, while in [49] the analysis of the author is restricted to source terms of the form σ(t)f (x), with σ known. More recently, the approach of [50] has been extended by [33] to hyperbolic equations with time-dependent second order coefficients and to less regular coefficients by [51]. We mention also the work of [17,37] using similar approach for inverse source problems stated for parabolic equations and the result of [48] proved by a combination of geometrical arguments and Carleman estimates. Concerning the Lamé system we refer to [25] where a uniqueness result has been stated for the recovery of time-independent source terms by mean of suitable Carleman estimate. We mention also the book [11] for an overview of the approach based on Carleman estimate for solving inverse source problems as well as the works [9,12,26,27,30] dealing with related problems as well as [10] where an inverse source problem for Biot's equations has been considered. We refer also to the recent work [6] where the recovery of a time-independent source term appearing in the Lamé system in all space has been proved from measurements outside the support of the source under consideration as well as the work of [34] dealing with this inverse source problem for fractional diffusion equations.
In all the above mentioned results the authors considered the recovery of time independent source terms (in other words, the spatial component of the source term). For the recovery of a source depending only on the time variable we refer to [21] where such problems has been considered for fractional diffusion equations and for the recovery of some class of sources depending on both space and time variable appearing in a parabolic equation on the half space, we refer to [28,Section 6.3]. For hyperbolic equations, we refer to [14,46] where the recovery of some specific time-dependent source terms have been considered. For Lamé systems, [6,Theorem 4.2] seems to be the only result available in the mathematical literature where such a problem has been addressed for time-dependent source terms. The result of [6,Theorem 4.2] is stated with source terms depending only on the time variable. To the best of our knowledge, except the result of [14], dealing with the recovery of discrete in time sources, and the result of the present paper, there is no result in the mathematical literature treating the recovery of a source term depending on both space and time variables appearing in hyperbolic equations.
1.4. Main results. In the present paper we consider three inverse problems related to the recovery of the source term F . In our first inverse problems we assume that the cavity D = ∅ is a domain with C 3 boundary ∂D, with connected exterior R 3 \D, and we consider source terms of the form where f is a real valued function and h = (h 1 , h 2 , h 3 ) : R 3 \D → R 3 is a vector valued function. Choose R > 0 sufficiently large such that B R also contains the support of h (i.e., supp(h) ⊂ B R ). We assume that f ∈ L 2 ([0, +∞)) is compactly supported and h ∈ L 2 (R 3 \D) 3 . Then, the problem (1)-(4) admits a unique solution The proof of this result can be carried out by combining the elliptic regularity properties of L λ,µ (see e.g., [ Inverse Problem 1 (IP1): Assume that f , D are both known in advance. Determine the spatially dependent function h from the radiated field U measured on the surface ∂B R × [0, T 1 ), T 1 ∈ (0, +∞]. Below we give a confirmative answer to the uniqueness issue for IP1 in two different cases. For source terms with low regularity we obtain 3 and F takes the form (5). Let also Ω := B R \D and let d j , j = 1, 2, be the Riemannian distance within Ω induced by the metric g j , where Then, for By considering measurements for all time (t ∈ (0, +∞)), we can remove the condition f (0) = 0 in the following way. 3 and F takes the form (5). Then the boundary data {U (x, t) : (x, t) ∈ ∂B R × R + } uniquely determine h.
For our last inverse problem, we consider the Lamé system with constant density and Lamé coefficients when the embedded cavity is absent (D = ∅). We assume here that F takes the form where the vector valued function f = (f 1 , f 2 , 0) is compactly supported onB R × [0, T ) and the scalar function g is supported in (−R, R) for some R > 0. Herẽ |x| < R}. Then our last inverse problem can be stated as follows.
Inverse Problem 2 (IP2): Assume that g is known in advance. Determine the time and space dependent function f from the radiated field U measured on the surface Γ × (0, T 1 ), with T 1 > 0, R 1 > 0 sufficiently large and Γ ⊂ ∂B R1 an open set with positive Lebesgue measurement.
In this paper we give a positive answer to (IP2) both in terms of uniqueness and stability. Our uniqueness result can be stated as follows.
and let Γ ⊂ ∂B R1 be an arbitrary open set with positive Lebesgue measurement. Then the source f can be uniquely determined by the data U (x, t) measured on Γ × (0, T 1 ).
