INVERSE PROBLEMS FOR SOME FRACTIONAL EQUATIONS WITH GENERAL NON-LINEARITY

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GENERAL NON-LINEARITY

PU-ZHAO KOW AND JENN-NAN WANG

Abstract. Inspired by some interesting equations modeling anomalous diffusion and non- linear phenomena, we will study the inverse problems of uniquely identifying coefficients in nonlinear terms from over-determined data. Precisely, we consider a semi-linear fractional Schrödinger operator (−∆)su + Q(x, u) = 0 in Ω with 0 < s < 1. The fractional Laplacian arises due to the anomalous diffusion, e.g. the motion of particles described by Lévy flights.

Here we consider the semi-linear term Q(x, u) = q(x, |u|)u, which appears naturally in the study of nonlinear optics with cubic Kerr-type nonlinearity, the complex Ginzburg-Landau equation with cubic-quintic nonlinearity, or even the Hartree equation with convolution-type nonlinearity, etc. In this article, we consider the time-independent and the time-evolution semi-linear fractional Schrödinger equations with “Dirichlet” condition given on the com- plement of Ω. We prove both the well-posedness of the forward problems and the unique determination of the inverse problems with measurements taken on the complement of Ω.

Contents

1. Introduction 1

2. Statements of Main results 3

2.1. Semi-linear fractional Schrödinger equations 3

2.2. Time-evolution semi-linear fractional Schrödinger equations 5

3. Reviews of related literature 8

4. Well-posedness of semi-linear fractional Schrödinger equation 9

5. Global uniqueness of the inverse problems 11

6. Well-posedness of time-evolution semi-linear equations 15 7. Global uniqueness of the time-dependent inverse problems 16 Appendix A. Space-time fractional Schrödinger equations 17

Acknowledgments 32

References 32

1. Introduction

In this paper, we study inverse problems for time-independent and time-evolution semi- linear fractional Schrödinger equations. We first describe the main equation considered in the paper

(1.1) (−∆)su + Q(x, u) = 0

1991 Mathematics Subject Classification. 35R11; 35R30; 47J05.

Key words and phrases. Fractional Laplacian, Nonlinear potentials, Kerr-type nonlinearity, Hartree po- tentials, Global uniqueness of inverse problems.

1

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with 0 < s < 1. Here the fractional Laplacian (−∆)s : Hs(Rn) → H−s(Rn) is defined by F ((−∆)su)(ξ) = |ξ|2sF u(ξ) for all ξ ∈ Rn, where F is the Fourier transform and |ξ|2 is the Fourier symbol of −∆. We assume that the semi-linear term Q(x, u) has the form

Q(x, u) = q(x, |u|)u,

i.e. q(x, |u|) can be regarded as a nonlinear potential. Let Ω be a bounded Lipschitz domain in Rn (n ≥ 1). Motivated by the result in [GKM22], we consider the nonlinear potential satisfying

Assumption 1.1. q(x, |z|) ∈ L(Ω × R+) and there exist CKerr > 0, 0 ≤ q0 ∈ L(Ω) and a strictly non-decreasing and continuous ω : R+ → R+ (modulus of continuity) with ω(0) = 0 such that

(1.2) kq(·, |z1|)z1− q(·, |z2|)z2− q0(·)(z1− z2)kL(Ω)

≤ CKerr ω(|z1|) + ω(|z2|)|z1− z2| for all z1, z2 ∈ C with |z1| ≤ 1 and |z2| ≤ 1.

Remark 1.2. Choosing z1 = z and z2 = 0 in (1.2) implies kq(·, |z|) − q0kL(Ω) ≤ CKerrω(|z|) for all z ∈ C with |z| ≤ 1. From this, we can easily see that q(x, 0) = q0(x) for all x ∈ Ω. We may refer CKerr the generalized Kerr constant.

Recall the following useful lemma, which can be found in [GKM22, Lemma A.1]:

Lemma 1.3. The inequality

|a|αa − |b|αb

≤ 2(|a| + |b|)α|a − b| holds for all a, b ∈ C, α > 0.

With the above lemma at hand, we can give an example that Assumption 1.1 holds (see [GKM22, Example 2.2]):

Example 1.4 (Generalized Kerr-type nonlinearity). Let (1.3) q(x, |z|) = q0(x) +

L

X

`=1

q`(x)|z|α` for all x ∈ Ω and z ∈ C,

where the exponents satisfy 0 < α1 < · · · < αL < ∞ and the functions q0, q1, · · · , qL∈ L(Ω) with q0 ≥ 0. We remark that q1, · · · , qL∈ L(Ω) can be complex-valued. Using Lemma1.3, it can be easily verified that (1.3) satisfies Assumption 1.1 with

Ckerr = 21+αL−min{1,α1}

L

X

`=1

kq`kL(D) and ω(t) = |t|α1. The well-known Kerr-type non-linearity corresponds to the special case when

L = 1 and α1 = 2,

which appears in the model of nonlinear optics, see, e.g. [Boy20, MN19] for the physical description. Here q0(x) is a trapped potential and q1 > 0 (q1 < 0) represents the defocusing (focusing) effect. Moreover, (1.1) with the non-linearity of type (1.3) can be derived as the continuum limit of a family of discrete non-linear Schrödinger equations with long-range lattice interactions, which models the charge transport in biopolymers like the DNA, see, e.g. [KLS13]. We also refer to [KSM14] for some numerical results concerning this non-linear equation.

