THE CALDER ´ON PROBLEM FOR THE FRACTIONAL WAVE EQUATION: UNIQUENESS AND OPTIMAL STABILITY

EQUATION: UNIQUENESS AND OPTIMAL STABILITY

PU-ZHAO KOW, YI-HSUAN LIN, AND JENN-NAN WANG

Abstract. We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and sta- bility estimate in the determination of the potential by the exterior Dirichlet- to-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial dimension n ∈ N.

Keywords. Calder´on problem, peridynamic, fractional Laplacian, nonlocal, fractional wave equation, strong uniqueness, Runge approximation, logarith- mic stability.

Mathematics Subject Classification (2020): 35B35, 35R11, 35R30

Contents

1. Introduction 2

2. The forward problems for the fractional wave equation 6

2.1. Sobolev spaces 6

2.2. The forward problem 7

2.3. The DN map and its duality 8

3. Global uniqueness for the fractional wave equation 9

3.1. Qualitative Runge approximation 10

3.2. Proof of Theorem 1.1 11

4. Stability for the fractional wave equation 11

4.1. Logarithmic stability of the Caffarelli-Silvestre extension 11

4.2. Quantitative unique continuation 14

4.3. Quantitative Runge approximation 16

4.4. Proof of Theorem 1.2 19

5. Exponential instability of the inverse problem 19 5.1. Matrix representation via an orthonormal basis 20

5.2. Special weak solutions 20

5.3. Matrix representation 22

5.4. Construction of a family of δ-net 23

5.5. Construction of an -discrete set 26

5.6. Proof of Theorem 1.3 27

Appendix A. Proofs related to the forward problem 31

References 36

1

1. Introduction

In this paper, we study an inverse problem for the fractional wave equation with a potential. The mathematical model for the fractional wave equation is formulated as follows. Let Ω ⊂ Rⁿ be a bounded Lipschitz domain, for n ∈ N. Given T > 0, s ∈ (0, 1) and q = q(x) ∈ L^∞(Ω), consider the initial exterior value problem for the wave equation with the fractional Laplacian,







∂_t²+ (−∆)^s+ q
u = 0 in ΩT := Ω × (0, T ), u = f in (Ω_e)_T := Ω_e× (0, T ), u = ∂tu = 0 in Rⁿ× {0},

(1.1)

where (−∆)^s is the standard fractional Laplacian¹, and Ω_e:= Rⁿ\ Ω

denotes the exterior domain. The fractional wave equation can be regarded as a special case of the peridynamics which models the nonlocal elasticity theory, see e.g. [Sil16].

In recent years, inverse problems involving the fractional Laplacian have received a lot of attention. Ghosh-Salo-Uhlmann [GSU20] first proposed the Calder´on prob- lem for the fractional Schr¨odinger equation, and the proof relies on the strong uniqueness of the fractional Laplacian ([GSU20, Theorem 1.2]) and the Runge ap- proximation ([GSU20, Theorem 1.3]). Based on these two useful tools, there are many related works appeared in past few years, such as [BGU21, CLL19, CL19, CLR20,GLX17,GRSU20,HL19,HL20,LL20,LL19,RS20,LLR20,Lin20] and the references therein.

Throughout this work, we assume that the (lateral) exterior data f is compactly supported in the set W_T := W × (0, T ) ⊂ (Ω_e)_T, where W ⊂ Ω_ewith W ∩ Ω = ∅ can be any nonempty open subset with Lipschitz boundary, and, to simplify our notations, we assume that both q and f are real-valued functions. Note that the initial boundary value problem (1.1) is a mixed local-nonlocal type equation. In order to study the inverse problem of (1.1), we will use the strong approximation property of (1.1), which is due to the nonlocality of the fractional Laplacian (−∆)^s, for 0 < s < 1. Hence, by the well-posedness of (1.1) (see Theorem 2.1), one can formally define the associated Dirichlet-to-Neumann (DN) map Λq

(1.2) Λ_q : C_c^∞((Ω_e)_T) → L²(0, T ; H^−s(Ω_e)), Λ_q : f 7→ (−∆)^su|_(Ω

e)_T , where u is the unique solution to (1.1). The precise definitions of the Sobolev spaces will be given in Section 2.1. Let us state the first main result of our work.

Theorem 1.1 (Global uniqueness). Consider T > 0, s ∈ (0, 1), and q_j= q_j(x) ∈ L^∞(Ω), for j = 1, 2. Assume that W₁, W₂ ⊂ Ω_e are arbitrary open sets with Lipschitz boundary such that W1∩ Ω = W2∩ Ω = ∅. Let Λq_j be the DN map of







∂_t²+ (−∆)^s+ q_j
u = 0 in Ω_T,

u = f in (Ωe)T,

u(x, 0) = ∂_tu(x, 0) = 0 in Rⁿ× {0}, (1.3)

for j = 1, 2. If

Λq₁(f )|_(W

2)T = Λq₂(f )|_(W

2)T, for any f ∈ C_c^∞((W1)T), (1.4)

then q1= q2 in ΩT2.

1A rigorous definition is given in Section2.

2Throughout this paper, we adapt the notation AT:= A × (0, T ), for any set A ⊂ Rⁿ.

The proof of Theorem 1.1 is based on the qualitative form of the Runge ap- proximation for the fractional wave equation: For any g ∈ L²(ΩT), there exists a sequence of functions {fk}_k∈N ∈ C_c^∞((W1)T) such that uk → g in L²(ΩT) as k → ∞, where uk is the solution to (1.1) with uk = fk in (Ωe)T, for all k ∈ N.