By assuming that Γ = ∂B R1 , we can extend this uniqueness result to a log-type stability estimate. For this purpose, we need a priori information on the regularity and upper bound of the source terms f and g.
√ µ , ρ, λ, µ be constants and assume Assume also that g is non-uniformly vanishing with a constant sign (g ≥ 0 or g ≤ 0) and that there exists M > 0 such that Then, there exists C > 0 depending on M , R 1 , ρ, λ, µ, T 1 , g L 1 (R) such that

1.5.
Comments about our results. To the best of our knowledge, in Theorems 1.1 and 1.2 we obtain the first results of recovery of source terms stated for the Lamé system with variable coefficients outside a cavity. Indeed, it seems that all other known results are stated on a bounded domain (e.g. [25]) or in the full space R 3 (e.g. [6]). We emphasize that Theorems 1.1 and 1.2 are valid even if the embedded cavity is unknown. In fact, the unique determination of the embedded cavity can be proved following Isakov's arguments [31, Theorem 5.1] by applying the unique continuation results of [19,20]; see also the proof of Theorem 1.1. We refer also to [32] for the determination of other impenetrable scatterers for the wave equation with a single measurement data. The main purpose of this paper is concerned with the identification of elastic sources in an unbounded domain. Let us observe that in Theorem 1.1, we manage to restrict our measurements to a finite interval of time. However, like in [25,48,50,51], we need to impose the additional condition f (0) = 0 for a source term F of the form (5). In contrast to Theorem 1.1 and results using Carleman estimates like [9,12,25,48,50,51], in Theorem 1.2 we state our result without assuming that the source under consideration is non-vanishing at t = 0. For a source term F of the form (5), such assumption will be equivalent to the requirement that f (0) = 0. From the practical point of view, this means that the results of [25,48,50,51], as well as Theorem 1.2, can only be applied to the determination of a source term associated with a phenomenon which has appeared before the beginning of the measurement. This restriction excludes applications where one wants to determine a phenomenon with measurements that start before its appearance. By removing this restriction in Theorem 1.2 we make our result more suitable for applications in that context. The approach of Theorems 1.1 and 1.2 consists in transforming our problem into the recovery of initial condition. Then, applying some results of unique continuation and global Holmgren theorem borrowed from [19,20] we complete the proof of Theorem 1.1, 1.2.
Let us remark that, although the main results of [9,12] correspond to the recovery of coefficients for Lamé system, their analysis can be applied to the stable determination of source terms. Indeed, using an observability inequality obtained by mean of the so called FBI transform, the results of [9] allow to prove the stable recovery of time independent source terms, appearing in a Lamé system on a bounded domain, from measurements on an arbitrary portion of the boundary provided that the source terms are known on a neighborhood of the boundary of the domain. In the same spirit, the result of [12] can be applied to the stable recovery of a source term from internal measurement on a subset corresponding to a neighborhood of the boundary of the domain. These two results can be compared to Corollary 5 where we prove the recovery of the source terms from measurements on a suitable subset. While the results of [9,12] are stated on a bounded domain, the present work is stated in an unbounded domain with the presence of an obstacle. In addition, in contrast to [9,12], in Corollary 5 the source terms under consideration are not known and no measurements are made on any neighborhood of the boundary ∂D of the domain R 3 \ D. Finally, as mentioned above the approach of [9,12] requires that the source under consideration is non-vanishing at t = 0, while this condition is removed in Corollary 5. However, the approach of [9,12] leads to the stable recovery of a source term stated at finite time while Corollary 5 is restricted to a uniqueness result stated at infinite time.
To the best of our knowledge, even for a bounded domain, Theorems 1.3 and 1.4 seem to be the first results of unique and stable recovery of some general source term depending on both time and space variables appearing in a hyperbolic equation. Indeed, it seems that only results dealing with recovery of source terms depending only on the time variable (see [6,46]) or space variable (see [6,25,48,50,51]) are available in the mathematical literature with the exception of [14] where the recovery of discrete in time sources has been considered. Therefore the results of Theorems 1.3 and 1.4 are not only new for the Lamé system but also more general for hyperbolic equations. We mention also that the stability result of Theorem 1.4 requires a result of stability in the unique continuation already considered by [13,16,36] for the recovery of time-dependent coefficients. Note also that, in contrast of Theorems 1.2 and 1.1, thanks to the strong Huygens principle we can state Theorems 1.3 and 1.4 at finite time.