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On the other hand, the complex Ginzburg-Landau equations are popular models for the light propagating in nonlinear dissipative media [QMM+20]. In this case, the nonlinear potential is of the form

q(x, |z|) = q0(x) + q1(x)|z|2+ q2(x)|z|4,

that is, α1 = 2, and α2 = 4 in (1.3). In other words, the fractional complex Ginzburg-Landau equation is given by

(−∆)su + q0(x)u + q1(x)|u|2u + q2(x)|u|4u = 0 with complex-valued functions q1 and q2.

2. Statements of Main results

2.1. Semi-linear fractional Schrödinger equations. Before stating our results, we first introduce some function spaces. For each γ ∈ R, let Hγ(Rn) be the standard L2-based fractional Sobolev space. Let U be an open set in Rn and denote ˜Hγ(U ) the closure of Cc(U ) in Hγ(Rn), where Hγ(U ) is the restriction of Hγ(Rn) in U equipped with the standard quotient norm. Moreover, if U is a bounded Lipschitz domain, then we have the following identifications:

( ˜Hγ(U ))0 = H−γ(U ), (Hγ(U ))0 = ˜H−γ(U ) for all γ ∈ R,

see, e.g. [GSU20, Section 2A], [McL00, Chapter 3], or [KLW21] and the references therein for more details.

2.1.1. Forward problems. Assume that q0 ≥ 0 a.e. in Ω. Given any g ∈ Cc(Ωe), using [GSU20, Lemma 2.3] and [LL19, Proposition 3.3], there exists a unique solution u0 ∈ L(Ω)∩

Hs(Rn) of (2.1)

((−∆)su0+ q0(x)u0 = 0 in Ω,

u0 = g in Ωe := Rn\ ¯Ω, and u0 satisfies

(2.2) ku0kL(Ω)∩Hs(Rn) ≤ CkgkL(Ωe)∩Hs(Rn), where C = C(n, s, Ω).

We now state results in the time-independent case. Consider the Dirichlet problem (2.3)

((−∆)su + q(x, |u|)u = 0 in Ω,

u = g in Ωe,

with g ∈ Cc(Ωe). We first state the well-posedness of the Dirichlet problem (2.3).

Theorem 2.1. Suppose that q(x, |u|) satisfies Assumption 1.1. There exists a sufficiently small δ0 > 0 such that if kgkL(Ωe)∩Hs(Rn) ≤ δ0, then there is a unique solution u ∈ L(Ω) ∩ Hs(Rn) to (2.3). In addition, the following estimate holds:

(2.4) ku − u0kL(Ω)∩ ˜Hs(Ω) ≤ Cω(Cδ)δ,

where δ = kgkL(Ωe)∩Hs(Rn) and u0 ∈ L(Ω) ∩ Hs(Rn) is the unique solution of (2.1).

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Remark 2.2. We may assume that ω(Cδ) ≤ ω(Cδ0) < 1. Then combining (2.2) and (2.4) implies

(2.5) kukL(Ω)∩Hs(Rn) ≤ CkgkL(Ωe)∩Hs(Rn).

Before discussing the inverse problem, it is interesting to point out that Theorem 2.1 can be extended the fractional Schrödinger equation with generalized Hartree nonlinearity.

Precisely, let n ≥ 2, 0 < d < n, 0 ≤ q0 ∈ L(Ω), and q1 ∈ L(Ω). We consider (2.6) q(x, |w(x)|) := q0(x) + q1(x)

Z

Rn

|x − y|−d|w(y)|2dy.

The nonlinear term w(x)R

R3|x − y|−1|w(y)|2dy, i.e. n = 3, d = 1, corresponds to a critical Hartree nonlinearity, which arises in the quantum theory of boson stars (see e.g. [Len07, (4)]

or [FJL07]). We want to remark that q of (2.6) does not satisfy Assumption1.1. However, the well-posedness of Theorem 2.1still holds for (2.3) with q given in (2.6) by following the same proof, see Remark 4.1 for more details. We also refer to the work [Sas12], which concerning the inverse scattering problems for the Hartree equation involving classical Laplacian.