The preceding characterization can be regarded as an exterior control approach, in the sense that one can always control the solution by choosing appropriate exterior data.

The second main result of the paper is a quantitative version of Theorem 1.1, which provides a stability estimate for our fractional Calder´on problem. Before we state the stability result, we introduce some notations.

Definition 1.1. Let H²(0, T ; eH^α(Ω)) be the Sobolev space equipped with the norm kuk_H2(0,T ; eH^α(Ω))= kuk_L2(0,T ; eH^α(Ω))+ k∂tuk_L2(0,T ; eH^α(Ω))+ k∂_t²uk_L2(0,T ; eH^α(Ω)). We also denote

H₀²(0, T ; eH^α(Ω)) :=n

u ∈ H²(0, T ; eH^α(Ω)) : u = ∂tu = 0 in Rⁿ× {0}o , and let H⁻²(0, T ; H^−α(Ω)) be the dual space of H₀²(0, T ; eH^α(Ω)). We shall explain the space eH^α(Ω) in more detail later in Section 2.

Definition 1.2. For each α > 0 and T > 0, we define kqk_Z^−α_{(Ω;T )}:= supn

  R

Ω_Tq(x, t)φ1(x, t)φ2(x, t) dx dt    o

, where the supremum is taken over all functions φ1, φ2∈ C_c^∞(ΩT) with

kφjk_H2(0,T ; eH^α(Ω))= 1 (j = 1, 2), and let Z^−α(Ω; T ) be the Banach space equipped with this norm.

Remark 1.3. Since kφ_jk_L2(Ω_T)≤ kφ_jk_H2(0,T ; eH^α(Ω))= 1 for all φ ∈ H₀²(0, T ; eH^α(Ω)), α > 0 and T > 0, it is easy to see that

kqk_Z−α(Ω;T )≤ kqk_L∞(ΩT)

for all q = q(x, t), which implies L^∞(Ω_T) ⊂ Z^−α(Ω; T ).

To shorten our notations, we denote the operator norm as k·k∗= k·k_L2(0,T ;H^2s

W)→L²(0,T ;H^−2s(W )), where the Sobolev space H^2s

W will be described in Section 2.1. We are now ready to state the second main result of our work.

Theorem 1.2 (Logarithmic stability). Let T > 0, s ∈ (0, 1), and q_j = q_j(x) ∈ L^∞(Ω), for j = 1, 2. Assume that W₁, W₂⊂ Ωebe arbitrary open sets with Lipschitz boundary such that W1∩ Ω = W2∩ Ω = ∅. Let Λq_j be the DN map of (1.3) for j = 1, 2. We also fix a regularizing parameter γ > 0. If q1 and q2 both satisfy the apriori bound

kqjk_L^∞_(Ω)≤ M for j = 1, 2, then

kq1− q2k_Z−s−γ(Ω;T )≤ ω (kΛq₁− Λq₂k_∗) where ω satisfies

ω(t) ≤ C| log t|^−σ, 0 ≤ t ≤ 1,

for some constants C and σ depending only on n, s, Ω, W1, W2, γ, T, M .

Inspired by Theorem 1.1, we will prove Theorem 1.2 by using a quantitative version of Runge approximation, which involves the well-known Caffarelli-Silvestre extension for the fractional Laplacian and the propagation of smallness. Moreover, Theorem 1.1and Theorem1.2are satisfied for any spatial dimension n ∈ N.

The third main result of this work studies the exponential instability of the Calder´on problem for the fractional wave equation. In other words, the stability result in Theorem 1.2is optimal. For brevity, we denote the operator norm

kAk⁰_∗= sup

06≡χ∈C^∞_c ((0,T ))

sup_{t∈(0,T )}kχAχk_L2(B₃\B₂)→L²(B₃\B₂)(t) kχk²_W2,∞(0,T )

, where Br with r > 0 stands for the ball of radius r centered at the origin.

Theorem 1.3 (Exponential instability I). Let Ω = B₁ ⊂ Rⁿ, for n ≥ 2, n ∈ N.

Given any T > 0, s ∈ (0, 1), α > 0 and R > 0. There exists a positive constant cR,T ,n,ssuch that: Given any 0 <  < cR,T ,n,s, there exist potentials q1, q2∈ C^α(Ω) such that kqjk_L∞(Ω)≤ R, j = 1, 2, satisfying

(1.5) kq1− q2k_L^∞_(Ω)≥ ,

but

(1.6) kΛq1− Λq2k⁰_∗≤ KR,T ,n,sexp

−^−(2n+1)αn  for some positive constant KR,T ,n,s.

For 1-dimensional case (n = 1), we can also establish the same estimate.

Theorem 1.4 (Exponential instability II). For n = 1, Theorem 1.3 is also valid with the norm k · k⁰_∗ being replaced by the following norm:

kAk⁰⁰_∗ := sup

06≡χ∈C_c^∞((0,T ))

sup_{t∈(0,T )}kχAχkL²((2,3))→L²((2,3))(t) kχk²_W2,∞(0,T )

.

For the local counterpart, let us consider the following initial boundary value problem for the local wave equation:







∂²_t− ∆ + q(x)
u = 0 in Ω_T,

∂νu(x, t) = g(x, t) in (∂Ω)T, u = ∂_tu = 0 in Ω × {0}, (1.7)

where q = q(x) ∈ L^∞(Ω). It is known that (1.7) is well-posed (for example, see [Eva98]) with suitable compatibility conditions. Assuming the well-posedness of (1.7), the corresponding (hyperbolic) Neumann-to-Dirichlet map of (1.7) is defined by

Λeqg := u|_{∂Ω×[0,T ]} for all g ∈ C_c^∞((∂Ω)T).