In Corollary 1 we prove that the results of Theorem 1.2 can be reformulated in terms of partial recovery of the source term from measurement on a subdomain where the source term or the initial data are known. This situation may for instance occur in several applications where the source under consideration has large support and the data considered in Theorem 1.2 is not accessible. What we prove in Corollary 1 is that even in such context one can expect the recovery of partial information of the source term under consideration by measurements located on some subdomain where the source is known.
Both Theorems 1.1, 1.2 and Corollary 1 remain valid if the cavity D is absent or if D is a rigid elastic body (i.e., U vanishes on ∂D). All the results of this paper can be applied to the wave equation. Actually, the proof for the wave equation will be easier in several aspects and the particular treatment for the Lamé system leads to some difficulties inherent to this type of systems (see for instance the proof of Theorems 1.3 and 1.4).

1.6.
Outline. This paper is organized as follows. In Section 2 we study the inverse problem (IP1). More precisely, we prove Theorems 1.1 and 1.2 as well as the Corollary 1. In Section 3 we treat the inverse problem (IP2). We start with the uniqueness result stated in Theorem 1.3. Then, we extend this result by proving the stability estimate stated in Theorem 1.4. We give also some results related to solutions of the problem (1)-(4) in the appendix.
2. Inverse source problem with traction-free boundary condition. This section is devoted to the proofs of Theorems 1.1 and 1.2 for our inverse problem (IP1). More precisely, we consider the radiation of an elastic source in an inhomogeneous medium in the exterior of a cavity D (see Figure 1): where L λ,µ U stands for the Lamé operator given by (2). Figure 1. Radiation of a source in an inhomogeneous isotropic elastic medium in the exterior of a cavity. Suppose that the cavity D is known. The inverse problem is to determine the source term from the data measured on ∂B R = {x ∈ R 3 : |x| = R}.
Together with the governing equation (9), we fix the initial conditions at t = 0: and the traction-free boundary condition T U on ∂D given by where ν = (ν 1 , ν 2 , ν 2 ) stands for the unit normal direction pointing into the exterior of D and the stress tensor σ(U ) is given by Note that I 3 means the 3-by-3 unit matrix and that the conormal derivative σ(U )ν corresponds to the stress vector or surface traction on ∂D. With these notations the Lamé operator (2) can be written as L λ,µ U = div σ(U ).
We suppose that D ⊂ R 3 is a bounded domain with C 3 -smooth boundary ∂D and with connected exterior R 3 \D. If the cavity D is absent (i.e., D = ∅) and the background medium is homogeneous and isotropic, it was shown in [6] via strong Huygens' principle and Fourier transform that the boundary data of Theorem 1.2 can be used to uniquely determine g. According to [35], in the context of Theorem 1.2, the strong Huygens principle is not valid and we can not even expect integrable local energy decay. For this purpose, we use a different approach based on application of Laplace transform for Theorem 1.2 and unique continuation properties for Theorem 1.1.
Let us first consider the proof of Theorem 1.1 stated with non-vanishing sources at t = 0.
Proof of Theorem 1.1. Assuming U (x, t) = 0 for |x| = R and t ∈ (0, T 1 ), we need to prove that h ≡ 0. Since Supp(h) ⊂ B R , the wave field U fulfills the homogeneous initial and boundary conditions of the Lamé system in the exterior of B R : Applying the elliptic regularity properties of L λ,µ (see e.g., [ (13).
Let us now consider the initial boundary value problem Analogous to the boundary value problem (13), one can prove that this exterior problem admits a unique solution V ∈ C([0, +∞); . Moreover, one can easily check that the solution U to (9) is connected with V via (which is well-known as Duhamel's principle) Combining (15) with the fact that Using the fact that f ∈ C 1 ([0, T ]), we can differentiate this expression with respect to t in order to get Combining this with the fact that f (0) = 0, we obtain Therefore, applying the Gronwall inequality, we deduce that From this, we deduce that Combining this with the fact that Combining the unique continuation result of [ (17) implies Combining this with (16), we get and differentiating with respect to t, we get Therefore, V restricted to (R 3 \D) × (0, T 2 + ) solves the initial boundary value problem The uniqueness of the solution of this problem implies that from which we deduce that h ≡ 0. Now let us consider Theorem 1.2, where we allow f (0) = 0 but we make measurements for all time.