2.1.2. Inverse problems. Fixing any open sets V, W ⊂ Ωe, let q satisfy Assumption 1.1 and uq ∈ L(Ω) ∩ Hs(Rn) be the unique solution to (2.3) provided g ∈ Cc(V ) and kgkL(V )∩Hs(Rn)≤ δ0. We now define the Dirichlet-to-Neumann map (DN-map):

Λq(g) := (−∆)su|W.

We remark that the DN-map is equivalent to the Cauchy data set, see [LL19, (2.2)] or [GSU20, Lemma A.2]. We prove that the nonlinear potential q can be uniquely determined from Λq when it is given by (1.3) in Example 1.4.

Theorem 2.3 (Global uniqueness). Assume that q(1), q(2) are given as in (1.3). If

(2.7) Λq(1)(g) = Λq(2)(g)

for all g ∈ Cc(V ) with kgkL(V )∩Hs(Rn) ≤ δ0, then q(1) ≡ q(2).

For the global uniqueness of the inverse problem, we only need to take measurements corresponding to small Dirichlet data. It turns out we can use one-dimensional Dirichlet data in the inverse problem when the potentials are continuous.

Theorem 2.4 (Uniqueness from 1-dimensional measurements). Let 0 < s < 1 and q(1), q(2) be defined as in (1.3). Choosing any g ∈ Cc(V ) such that g 6≡ 0. If

(2.8) Λq(1)(g) = Λq(2)(g) for all sufficiently small  > 0, then q(1) ≡ q(2).

Applying similar arguments, we can establish the following uniqueness theorem of the inverse problem with the Hartree nonlinearity using one single measurement.

Theorem 2.5 (Uniqueness for Hartree nonlinearity). Let n ≥ 2 be an integer, 0 < d < n and q(j) be given by

q(j)(x, |w(x)|) := q1(j)(x) Z

Rn

|x − y|−d|w(y)|2dy, j = 1, 2

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with q1(j) ∈ L(Ω). Let u(j) ∈ L(Ω) ∩ Hs(Rn) be the unique solution of (2.3) with u(j) = g in Ωe, where 0 6≡ g ∈ Cc(V ). If

Λq(1)(g) = Λq(2)(g) for any fixed g, then q(1)1 ≡ q1(2).

2.2. Time-evolution semi-linear fractional Schrödinger equations. In this section, we want to state analogous results for the time-evolution semi-linear fractional Schrödinger equations. In particular, the global uniqueness results in Theorem 2.3 and Theorem 2.4 can be extend to a class of time-evolution semi-linear equations. The following assumption is similar to Assumption 1.1.

Assumption 2.6. Let T > 0. We assume that q ∈ L((0, T ) × Ω × R) and there exist CKerr > 0, q0 ∈ L((0, T ) × Ω) and a monotone non-decreasing, continuous ω : R≥0 → R≥0 (modulus of continuity) with ω(0) = 0 such that

kq(·, |z1|)z1 − q(·, |z2|)z2− q0(·)(z1− z2)kL((0,T )×Ω)

≤ CKerr ω(|z1|) + ω(|z2|)|z1− z2| for all z1, z2 ∈ C with |z1| ≤ 1 and |z2| ≤ 1.

2.2.1. Forward problems. We now consider the following initial boundary value time- evolution equation:

(2.9)





P(∂t) + (−∆)su + q(·, |u|)u = 0 in ΩT := (0, T ) × Ω, Q(1)(∂t), · · · , Q(k)(∂t)u = 0 on {0} × Ω,

u = g in (Ωe)T := (0, T ) × Ωe

with g ∈ Cc((Ωe)T). Here, P(∂t), Q(1)(∂t), · · · , Q(k)(∂t) are linear differential operators, which will be specified later. We assume that for any open set O in Ω

(2.10) w = 0 in OT implies P(∂t)w = 0 in OT.

We want to remark that (2.10) is not too restrictive. It is satisfied for all local differential operators of ∂t and even non-local operators of Caputo’s type. The corresponding linear approximation of (2.9) is given by

(2.11)





P(∂t) + (−∆)su0+ q0(·)u0 = 0 in ΩT, Q(1)(∂t), · · · , Q(k)(∂t)u0 = 0 on {0} × Ω,

u0 = g in (Ωe)T.

For (2.11), we assume that the following well-posedness holds, i.e.

Assumption 2.7. Let q0 ∈ L(ΩT). For each F ∈ L(ΩT) and g ∈ Cc((Ωe)T), there exists a unique solution w of

(2.12)





P(∂t) + (−∆)sw + q0(·)w = F in ΩT, Q(1)(∂t), · · · , Q(k)(∂t)w = 0 on {0} × Ω,

w = g in (Ωe)T,

and satisfying

kwkL(0,T ;Hs(Rn))∩L(ΩT) ≤ C kF kL(ΩT)+ kgk

 with some suitable norm k · k.