In fact, eΛq : L²(∂Ω × (0, T )) → H¹(0, T ; L²(∂Ω)) is a bounded linear operator, which can be proved by the energy estimate of (1.7), see e.g. [CP82, Section 6.7.5].

Now we assume

(1.8) T > diam (Ω).

Under assumption (1.8), in [RW88], they showed the global uniqueness result for time-independent potentials:

Λe_q1 = eΛ_q2 implies q₁= q₂in Ω.

In [Sun90], the author showed that, if (1.8) holds, under some apriori assumptions, the following estimate hold:

(1.9) kq₁− q₂k_L2(Ω)≤ C

Λe_q1− eΛ_q2

α L

for some constants C and α, where k·k_L stands for the operator norm for the Neumann-to-Dirichlet map. A similar estimate also holds for the hyperbolic Dirichlet- to-Neumann map [AS90]. In other words, the stability of the inverse problem for the local wave equation is of H¨older-type. We also mention other related results of inverse problems for the local wave equation with potentials [Esk06,Esk07, Isa91, Kia17,RS91,Sal13].

Similar to the local version, we can prove the global uniqueness result for time- independent potentials for the fractional wave equation (see Theorem 1.1). How- ever, in the nonlocal counterpart of (1.9), we show that the stability of the inverse problem for the fractional wave equation is of (optimal) logarithmic-type in view of Theorem 1.2and Theorem1.3. We also want to point out that we do not need to assume the large influence time condition (1.8). One possible explanation is that while the speed of propagation of the local wave equation is finite, the speed of prop- agation of the fractional wave operator is infinite due the nonlocal nature of the fractional Laplacian (−∆)^s, for 0 < s < 1. We will offer some detailed arguments in Section2.

Before ending this section, we would like to discuss some interesting results for the time-harmonic wave equation. Consider the time-harmonic wave equation with a potential (a.k.a. Schr¨odinger equation):

(1.10) −∆ + q(x) − κ²
v = 0 in Ω.

Ignoring the effect of the frequency κ > 0, Alessandrini [Ale88] proved the well- known logarithmic stability estimate for the inverse boundary value problem of (1.10), and Mandache [Man01] established that this logarithmic estimate is optimal by showing that the inverse problem is exponentially unstable. Nonetheless, by taking the frequency into account, it was shown in [INUW14] that

(1.11) kq₁− q₂k_H−α(Rⁿ)≤ C



κ + log 1

dist (Cq1, Cq2)

−2α−n

+ Cκ⁴dist (C_q1, C_q2) , where C_q1, C_q2are the Cauchy data of the Schr¨odinger equation (1.10) corresponding to q₁, q₂, and dist (C_q1, C_q2) is the Hausdorff distance between C_q1 and C_q2. Isakov [Isa11] proved a similar estimate in terms of the DN maps.

The estimate (1.11) is shown to be optimal in the recent paper [KUW21]. The estimate (1.11) clearly indicates that the logarithmic part decreases as the frequency κ > 0 increases and the estimate changes from a logarithmic type to a H¨older type.

This phenomena is termed as the increasing stability. It is interesting to compare the stability estimate (1.11) of the time-harmonic wave equation (1.10) with the stability estimate (1.9) of the local wave equation (1.7).

Similarly to the local wave equation, we consider the following time-harmonic fractional wave equation

(1.12) (−∆)^s+ q(x) − κ²
v = 0 in Ω,

which is a fractional Schr¨odinger equation. Without considering the effect of the frequency κ > 0, R¨uland and Salo [RS20] obtained a logarithmic type stability estimate for the inverse boundary value problem of the time-harmonic fractional wave equation (1.12) and, in [RS18], they proved that such logarithmic estimate is optimal by showing the exponential instability phenomenon. These results give rise to a natural question: in the inverse boundary value problem for (1.12), if we take the frequency κ into account, does the increasing stability estimate similar to (1.11) hold? In view of the optimal logarithmic stability results in Theorem1.2and Theorem 1.3, we have a strong reason to believe that the answer to this question is negative.

The paper is organized as follows. We discuss and prove the well-posedness of the fractional wave equation in Section 2 and in Appendix A, respectively. We then prove Theorem 1.1 in Section 3, and prove Theorem1.2 in Section 4. The approach is mainly based on the qualitative and quantitative Runge approximation properties for the fractional wave equation. Finally, we prove Theorem 1.3 and Theorem 1.4in Section5.

2. The forward problems for the fractional wave equation In this section, we provide all preliminaries that we need in the rest of the paper.

Let us first recall (fractional) Sobolev spaces and prove the well-posedness of the fractional wave equation (1.1).

2.1. Sobolev spaces. Let F , F⁻¹ be Fourier transform and its inverse, respec- tively. For s ∈ (0, 1), the fractional Laplacian is defined via

(−∆)^su := F⁻¹ |ξ|^2sF (u)
, for u ∈ H^s(Rⁿ),

where H^s(Rⁿ) stands for the L²-based fractional Sobolev space (see [DNPV12, Kwa17, Ste16]). The space H^a(Rⁿ) = W^a,2(Rⁿ) denotes the (fractional) Sobolev space equipped with the norm

kukH^a(Rⁿ):=

F⁻¹{hξi^aF u}

L²(Rⁿ),

for any a ∈ R, where hξi = (1 + |ξ|²)¹². It is known that for s ∈ (0, 1), k · k_Hs(Rⁿ)

has the following equivalent representation

kukH^s(Rⁿ):= kuk_L2(Rⁿ)+ [u]_Hs(Rⁿ)

where

[u]²_Hs(O):=

Z

O×O

|u(x) − u(y)|²

|x − y|^n+2s dxdy, for any open set O ⊂ Rⁿ.