Proof of Theorem 1.2. Assuming U (x, t) = 0 for |x| = R and t ∈ R + , we need to prove that h ≡ 0. Repeating the arguments used at the beginning of Theorem 1.1, one can check that Then, repeating the arguments used in Theorem 1.1 with some minor modifications, we can prove that with V the solution of (14). Since f (t) ≡ 0 in t < 0, the identity (18) can be rewritten as (19) f where the operator * denotes the convolution. For R 1 > R, we fix Ω 1 := B R1 \B R . Using standard idea for deriving energy estimates, one can prove that t → V (·, t) H 1 (Ω1) has a long time behavior which is at most of polynomial type (see Proposition 3 in the Appendix). This allows us to define the Laplace transform of t → V (·, t) |Ω1 with respect to the time variable as following: and z →V (·, z) |Ω1 is a holomorphic function on C + := {z ∈ C : Rz > 0} taking values in H 1 (Ω 1 ) 3 . Therefore applying the Laplace transform to both sides of (19), we get (20) Using the fact that f ∈ L 1 (R + ) is supported in [0, T ] and it does not vanish identically, we deduce that the functionf is holomorphic in C and not identically zero. Thus, there exists an interval I ⊂ (0, +∞) such that |f (τ )| > 0 for τ ∈ I. Combining this with (20), we deduce that V (x, τ ) = 0, x ∈ Ω 1 , τ ∈ I and using the fact that z →V (·, z) |Ω1 is a holomorphic function on C + := {z ∈ C : Rz > 0}, we deduce thatV Then, the injectivity of the Laplace transform, implies Combining this with the arguments used at the end of Theorem 1.1, we deduce that h ≡ 0.
We remark that surface data are utilized in the proof of Theorem 1.2. As a corollary, we prove that interior volume observations can also be used to extract partial information of the spatial source term. Below we consider again the problem (9)-(11), with f, g being given as in Theorem 1.2. Proof. We need to prove that the condition U (x, t) = 0 for (x, t) ∈ (ω × R + ) ∪ (∂Ω × R + ) implies that h| Ω ≡ 0. Repeating the arguments used in Theorem 1.2, for V the solution of (14), we have In a similar way to Theorem 1.1, combing the unique continuation result of [19,Theorem 5.5] with the global Holmgren theorem stated in [20, Theorem 1.2], we deduce that the condition V (x, t) = 0, x ∈ ω, t > 0 implies that there exists T 3 > 0 such that Combining this with the fact that we deduce that the restriction of V to Ω × R + solves the problem Then, the uniqueness of the solution of this problem implies that In particular, we obtain h(x) = ∂ t V (x, t)| t=0 = 0 in Ω.
Remark 1. Corollary 1 shows that, the volume observation data on ω and the surface measurements on ∂Ω determine uniquely the source term h on Ω. This gives partial information of h only. However, in the special case that Supp(h) ⊂ Ω (for instance, Ω = B R ), one may deduce from Corollary 1 that h can be uniquely determined by the data of U on ω × R + .

Remark 2.
Assuming that f (0) = 0, in a similar way to Theorem 1.1, we can restrict the measurements in Corollary 1 to a finite time depending on the coefficients ρ, λ, µ and the domain Ω, ω.
3. Determination of the source term g(x 3 )f (x, t). In the previous section, we established uniqueness of recovering a spatial source term in an inhomogeneous background medium with or without embedded obstacles. However, the dependance of the source term on time and spatial variables are completely separated. The counterexamples constructed in [6] show that it is impossible to recover general source terms of the form F (x, t) from the boundary observation on ∂B R × (0, ∞). This implies that a priori information on the source term is always necessary in proving uniqueness. In this section we restrict our discussions to the inverse problem (IP2) for alternative source terms of the form g( where the vectorial function f = (f 1 , f 2 , 0) is compactly supported on [0, T )×B R and the scalar function g is supported in (−R, R) for some R > 0. We recall that herex = ( For simplicity, we assume in this section that D = ∅ and the background medium is homogeneous with constant Lamé coefficients λ, µ and a constant density function ρ. Below we shall consider the initial value problem The function g(x 3 ) f (x, t) can be used to model source terms which mainly radiate over the ox 1 x 2 -plane and g(x 3 ) can be regarded as an approximation of the delta function δ(x 3 ) in the x 3 -direction. Suppose that the function g is known in advance. Our inverse problem in this section is concerned with the recovery of f from U (x, t) measured on Γ × (0, T 1 ) for some T 1 > 0, R 1 > √ 2R and Γ an open subset of ∂B R1 . The proofs of the uniqueness and stability results (Theorems 1.3 and 1.4) will be presented in the subsequent two subsections.