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Remark 2.8. In view of Assumption 2.7, the unique solution u0 of (2.11) satisfies (2.13) ku0kL(0,T ;Hs(Rn))∩L(ΩT) ≤ Ckgk.

We now provide some examples that satisfy Assumption 2.7.

Example 2.9 (Non-local diffusion). Let P(∂t) = ∂t and k = 1 with Q(1)(∂t) = Id. Then (2.12) is written as





t+ (−∆)sw + q0(·)w = F in ΩT,

w = 0 on {0} × Ω,

w = g in (Ωe)T.

If q0 is real-valued, from [KMS22, Proposition 2.9], we can see that Assumption 2.7 holds with

(2.14) kgk = kgkL(0,T ;Hs(Rn))∩L((Ωe)T)+ k(−∆)sgkL2(ΩT).

Example 2.10 (Non-local wave). Let P(∂t) = ∂t2 and k = 2 with (Q(1)(∂t), Q(2)(∂t)) = (Id, ∂t). Then (2.12) becomes





t2+ (−∆)sw + q0(·)w = F in ΩT,

w = ∂tw = 0 on {0} × Ω,

w = g in (Ωe)T.

Repeating the argument in the proof of [KLW21, Theorem 2.1] where q0 is independent of t, we can show that Assumption 2.7 holds with (2.14) for n = 1 and 12 < s < 1 (since the proof utilized the continuous embedding of Hs ⊂ L, see e.g. [DNPV12, Theorem 8.2]). The required estimate also follows from Corollary A.7 for β = 2.

Example 2.11 (Space-time fractional equation). Let P(∂t) = ∂tβ with 1 < β < 2 be the Caputo derivative. Then (2.12) becomes





tβ + (−∆)sw + q0(·)w = F in ΩT,

w = ∂tw = 0 on {0} × Ω,

w = g in (Ωe)T.

The time-fractional derivative models particle sticking and trapping phenomena, while the fractional space derivative is used to describe particles performing long jumps. From Corol- lary A.7, we see that Assumption 2.7 is satisfied with

kgk = kgkL(0,T ;Hs(Rn))+ k(−∆)sgkL2(ΩT)

for n = 1 and 12 < s < 1 (since the proof utilized the continuous embedding of Hs ⊂ L).

We now state the well-posedness of (2.9).

Theorem 2.12. Suppose that Assumption 2.6 and Assumption 2.7 hold. There exists a sufficiently small δ0 > 0 such that if kgk ≤ δ0, then there exists a unique solution u ∈ L(0, T ; Hs(Rn)) ∩ L(ΩT) of (2.9) having the following estimate

(2.15) ku − u0kL(0,T ; ˜Hs(Ω))∩L(ΩT) ≤ Cω(Ckgk)kgk, where u0 is the unique solution of (2.11).

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Remark 2.13. We may restrict ω(Ckgk) ≤ ω(C(δ0)) < 1. Combining (2.13) and (2.15), we have

(2.16) kukL(0,T ;Hs(Rn))∩L(ΩT)≤ Ckgk.

2.2.2. Inverse problems. Fixing any open sets V, W ⊂ Ωe and let q(1), q(2) satisfy Assump- tion 2.6 with linear approximations q0(1), q0(2) respectively. Under Assumption 2.7, the above well-posedness result enables us to define the following DN-map:

Λq(j)(g) := (−∆)su(j)|WT

for all g ∈ Cc(VT) provided kgk ≤ δ, where u(j) is the unique solution of (2.9). We now state an analogy to Theorem 2.4 when the nonlinear potential is time-independent.

Theorem 2.14 (Uniqueness from 1-dimensional measurements). Assume that q(1), q(2) have the form

q(j)(x, |z|) = q0(j)(x) +

L

X

`=1

q`(j)(x)|z|α` for x ∈ Ω and z ∈ C,

where the exponents satisfy 0 < α1 < · · · < αL < ∞ and the functions q(j)0 , q1(j), · · · , qL(j) ∈ C0(Ω). Suppose that Assumption 2.7 holds. Let g ∈ Cc(VT) be given satisfying g(t, x) 6≡ 0, then

Λq(1)(g) = Λq(2)(g) for all sufficiently small  > 0 implies q(1) ≡ q(2).

Remark 2.15. Here we impose the C0(Ω) assumption since we do not know whether the analogue MUCP result (see Lemma 5.4) holds for (2.11).

In order to recover the time-dependent nonlinear term q(j), we need another property.

Assumption 2.16. Suppose that Assumption2.7holds. For each g ∈ Cc(VT), let Pq0g := w, where w is the unique solution of (2.12) with F ≡ 0. We assume the set

{ Pq0g|T g ∈ Cc(VT) } is dense in L2(ΩT).