Given any open set O of Rⁿand a ∈ R, let us define the following Sobolev spaces, H^a(O) := {u|_O; u ∈ H^a(Rⁿ)},

He^a(O) := closure of C_c^∞(O) in H^a(Rⁿ), H₀^a(O) := closure of C_c^∞(O) in H^a(O), and

H_O^a := {u ∈ H^a(Rⁿ); supp(u) ⊂ Ω}.

In addition, the Sobolev space H^a(O) is complete under the norm kuk_Ha(O):= infkvkH^a(Rⁿ); v ∈ H^a(Rⁿ) and v|_O= u . It is not hard to see that eH^a(O) ⊆ H₀^a(O), and that H^a

O is a closed subspace of H^a(Rⁿ). We also denote H^−s(O) to be the dual space of eH^s(O). In fact, H^−s(O) has the following characterization:

H^−s(O) =u|_O : u ∈ H^−s(Rⁿ)

with inf

w∈H^s(Rⁿ),w|O=ukwkH^s(Rⁿ), see e.g. [GSU20, Section 2.1], [McL00, Chapter 3], or [Tri02] for more details about the fractional Sobolev spaces. Moreover, we will use

(f, g)_L2(A):=

Z

A

f g dx, (F, G)_L2(A_T):=

Z T 0

Z

A

F G dxdt, in the rest of this paper, for any set A ⊂ Rⁿ.

2.2. The forward problem. We first state the well-posedness of the fractional wave equation. As above, let Ω ⊂ Rⁿ be a bounded Lipschitz domain with n ∈ N.

Given T > 0, s ∈ (0, 1), and q = q(x) ∈ L^∞(Ω), consider the initial exterior value problem for the fractional wave equation







∂²_t+ (−∆)^s+ q
u = F in ΩT,

u = f in (Ω_e)_T,

u = ϕ, ∂tu = ψ in Rⁿ× {0}, (2.1)

where f ∈ C_c^∞(W_T) for some open set with Lipschitz boundary W ⊂ Ω_esatisfying W ∩ Ω = ∅, ϕ ∈ eH^s(Ω), and ψ ∈ L²(Rⁿ) with supp (ψ) ⊂ Ω. We want to show the well-posedness of (2.1). Setting v := u − f , we then consider the fractional wave equation with zero exterior data







∂_t²+ (−∆)^s+ q
v =Fe in ΩT,

v = 0 in (Ω_e)_T,

v =ϕ,e ∂tv = eψ in Rⁿ× {0}, (2.2)

where eF := F − (−∆)^sf , ϕ(x) = ϕ(x) − f (x, 0) = ϕ(x) and ee ψ(x) = ψ(x) −

∂tf (x, 0) = ψ(x). Hence, we simply denote the initial data as (ϕ, ψ) in the rest of the paper. Now it suffices to study the well-posedness of (2.1).

Let us introduce the following notations. Define u : [0, T ] → eH^s(Ω) by

[u(t)](x) := u(x, t), for x ∈ Rⁿ, t ∈ [0, T ].

Similarly, the function eF : [0, T ] → L²(Ω) can be defined analogously by [ eF (t)](x) := eF (x, t), for x ∈ Rⁿ, t ∈ [0, T ].

With these notations at hand, we can define the weak formulation for the fractional wave equation. Let φ ∈ eH^s(Ω) be any test function, multiplying (2.1) with φ gives

(v⁰⁰, φ)_L2(Ω)+ B[v, φ; t] = ( eF , φ)_L2(Ω), for 0 ≤ t ≤ T, where B[v, φ; t] is the bilinear form defined via

B[v, φ; t] :=

Z

Rⁿ

(−∆)^s/2v(−∆)^s/2φ dx + Z

Ω

qvφ dx.

Definition 2.1 (Weak solutions). A function

v ∈ L²(0, T ; eH^s(Ω)), with v⁰∈ L²(0, T ; L²(Ω)) and v⁰⁰∈ L²(0, T ; H^−s(Ω)) is a weak solution of the initial exterior value problem (2.2) if

(1) (v⁰⁰(t), φ)_L2(Ω)+ B[v, φ; t] = F , φe 

L²(Ω), for all φ ∈ eH^s(Ω), and for 0 ≤ t ≤ T a.e.

(2) v(0) =ϕ and ve ⁰(0) = eψ.

Theorem 2.1 (Well-posedness). For any eF ∈ L²(0, T ; L²(Ω)), ϕ ∈ ee H^s(Ω), and ψ ∈ Le ²(Rⁿ) with supp ( eψ) ⊂ Ω, there exists a unique weak solution v to (2.2).

Moreover, the following estimate holds:

kvk_L∞(0,T ; eH^s(Ω))+ k∂tvk_L∞(0,T ;L²(Ω))

≤ C

k eF k_L2(0,T ;L²(Ω))+ kϕke

He^s(Ω)+ k eψk_L2(Ω)

 (2.3) .