we introduce the test functions The numbers k p and k s denote respectively the compressional and shear wave numbers in the frequency domain. One can easily check that ∇ ⊥ x · V p ≡ 0, ∇x · V s ≡ 0 in R 3 and, using the fact that we deduce that V α (α = p, s) satisfies the homogeneous Lamé system in the frequency domain for any fixed ω ∈ R. Now, multiplying V p to both sides of the equation (23) and applying Betti's formula, we obtain where we have used the vanishing of the Cauchy data ofÛ on ∂B R1 . On the other hand, making use of (24) together with the relation ∇ ⊥ x · V p ≡ 0 yields for all ω ∈ R and ξ ∈ R 2 satisfying |ξ| > k s . Since g is compactly supported and lies in the space L 1 ((−R, R)), the function is holomorphic in C. Then, using the fact that g is not uniformly vanishing, for every ω ∈ R, we can find an open and not-empty interval I ω ⊂ (k s , +∞) such that Hence, for every ω ∈ R, we have This implies that, for ω ∈ R and forf p (·, ω) :x →f p (x, ω), the Fourier transform Fx[f p ](ξ) off p (·, ω) with respect tox ∈ R 2 vanishes for ξ ∈ {η ∈ R 2 : |η| ∈ I ω }. On the other hand, since, for all ω ∈ R,f p (·, ω) is supported inB R , the function is real analytic with respect to ξ ∈ R 2 . Then, using the fact that the set {ξ ∈ R 2 : |ξ| ∈ I ω } is an open subset of R 2 , it follows from (25) that Applying the inverse Fourier transform inx, we getf p (·, ω) = 0 for all ω ∈ R. Further, applying the inverse Fourier transform in t yields f p (x, t) ≡ 0 for allx ∈B R and t > 0. The fact that f s ≡ 0 can be verified analogously by multiplying V s to both sides of (23). This finishes the proof of the relation f ≡ 0 inB R × (0, T ).
To derive a stability estimate of f , we need the dynamic data measured over the whole boundary ∂B R . In contrast with the proof of Theorem 1.3 , we shall carry out the proof of Theorem 1.4 in the time domain without using the Fourier transform in the time variable. (24), we can split f via Hodge decomposition into the form

Proof of Theorem 1.4. As done in
where f p (·, t) and f s (·, t) are scalar functions compactly supported onB R . Fixing ω > 0 and ξ ∈ R 2 such that (27) |ξ| 2 > k 2 p := we introduce the time-dependent test function In the same way, for we introduce the function Then, in a similar way to the proof of the uniqueness result, one can check that V α (α = p, s) are solutions to the homogeneous elastodynamic equation for any fixed ξ ∈ R 2 and ω ∈ R satisfying (27) or (28). Moreover, one can easily check that Therefore, multiplying V p (x, t; ξ, ω) to the right hand side of the equation (22), using (29) and applying integration by parts yield Again recalling Huygens principle, we know U (x, T 1 ) = ∂ t U (x, T 1 ) = 0 for all x ∈ B R and T 1 > T + 2R1 √ ρ √ µ . Hence, the integral over B R1 on the right hand side of the previous identity vanishes. Following estimate (42) of Proposition 2 in the appendix, the traction of U on the boundary ∂B R1 can be bounded by the trace of U itself. Hence, the left hand side can be bounded by for all |ξ| > k p , where C > 0 depends on M , R 1 , T 1 , ρ, λ and µ. On the other hand, using the governing equation (22) together with the relations (26), (30) and using the fact that the sign of g is constant, we obtain a lower bound of the left hand side of (31): for all |ξ| > k p . Since f p is supported onB R × (0, T 1 ), the first integral on the right hand of the last identity is the Fourier transform of f p with respect to (x, t) at the value (ξ, ω), which we denote byf p (ξ, ω). Combining the previous two relations we obtain for all |ξ| > k p (ω). We note that (32) gives the estimate off p over the cone {(ξ, ω) ∈ R 3 : |ξ| 2 > ω 2 ρ/(λ + 2µ)}. In order to derive from (32) a stability estimate off p on B r for a large r > 0, we will use a result of stability in the analytic continuation, following the arguments presented in [16,36]. Below we state a stability estimate for analytic continuation problems; see [5,Theorem 4] (see also [43,45], where similar results were established).