Example 2.17. Assumption 2.16 is satisfied for non-local diffusion equation, see [KMS22, Proposition 3.2] and [Li22, Proposition 2.4]. This assumption also holds for non-local wave, see [KLW21, Theorem 3.1].

Theorem 2.18 (Uniqueness from the DN-map). Assume that q(1), q(2) have the form q(j)(t, x, |z|) = q(j)0 (t, x) +

L

X

`=1

q(j)` (t, x)|z|α` for all (t, x) ∈ ΩT and z ∈ C,

where 0 < α1 < · · · < αL < ∞ and q(j)0 , q1(j), · · · , qL(j)∈ L(ΩT). Suppose that Assumption2.7 and Assumption 2.16 hold. If

Λq(1)(g) = Λq(2)(g)

for all g ∈ Cc(VT) with kgk ≤ δ0, where δ0 is defined in Theorem 2.12, then q(1) ≡ q(2).

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3. Reviews of related literature

Before proving our results stated above, we first discuss some existing literature. Here we only mention results involving the fractional Laplacian. We begin with the linear equations.

The first uniqueness of the inverse problem was proved in [GSU20] for the case q(x, |u|) = q(x)u. In [HL19, HL20], the authors provided a constructive proof using a monotonicity- relation between the potentials. Further results are listed in the following:

• [GRSU20]: determining the potential using a single measurement.

• [CLR20]: linear equation with drift and potential terms.

• [CLL19, GLX17]: anisotropic fractional Schrödinger equation.

• [Cov20]: fractional conductivity equation.

• [RS20, RS18]: optimal logarithmic stability estimate of determining the potential.

• [LLR20]: Space-time fractional parabolic equation.

• [KLW21]: uniqueness, optimal logarithmic stability estimate in the fractional (non-local) wave equation.

Next we would like to mention some results in the time-independent and time-evolution fractional Schrödinger equations with nonlinear potentials. Those results are more closely related to this work. We will comment on those results.

• [LL19]. The authors studied (1.1) with the nonlinear potential Q(x, z). They assumed that Q > 0 and continuous in Ω × R, lim

z→0

Q(x, z)

z = 0 uniformly in x ∈ Ω and the existence of ∂zQ(x, z). In addition, they assumed that

zQ(x, z) ≥ 0

and it is continuous in (x, z) ∈ Ω × R. It is clear that the Kerr-type nonlinearity (Exam- ple 1.4) does not satisfy the assumptions.

• [LL22a, LO22, Li21]. In these papers, the authors considered the nonlinear potential Q(x, z) satisfying

(3.1) ( Q(·, 0) = 0, ∂zQ(·, z) ≥ 0,

z → Q(·, z) is holomorphic with value in Cs( ¯Ω).

It follows from (3.1) that Q(x, z) =

X

k=1

qk(x)zk

k!, q1 ≥ 0 and all qk ∈ Cs( ¯Ω).

In [LL22b] (and [Lin20]), the authors considered nonlinear potentials of power type, that is,

Q(x, z) =

L

X

k=0

qk(x)zk

and qk ∈ Cs( ¯Ω), for k = 0, · · · , L. Since the Kerr-type nonlinearity is not holomorphic with respect to z, hence our setting is different with these works.

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• [Li22]. The author studied the following non-local diffusion equation:

(3.2) ∂tu + (−∆)su + q(t, x, u)u = 0 with q(t, x, z) =

L

X

`=1

q`(t, x)|z|α`, where

0 ≤ q` ≤ C([0, T ] × Ω), 0 < α1 < · · · < αL.

It should be noted that the potential considered in (3.2) does not include the linear one.

Our results generalize the well-posedness of the forward problem obtained in [Li22, Propo- sition 3.2] and the global uniqueness of the inverse problem in [Li22, Theorem 1.1] to more general potentials. Furthermore, for the case where q is independent of variable t, we are able to prove the uniqueness using less data, see Theorem 2.14.

• [KMS22]. Nonlocal diffusion and nonlocal wave equations were studied there. The nonlin- ear potentials considered in that paper mimick assumption (3.1).

Finally, we want to point out that the global uniqueness of determining the Hartree po- tential (Theorem 2.5) and the determination of the nonlinear potential in the space-time fractional equation (Example 2.11 and Theorem 2.14) have not been studied before.

The paper is organized as follows. The well-posedness question stated in Theorem 2.1 is proved in Section 4, and then the proofs of global uniqueness, Theorem 2.3and Theorem2.4, are given in Section5. For the time-evolution equations, the well-posedness in Theorem2.12 is proved in 6. The corresponding inverse problems, Theorem 2.14 and Theorem 2.18, are studied in Section7. Finally, we elaborate the details given in Example 2.11 in AppendixA.