Corollary 2.2. Let Ω ⊂ Rⁿ be a bounded Lipschitz domain for n ∈ N, and W ⊂ Ωe

be any open set with Lipschitz boundary satisfying W ∩ Ω = ∅. Then for any F = F (x, t) ∈ L²(0, T ; L²(Ω)), f = f (x, t) ∈ C_c^∞(WT), ϕ ∈ eH^s(Ω), and ψ ∈ L²(Rⁿ) with supp (ψ) ⊂ Ω, there exists a unique weak solution u = v + f of (2.1), where v ∈ L²(0, T ; eH^s(Ω)) ∩ H¹(0, T ; L²(Ω)) is the unique weak solution of (2.2).

Furthermore, we have the following estimate

ku − f k_L∞(0,T ; eH^s(Ω))+ k∂t(u − f )k_L^∞_{(0,T ;L}2(Ω))

≤ C

kF − (−∆)^sf k_L2(0,T ;L²(Ω))+ kϕk

He^s(Ω)+ kψk_L2(Ω)

 . (2.4)

The proof of Theorem 2.1is similar to the well-posedness of the classical wave equation (i.e., s = 1) and, for the sake of completeness, we will give a comprehen- sive proof in Appendix A. In this article, we only consider the time-independent potential q = q(x) ∈ L^∞(Ω). In fact, the well-posedness for a space-time dependent potential q = q(x, t) ∈ L^∞(ΩT) has been studied. We refer to [Bre11, Theorem 10.14] for the well-posedness of the abstract wave equations, and to [DFA20] for the well-posedness result for non-local semi-linear integro-differential wave equations which involve both the fractional Laplacian (in space) and the Caputo fractional derivative operator (in time).

2.3. The DN map and its duality. With the well-posedness at hand, one can define the corresponding DN map (1.2) for the fractional wave equation (1.1). Let us define the solution operator

Pq : C_c^∞(WT) → L²(0, T ; H^s(Ω)), f 7→ u|ΩT, (2.5)

where W ⊂ Ωe is a Lipschitz set with W ∩ Ω = ∅, and u is the solution of (1.1).

Given any ϕ(x, t) defined in (Ωe)T, we define

ϕ^∗(x, t) := ϕ(x, T − t) for all (x, t) ∈ (Ω_e)_T, and we define the following backward DN-map:

Λ^∗_q(f ) := (Λ_q(f ))^∗ for any f ∈ C_c^∞((Ω_e)_T).

Lemma 2.3. Given any q ∈ L^∞(Ω), Λ^∗_q is self-adjoint, that is, Z

(Ωe)T

Λ^∗_q(f1)f2dxdt = Z

(Ωe)T

f1Λ^∗_q(f2) dxdt, for all f1, f2∈ C_c^∞((Ωe)T).

Proof. Let u1= Pqf1and u2= Pqf2. Using integration by parts, we have (2.6)

Z

Ω_T

u1(∂_t²u^∗₂) − (∂_t²u₁)u^∗₂ dxdt = 0.

Therefore, 0 =

Z

Ω_T

u1 ∂_t²u^∗₂+ (−∆)^su^∗₂+ qu^∗₂
− ∂_t²u₁+ (−∆)^su₁+ q(x)u₁
u^∗₂ dxdt

= Z

Ω_T

[u1((−∆)^su^∗₂) − ((−∆)^su1)u^∗₂] dxdt

= Z

(Rⁿ)T

− Z

(Ωe)T

!

[u1((−∆)^su^∗₂) − ((−∆)^su1)u^∗₂] dxdt

= − Z

(Ωe)T

[u1((−∆)^su^∗₂) − ((−∆)^su1)u^∗₂] dxdt

= − Z

(Ω_e)_T

f1Λ^∗_q(f₂) − Λ_q(f₁)f₂^∗ dxdt.

Finally, changing the variable t 7→ T − t, we have (2.7)

Z

(Ωe)T

f1Λ^∗_q(f2) dxdt = Z

(Ωe)T

Λq(f1)f₂^∗dxdt = Z

(Ωe)T

Λ^∗_q(f1)f2dxdt,

which is our desired lemma. 

Since Λ^∗_q is self-adjoint, we can derive the following identity immediately.

Lemma 2.4 (Integral identity). Let q₁, q₂ ∈ L^∞(Ω), and given any f₁, f₂ ∈ C_c^∞((Ω_e)_T). Let u₁ := P_q1f₁ and u₂ := P_q2f₂, where the operator P_q is given in (2.5), for q = q1 and q = q2, respectively. Then

(2.8)

Z

ΩT

(q1− q2)u1u^∗₂dxdt = Z

(Ωe)T

((Λq₁− Λq₂)f1)f₂^∗dxdt.

Proof. Using (2.6), we have Z

Ω_T

(q₁− q2)u₁u^∗₂dxdt

= Z

ΩT

[q1u1u^∗₂− u1(q2u^∗₂)] dxdt

= − Z

Ω_T

(∂_t²u₁+ (−∆)^su₁)u^∗₂− u₁(∂²_tu^∗₂+ (−∆)^su^∗₂) dxdt

= − Z

Ω_T

[((−∆)^su1)u^∗₂− u1((−∆)^su^∗₂)] dxdt

= Z

(Ωe)T

− Z

(Rⁿ)T

!

[((−∆)^su1)u^∗₂− u1((−∆)^su^∗₂)] dxdt

= Z

(Ωe)T

[((−∆)^su1)u^∗₂− u1((−∆)^su^∗₂)] dxdt

= Z

(Ω_e)_T

Λq1(f1)f₂^∗− f1Λ^∗_q2(f2) dxdt.