Proposition 1. Let s > 0 and assume that g : B 2s ⊂ R 3 → C is a real analytic function satisfying for some N > 0 and 0 < τ ≤ 1. Further let E ⊂ B s be a measurable set with strictly positive Lebesgue measure. Then, , where b ∈ (0, 1), C > 0 depend on τ , |E| and s.
Following [36], we introduce the function for some r > 1 and |(ξ, ω)| ≤ 2s. In a similar way to [36], we fix s = [max(T 1 , 2R)] −1 + 1, choose N = Ce 3r , with C some constant independent of r, and take τ = Moreover, fixing c := ρ λ+2µ , d := s 2 √ 1+c −1 and a r ∈ 0, d √ c , we define It is easy to check that E r is a subset of B s in R 3 , and it is also a subset of the cone {(ξ, ω) ∈ R 3 : |ξ| 2 > ω 2 ρ/(λ + 2µ)}. We remark that |E r | = κ r (−a r ), where Note that κ r − d √ c = 2κ r (0) and one can check that Thus, there exists r 0 > 1 depending only on R, ρ, λ, µ, T , such that Therefore, we have and, from the continuity of the map κ r , we deduce that we can choose a r ∈ 0, d √ c in such way that This implies that, with such choice of a r , the volume |E r | depends only on R, ρ, λ, µ and T 1 . Consequently, combining (33) with Proposition 1, we deduce that where C > 0, b ∈ (0, 1) depend only on R, ρ, λ, µ and T 1 . In addition, applying (32), we get where C and c 1 depend only on R, ρ, λ, µ and T 1 . Therefore, we can find C, c depending only on R, ρ, λ, µ and T 1 such that It follows that by eventually replacing the constants C and c. On the other hand, using (7) and the fact that ∆xf p = ∇x · f , we deduce that f p ∈ H 2 (R 3 ) and f p H 2 (R 3 ) ≤ CM , with C depending only on T and R. Thus, we find Combining this with (34), we find Recalling the Plancherel formula, it holds that which can be obtained by applying the classical arguments of optimization (see for instance the end of the proof of [36,Theorem 1]). This gives the estimate of f p by our measurement data taken on ∂B R1 . Using similar arguments, we can prove (36) On the other hand, by interpolation and the upper bound (7) we have where both f and g are compactly supported scalar functions and c is a constant. If the wave speed c and the function g(x 3 ) are known, one can determine the source term f (x, t) from partial boundary data. In particular, f is allowed to be a moving source with the orbit lying on the ox 1 x 2 -plane. In the frequency domain, the above wave equation gives rise to an inverse problem of recovering the wave-numberdependent source term f (x, k) from the multi-frequency boundary observation data of the inhomogeneous Helmholtz equation Progress along these directions will be reported in our forthcoming publications (see e.g., [24]).

Appendix.
4.1. Well-posedness result and estimation of surface traction. In this subsection, we consider the inhomogeneous Lamé system where the operator L λ,µ is given by (9). We assume that supp(F ) ⊂ [0, T ) × B R , with B R := {x ∈ R 3 : |x| < R}. It is well known that the operator L λ,µ is an elliptic operator and the standard elliptic regularity holds; see e.g., [44,Chapters 4 and 10] and [22,Chapter 5]. The quadratic form corresponding to L λ,µ is given by where the stress tensor E is defined via (12), with the notation A : for all U, V ∈ H 2 (D) 3 . By the well-known Korn's inequality (see e.g. [44,Theorem 10.2], [18,Chapter 3]), it holds that for some constants c 1 , c 2 > 0. In the particular case of constant Lamé coefficients, we have In this case, the surface traction can be simplified to be We refer to the monograph [40] for comprehensive studies on the Lamé system. Below we state a well-posedness result to the elastodynamic system in unbounded domains by applying the standard arguments of [41,Chapter 8]. 3 . Then problem (37) admits a unique solution Under the additional condition F (·, 0) = x → F (0, x) = 0, the unique solution lies in the space U ∈ C 2 ([0, +∞); L 2 (R 3 )) 3 ∩ C([0, +∞); H 2 (R 3 )) 3 .
Combining the last two estimates we finally obtain (42).