4. Well-posedness of semi-linear fractional Schrödinger equation Note that if Ω is Lipschitz bounded, then L(Ω) ⊂ H−s(Ω). Let 0 ≤ q0 ∈ L(Ω). From [GSU20, Lemma 2.3] and [LL19, Proposition 3.3], we can define the bounded linear operator S : L(Ω) → L(Ω) ∩ ˜Hs(Ω) by SF := v, where v is the unique solution of

((−∆)sv + q0v = F in Ω,

v = 0 in Ωe,

and there exists a constant C = C(n, s, Ω) such that

(4.1) kSF kL(Ω)∩ ˜Hs(Ω) ≤ CkF kL(Ω).

We now prove the well-posedness result for the Dirichlet problem (2.3).

Proof of Theorem 2.1. The proof is based on the contraction mapping theorem. Let u be a solution of (2.3), then the remainder function v := u − u0 satisfies

(4.2)

((−∆)sv + q0v = −F (v) in Ω,

v = 0 in Ωe,

where

F (v) := (q(·, |v + u0|) − q0)(v + u0).

Let g ∈ Cc(Ωe) satisfying kgkL(Ωe)∩Hs(Rn) = δ ≤ δ0, where δ0 > 0 is a small constant to be determined later. We consider the Banach space

Xδ :=

n

v ∈ L(Ω) ∩ ˜Hs(Ω) kvkL(Ω)∩ ˜Hs(Ω) ≤ δ o .

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We first show that

(4.3) SF (v) ∈ Xδ for all v ∈ Xδ.

For each v ∈ Xδ, from (4.1), Assumption 1.1 and (2.2), we know that

(4.4)

kSF (v)kL(Ω)∩ ˜Hs(Ω) ≤ CkF (v)kL(Ω)

≤ Cω(kv + u0kL(Ω))kv + u0kL(Ω)

≤ Cω(Cδ)δ.

We now choose sufficiently small δ0 > 0 such that Cω(Cδ) ≤ Cω(Cδ0) ≤ 1, and we conclude the claim (4.3).

Next, we want to show that

(4.5) SF is a contraction map on Xδ.

Let v1, v2 ∈ Xδ and compute

F (v1) − F (v2)

= q(·, |v1+ u0|)(v1+ u0) − q(·, |v2+ u0|)(v2+ u0)

− q0((v1+ u0) − (v2 + u0)).

Since S is linear, by (4.1), Assumption 1.1, (2.2), and (2.2), we have that

(4.6)

kSF (v1) − SF (v2)kL(Ω)∩ ˜Hs(Ω)

≤ CkF (v1) − F (v2)kL(Ω)

≤ C ω(kv1+ u0kL(Ω)) + ω(kv2 + u0kL(Ω))kv1− v2kL(Ω)

≤ Cω(Cδ)kv1− v2kL(Ω)∩ ˜Hs(Ω).

Possibly replacing a smaller δ0 > 0 (such that Cω(Cδ) ≤ Cω(Cδ0) ≤ 12), we obtain that kSF (v1) − SF (v2)kL(Ω)∩ ˜Hs(Ω) ≤ 1

2kv1− v2kL(Ω)∩ ˜Hs(Ω)

and therefore (4.5).

From (4.3) and (4.5), the Banach fixed point theorem guarantees that there exists a unique v ∈ Xδ such that

(4.7) v = SF (v),

that is, there exists a unique solution v ∈ Xδ of (4.2). In other words, u = v + u0 ∈ L(Ω) ∩ Hs(Rn) is the unique solution to (2.3). Finally, combining (4.4), (4.7) and (2.2) yields

kvkL(Ω)∩ ˜Hs(Ω)≤ Cω(kv + u0kL(Ω))kv + u0kL(Ω)

≤ Cω(Cδ)kvkL(Ω)+ Cω(Cδ)ku0kL(Ω)

≤ Cω(Cδ)kvkL(Ω)∩ ˜Hs(Ω)+ Cω(Cδ)δ.

Choosing a smaller δ > 0, if necessary, such that Cω(Cδ) ≤ 12, we then have (2.4). 

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Remark 4.1. Before ending this section, we would like to show that the well-posedness result in Theorem 2.1 also holds for the Hartree nonlinearity q given in (2.6). It suffices to verify Assumption 1.1 for such q. If supp (w1− w2) ⊂ Ω, we have that

|q(x, |w1(x)|) − q(x, |w2(x)|)|

≤ kq1kL(Ω) Z

Rn

|x − y|−d

|w1(y)|2− |w2(y)|2 dy

≤ kq1kL(Ω) kw1kL(Rn)+ kw2kL(Rn) Z

Rn

|x − y|−d

|w1(y)| − |w2(y)|

dy

≤ kq1kL(Ω) kw1kL(Rn)+ kw2kL(Rn) Z

Rn

|x − y|−d|w1(y) − w2(y)| dy

≤ kq1kL(Ω) kw1kL(Rn)+ kw2kL(Rn)kw1− w2kL(Ω) Z

|x − y|−ddy for all x ∈ Ω, which implies

(4.8) kq(·, |w1|) − q(·, |w2|)kL(Ω)