Combining with (2.7), we obtain Z

Ω_T

(q₁− q₂)u₁u^∗₂dxdt = Z

(Ω_e)_T

[(Λ_q1f₁)f₂^∗− (Λ_q2f₁)f₂^∗] dxdt,

which is our desired lemma. 

3. Global uniqueness for the fractional wave equation

In this section, let us state and prove a qualitative Runge type approximation for the fractional wave equation, and then prove Theorem 1.1. Before further discussion, let us comment on the speeds of propagation of the local and nonlocal wave equations. Given V = V (x) ∈ L^∞(Rⁿ), let u be a solution of

∂_t²− ∆ + V
u = 0 in Rⁿ× (0, ∞).

It is known that if u(x, 0) = φ(x), for x ∈ Rⁿ, such that φ 6≡ 0 and φ is compactly supported, then for every t > 0, the solution u(·, t) has compact support.

On the other hand, the speed of propagation for the fractional wave equation is infinite due to the nonlocal nature of the fractional Laplacian. To prove this rigorously, let us recall the strong uniqueness property for the fractional Laplacian.

Given 0 < s < 1, r ∈ R, if u ∈ H^r(Rⁿ) satisfies u = (−∆)^su = 0 in any nonempty open subset of Rⁿ, then u ≡ 0 in Rⁿ. By this property, we can prove the following lemma.

Lemma 3.1. Given V = V (x) ∈ L^∞(Rⁿ), let u be a solution of (3.1) ∂_t²+ (−∆)^s+ V
u = 0 in Rⁿ× (0, T ), then u does not have a finite speed of propagation.

Proof. Suppose the contrary, that the speed of propagation of (3.1) is finite. If we choose u(x, 0) = φ(x) for some 0 6≡ φ ∈ C_c^∞(Rⁿ), given any T > 0, there exists a bounded set Ω such that

u = 0 in (Ωe)T, therefore, ∂_t²u = 0 in (Ωe)T. Using (3.1), we also have

(−∆)^su = 0 in (Ω_e)_T.

Using the strong uniqueness for the fractional Laplacian, we conclude that u ≡ 0,

which implies φ ≡ 0, this is a contradiction. 

3.1. Qualitative Runge approximation. The qualitative approximation prop- erty is based on the strong uniqueness for the fractional Laplacian ([GSU20, The- orem 1.2]).

Theorem 3.1 (Qualitative Runge approximation). Let Ω ⊂ Rⁿ be a bounded Lip- schitz domain for n ∈ N, and W ⊂ Ωe be an open set with Lipschitz boundary satisfying W ∩ Ω = ∅. For s ∈ (0, 1), let P_q be the solution operator given by (2.5), and define

D := {u|ΩT : u = P_qf, f ∈ C_c^∞(W_T)} . Then D is dense in L²(Ω_T).

Remark 3.2. The Runge approximation plays an essential role in the study of fractional inverse problems, for example, see [GSU20,GLX17, RS20, CLR20] and references therein.

Proof of Theorem 3.1. By using the Hahn-Banach theorem and the duality argu- ments, it suffices to show that if v ∈ L²(ΩT), which satisfies

(Pqf, v)_L2(Ω_T)= 0, for any f ∈ C_c^∞(WT), (3.2)

then v ≡ 0 in ΩT. Now, consider the adjoint wave equation







∂²_t+ (−∆)^s+ q
w = v in Ω_T,

w = 0 in (Ωe)T,

w = ∂tw = 0 in Rⁿ× {T }.

(3.3)

Similar to the proof of Theorem2.1, it is easy to see that (3.3) is well-posed.

For f ∈ C_c^∞(WT), let u and w be the solutions of (1.1) and (3.3), respectively.

Note that u − f is only supported in Ω_T, then we have (Pqf, v)_L2(Ω_T)= u − f, (−∂_t²+ (−∆)^s+ q)w

L²(Ω_T)

= − (f, (−∆)^sw)_L2(W_T), (3.4)

where we have used u is the solution of (1.1), u(x, 0) = ∂tu(x, 0) = 0 and w(x, T ) =

∂tw(x, T ) = 0 for x ∈ Rⁿ in last equality of (3.4) . By using the conditions (3.2) and (3.4), one must have (f, (−∆)^sw)_L2(W_T) = 0, for any f ∈ C_c^∞(WT), which implies that

w = (−∆)^sw = 0 in WT.

Fix any fixed t ∈ (0, T ), the strong uniqueness for the fractional Laplacian (see [GSU20, Theorem 1.2]) yields that w(·, t) = 0 in Rⁿ × {t}, for all t ∈ (0, T ).

Therefore, we derive v = 0 as desired, and the Hahn-Banach theorem infers the

density property. This proves the assertion. 

Remark 3.3. By using similar arguments, one can also consider the well-posedness (Theorem 2.1) and the Runge approximation (Theorem 3.1) also hold for the case q = q(x, t) ∈ L^∞(ΩT). In this work, we are only interested in time-independent potentials q = q(x).

Remark 3.4. For other unique continuation property for the fractional elliptic op- erators, we refer the reader to [FF14,GFR19,R¨ul15,Yu17] and references therein.

3.2. Proof of Theorem 1.1. With the help of Lemma2.4 and Theorem3.1, we can prove the global uniqueness of the inverse problem for the fractional wave equation.

Proof of Theorerm 1.1. Given any g ∈ L²(ΩT), using Theorem3.1, there exists a sequence f1,k∈ C_c^∞((W1)T) such that

k→∞lim ku1,k− gk_L2((0,T )×Ω)= 0, where u1,k= Pqf1,k.