≤ Ckq1kL(Ω) kw1kL(Rn)+ kw2kL(Rn)kw1− w2kL(Ω), where C = C(d, n, Ω). Similarly, if wj has compact support, we can also estimate (4.9) kq(·, |wj|) − q0kL(Ω) ≤ C(d, n, Ω, supp (wj))kq1kL(Ω)kwjk2L(Rn). Combining (4.8) and (4.9), we obtain

(4.10)

kq(·, |w1|)w1− q(·, |w2|)w2− q0(·)(w1− w2)kL(Ω)

≤ kw1kL(R3)kq(·, |w1|) − q(·, |w2|)kL(Ω) + kq(·, |w2|) − q0kL(Ω)kw1− w2kL(Ω)

≤ ˜C kw1k2L(R3)+ kw2k2L(R3)kw1− w2kL(Ω)

for all wj ∈ L(R3) with compact support and satisfying supp (w1− w2) ⊂ Ω, where ˜C = Ckq1kL(Ω). Let wj = vj + u0, then wj is compactly supported and supp (w1 − w2) ⊂ Ω.

It follows from (4.9) and (4.10) that (4.4) and (4.6) remain true. Therefore, by the same contraction argument, we can show that Theorem 2.1 also holds for q given in (2.6).

5. Global uniqueness of the inverse problems We now recall the following anti-locality property of the fractional Laplacian.

Lemma 5.1. ([GSU20, Theorem 1.2]) Let 0 < s < 1 and u ∈ Hγ(Rn) for some γ ∈ R. If both u and (−∆)su vanish in some open set, then u ≡ 0.

Remark 5.2. See also [GRSU20, Theorem 2] for a constructive version of Lemma5.1, which is an application of the standard Tikhonov regularization scheme.

The following Runge approximation property for the fractional Schrödinger equation plays an essential role in our proof, which is a special case of [GSU20, Lemma 5.1].

Lemma 5.3. Suppose that Ω ⊂ Rn is a bounded open set, 0 < s < 1, and q0 ∈ L(Ω) is non-negative. Let V be an open subset of Ωe, then the set

R :=



v| ((−∆)s+ q0)v = 0 in Ω v = g ∈ Cc(V )



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is dense in L2(Ω).

We now prove our main results by modifying some ideas in [GKM22, Li22].

Proof of Theorem 2.3. We first establish q(1)0 ≡ q0(2)by showing that the equivalence of the DN maps for nonlinear potentials implies the equivalence of the DN maps for the corresponding linear potentials. To begin, let us define the linear operator Λq(j)

0

: L(V ) ∩ Hs(Rn) → H−s(W ):

Λq(j) 0

(g) := (−∆)su(j)0 |W for g ∈ Cc(V ), where u(j)0 ∈ L(Ω) ∩ Hs(Rn) is the unique solution of (2.1) with (5.1) u(1)0 = u(2)0 = g in Ωe.

Likewise, let u(j) ∈ L(Ω) ∩ Hs(Rn) be the unique solution of (2.3) with

(5.2) u(1) = u(2) = g in Ωe,

where g ∈ Cc(V ) satisfies kgkL(Ωe)∩Hs(Rn)≤ δ0 as in Theorem 2.1. By Lemma 5.1, in view of (2.7) and (5.2), we have

(5.3) u(1) ≡ u(2) =: u in Rn.

From (2.4), it is easy to see that kΛq(j)(g) − Λq(j)

0

(g)kH−s(W ) ≤ k(−∆)s(u − u(j)0 )kH−s(Rn)

≤ Cku − u(j)0 kHs(Rn) = Cku − u(j)0 kH˜s(Ω) ≤ Ckgk1+αL(V )∩H1 s(Rn)

since ω(t) = |t|α1. Therefore, from (2.7), it yields kΛq(1)

0

(g) − Λq(2) 0

(g)kH−s(W )

2

X

j=1

q(j)(g) − Λ

q(j)0 (g)kH−s(W )≤ Ckgk1+αL(V )∩H1 s(Rn). Observe that Λq(j)

0

: L(V ) ∩ Hs(Rn) → H−s(W ) is linear. Hence, for each  > 0, we can derive

(5.4)

q(1) 0

(g) − Λq(2) 0

(g)kH−s(W )

= −1q(1) 0

(g) − Λq(2) 0

(g)kH−s(W )

≤ −1Ckgk1+αL(V )∩H1 s(Rn) = α1Ckgk1+αL(V )∩H1 s(Rn). Taking  → 0 in (5.4), we conclude that

Λq(1)

0 (g) = Λq(2)

0 (g) for all g ∈ Cc(V ).