Since 1 ∈ L²(Ω_T), similarly, we can choose a sequence f_2,k∈ C_c^∞((W₂)_T) such that lim

k→∞

u^∗_2,k− 1

L²((0,T )×Ω)= 0, where u_2,k= P_qf_2,k. Combining (1.4) and (2.8), we know that

Z

Ω_T

(q₁− q2)u_1,ku^∗_2,kdxdt = 0.

Taking the limit k → ∞, we obtain Z

Ω_T

(q1− q2)g dxdt = 0.

Finally, by the arbitrariness of g ∈ L²(ΩT), we conclude that q1= q2 in ΩT. 

4. Stability for the fractional wave equation

In order to understand the stability estimate for the fractional wave equation, let us recall the famous Caffarelli-Silvestre extension [CS07] for the fractional Lapla- cian. For each x⁰ ∈ Rⁿ and xn+1 ∈ Rⁿ⁺¹+ , we denote x = (x⁰, xn+1) ∈ Rⁿ× R+ = Rⁿ⁺¹+ . Fixing any 0 < s < 1 and t ∈ (0, T ). If there exists γ ∈ R such that v(t) = v(x⁰, t) ∈ H^γ(Rⁿ), using [RS20, Lemma 4.1], there exists a Caffarelli- Silvestre extension v^cs(t) = v^cs(x⁰, x_n+1, t) ∈ C^∞(Rⁿ⁺¹+ ) of v satisfies

∇ · x^1−2s_n+1∇v^cs= 0 in Rⁿ⁺¹+ , v^cs= v on Rⁿ× {0}, lim

xn+1→0x^1−2s_n+1∂_n+1v^cs= −a_n,s(−∆)^sv, where an,s:= 21−2s Γ(1−s)

Γ(s) and ∇ = (∇x⁰, ∂x_n+1) = (∇⁰, ∂n+1).

4.1. Logarithmic stability of the Caffarelli-Silvestre extension. We now define

Ω :=ˆ



x ∈ Rⁿ× {0} : dist (x, Ω) < 1

2dist (Ω, W )

 .

We now prove a lemma, which concerns the propagation of smallness for the Caffarelli-Silvestre extension. By using similar ideas as in [RS20, Section 5], we can derive the following boundary logarithmic stability estimate.

Lemma 4.1. Let W ⊂ Ω_e be an open bounded Lipschitz set such that W ∩ Ω = ∅.

Let v^cs(x⁰, xn+1, t) be the Caffarelli-Silvestre extension of v(x⁰, t). Define η(t) :=

x_n+1lim→0x^1−2s_n+1∂n+1v^cs(t) _H−s(W )

= ask(−∆)^sv(t)k_H^−s_{(W )}

Suppose that there exist constants C₁> 1 and E > 0 such that η(t) ≤ E and (4.2)

x

1−2s 2

n+1v^cs

_{L∞(0,T ;L2}

(Rⁿ×[0,C1]))+ x

1−2s 2

n+1∇v^cs

_{L∞(0,T ;L2}

(Rⁿ⁺¹₊ ))≤ E, then

(4.3)

x

1−2s 2

n+1v^cs(t)

_{L2( ˆΩ×[0,1])}≤ CE log^−µ CE η(t)



for some constants C > 1 and µ > 0, both depending only on n, s, C1, Ω, W . More- over, given any γ > 0, we have

(4.4)

x

1−2s 2 +γ

n+1 ∇v^cs(t)

_{L2( ˆΩ×[0,1])}≤ CE log^−µ CE η(t)

 ,

for some constants C > 1 and µ > 0, both depending only on n, s, C1, Ω, W , as well as γ.

Proof. Estimate (4.3) is an immediate consequence of [RS20, (5.3) of Theorem 5.1].

Replacing [RS20, (5.67) of Theorem 5.1] by the following inequality:

x

1−2s 2 +γ

n+1 ∇⁰v^cs(t)

_{L2( ˆΩ×[0,h])}

≤C x

1−2s 2

n+1∇⁰v^cs(t)

_{L2( ˆΩ×[0,h])}

x^γ_n+1

_{L∞(Ω×[0,h])}

≤Ch^γE, and

x

1−2s 2 +γ

n+1 ∂n+1v^cs(t)

_{L2( ˆΩ×[0,h])}

≤C x

1−2s 2

n+1∂n+1v^cs(t)

_{L2( ˆΩ×[0,h])}

x^γ_n+1

_{L∞(Ω×[0,h])}

≤Ch^γE,

we can prove (4.4) using the similar argument as in the proof of [RS20, (5.5), Theorem 5.1], with a slight modification as indicated above.  Remark 4.2. In view of [RS20, Lemma 4.2], we have

x

1−2s 2

n+1v^cs

_L∞(0,T ;L²(Rⁿ×[0,C1]))

+ x

1−2s 2

n+1∇v^cs

_L∞(0,T ;L²(Rⁿ⁺¹+ ))

≤CkvkL^∞(0,T ;H^s(Rⁿ)),

therefore, (4.2) can be achieved by the following sufficient condition:

kvkL^∞(0,T ;H^s(Rⁿ)) ≤ E.

We now define

ω₁(z) := log^−µ C z



, for 0 < z < 1.

Note that ω₁²(z) is concave on z ∈ (0, z₀) for some sufficiently small z₀= z₀(µ) > 0.

From (4.3) and (4.4), together with [RS20, Lemma 4.4], we obtain 1{η(t)<z0E}kv(t)k_Hs−γ(Ω)

≤1{η(t)<z0E}C

 x

1−2s 2

n+1v^cs(t)

_{L2( ˆΩ×[0,1])}+ x

1−2s 2 +γ

n+1 ∇v^cs(t)

_{L2( ˆΩ×[0,1])}



≤1{η(t)<z0E}CEω₁ η(t) E



=CEω₁



1{^η(t)_E ^<z0} η(t)

E

 . (4.5)

Using Jenson’s inequality for concave functions, we have

(4.6) 1

T Z T

0

ω₁²



1{^η(t)E <z₀} η(t)

E



dt ≤ ω²₁ 1 T

Z T 0

1{^η(t)E <z₀} η(t)

E dt

! .

Combining (4.5) and (4.6) implies

(4.7) Z T

0

1{η(t)<z0E}kv(t)k²_Hs−γ(Ω)dt ≤ C²E²T ω₁² 1 T E

Z T 0

1{^η(t)E <z0}η(t) dt

! .

We extend ω1so that it is continuous and monotone increasing on (0, ∞). Therefore, (4.7) gives

Z T 0

1{η(t)<z₀E}kv(t)k²_Hs−γ(Ω)dt ≤C²E²T ω₁² 1 T E

Z T 0

η(t) dt

!

≤C²E²T ω₁²

kηk_L2(0,T )

T¹²E



=C²E²T ω₁² ask(−∆)^svkL²(0,T ;H^−s(W ))

T¹²E

 . (4.8)

On the other hand, from [RS20, Lemma 4.4], it follows that

1{η(t)≥z0E}kv(t)kH^s−γ(Ω)

≤1{η(t)≥z0E}C

 x

1−2s 2

n+1v^cs(t)

_{L2( ˆΩ×[0,1])}+ x

1−2s 2 +γ

n+1 ∇v^cs(t)

_{L2( ˆΩ×[0,1])}



≤1{η(t)≥z₀E}C

 x

1−2s 2

n+1v^cs

_L∞(0,T ;L²(Rⁿ×[0,C1]))

+ x

1−2s 2

n+1∇v^cs

_L∞(0,T ;L²(Rⁿ⁺¹+ ))



≤1{η(t)≥z0E}CE

≤Cz₀⁻¹η(t)

=Cz₀⁻¹ask(−∆)^sv(t)k_H−s(W ). (4.9)

Squaring both sides of (4.9) and subsequently integrating it, we obtain Z T

0

1{η(t)≥E}kv(t)k²_Hs−γ(Ω)dt

≤C²z⁻²₀ a²_sk(−∆)^svk²_L2(0,T ;H^−s(W ))

=C²z⁻²₀ E²a²_s

k(−∆)^svk_L2(0,T ;H^−s(W ))

E

² . (4.10)

Summing (4.8) and (4.10) yields kvk²_L2(0,T ;H^s−γ(Ω))=

Z T 0

kv(t)k²_Hs−γ(Ω)dt

≤C²E²



T ω²₁ a_sk(−∆)^svk_L2(0,T ;H^−s(W ))

T¹²E



+z₀⁻²a²_s

k(−∆)^svk_L2(0,T ;H^−s(W ))

E

²# . (4.11)

We now define

ω(z) :=

T ω₁²(T⁻¹²a_sz) + z₀⁻²a²_sz²¹₂ .

Note that ω(z) is of logarithmic type when z is small. Therefore, (4.11) can be written as

kvk_L2(0,T ;H^s−γ(Ω))≤ CEω

k(−∆)^svk_L2(0,T ;H^−s(W ))

E

 . We summarize the above discussions in the following corollary.

Corollary 4.3. Let W ⊂ Ωe be an open bounded Lipschitz set and W ∩ Ω = ∅. If there exists a constant E > 0 such that

(4.12) kvk_L∞(0,T ;H^s(Rⁿ)) ≤ E,

then there exists a constant C > 1 and a function of logarithmic type ω, both depending only on n, s, Ω, W, γ, T , such that

kvk_L2(0,T ;H^s−γ(Ω))≤ CEω k(−∆)^svk_L2(0,T ;H^−s(W ))

E

! .

4.2. Quantitative unique continuation. Given any F ∈ L²(Ω_T), by Corol- lary 2.2, there exists a unique solution v_F of the backward wave equation

(4.13)







∂_t²+ (−∆)^s+ q
vF = F in ΩT,

v_F = 0 in (Ω_e)_T,

vF = ∂tvF = 0 in Rⁿ× {T }, such that

kvFk_L∞(0,T ; eH^s(Ω))≤ C0kF kL²(Ω_T),

for some constant C0> 0 independent of vF and F . Choosing E = C0kF k_L2(ΩT), the condition (4.12) satisfies, and then we can employ Corollary 4.3with v = vF

to obtain

(4.14) kvFk_L2(0,T ;H^s−γ(Ω))≤ CkF kL²(Ω_T)ω k(−∆)^svFk_L2(0,T ;H^−s(W ))

kF k_L2(ΩT)

! .

Meanwhile, for any function u ∈ H⁻²(0, T ; H^s−γ(Ω)), by the duality argument, one has

(4.15) kukH⁻²(0,T ;H^s−γ(Ω))≤ kukL²(0,T ;H^s−γ(Ω)). Likewise, if u(T ) = ∂tu(T ) = 0, we can see that

(4.16)

∂_t²u

_{H−2(0,T ;H−s−γ(Ω))}≤ kuk_L2(0,T ;H^−s−γ(Ω)).