Now, using [GSU20, Theorem 1.1], we conclude that q0(1) ≡ q0(2), and thus (5.5) u(1)0 ≡ u(2)0 =: u0 in Rn.

Our next task is to prove q(1)` ≡ q`(2) inductively. We assume that (5.6) q`(1) ≡ q`(2) =: q` for ` = 0, · · · , m − 1,

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where m ∈ {1, · · · , L} and aim to show

(5.7) qm(1) ≡ qm(2).

Using the equation in (2.3), (5.3), and (5.6), we have (5.8)

L

X

`=m

(q`(1)− q(2)` )|u|α`u = 0.

Combining (5.8) and (2.5) yields

|q(1)m − qm(2)||u|1+αm =

(qm(1)− q(2)m )|u|αmu

=

L

X

`=m+1

(q`(1)− q`(2))|u|α`u

≤ Ckgk1+αL(Ωm+1e)∩Hs(Rn), and thus

|q(1)m − qm(2)|1+αm1 |u| ≤ Ckgk

1+αm+1 1+αm

L(Ωe)∩Hs(Rn). Consequently, from (2.4), it follows that

|qm(1)− q(2)m |1+αm1 |u0|

≤ |q(1)m − qm(2)|1+αm1 |u0− u| + |qm(1)− qm(2)|1+αm1 |u|

≤ C kgk1+αL(Ω1 e)∩Hs(Rn)+ kgk

1+αm+1 1+αm

L(Ωe)∩Hs(Rn).

Since u0 satisfies a linear equation with (5.1), for each  > 0, we obtain

(5.9)

|q(1)m − qm(2)|1+αm1 |u0|

= −1|q(1)m − qm(2)|1+αm1 |u0|

≤ −1C kgk1+αL(Ω1e)∩Hs(Rn)+ kgk

1+αm+1 1+αm

L(Ωe)∩Hs(Rn)



≤ αC kgk1+αL(Ω1e)∩Hs(Rn)+ kgk

1+αm+1 1+αm

L(Ωe)∩Hs(Rn)

 with

α = min



α1,1 + αm+1

1 + αm − 1



> 0.

Taking  → 0 in (5.9) gives

(5.10) |qm(1)− qm(2)|1+αm1 |u0| = 0 in Ω

⇐⇒ |qm(1)− q(2)m |1+αm1 u0 = 0 in Ω.

Then we have

Z

|qm(1)− q(2)m |1+αm1 u0dx = 0 for all g ∈ Cc(V ).

By arbitrariness of g ∈ Cc(V ) and using Lemma5.3, we conclude that

|q(1)m − qm(2)|1+αm1 ≡ 0,

which implies (5.7). 

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We now prove Theorem 2.4 in which less data are used. To begin, we recall the follow- ing measurable unique continuation property (MUCP) of the linear fractional Schrödinger equation.

Lemma 5.4. (A special case of [GFR19, Theorem 4]) Let Ω be a bounded open set in Rn and q0 ∈ L(Ω). Assume that 0 < s < 1 and u0 ∈ Hs(Rn) satisfies

(5.11) (−∆)s+ q0u0 = 0 in Ω.

If there exists some measurable set E ⊂ Ω with |E| > 0 such that u|E = 0, then u ≡ 0 in Rn. Remark 5.5. In [GRSU20, Proposition 5.1], the authors proved Lemma 5.4 for the regime

1

4 ≤ s < 1. The case when q0 ∈ C1(Ω) was proved in [FF14, Theorem 1.4]. In [FF14], the MUCP is shown to be true even for fractional equations with nonlinear terms. It should be noted that (5.11) holds for q0 ∈ L(Ω) if and only if

(5.12) |(−∆)su0| ≤ C|u0| in Ω

for some constant C > 0. In view of the Caffarelli-Silvestre extension for the fractional Laplacian [CS07,GFR19] and the counterexample constructed in [BW90], it should be point out that the equation (5.11) is essential in Lemma 5.4, see also [AE97, TZ05] for related results.

The proof of Theorem 2.4 is very similar to that of Theorem2.3.

Proof of Theorem 2.4. As above, we first prove q(1)0 ≡ q0(2). The inequality (5.4) implies Λq(1)

0

(g) = Λq(2) 0

(g)

for any fixed 0 6= g ∈ Cc(V ). The unique continuation property for the fractional Laplacian implies that u(1)0 = u(2)0 = u0 in Rn, where u(j)0 is the unique solution of (2.1) corresponding to q(j)0 . In other words, we have

(5.13) (q(1)0 (x) − q(2)0 (x))u0(x) = 0 in Ω.

By Lemma 5.4 and the fact that g 6= 0, we see that u0 6= 0 a.e. in Rn. Therefore, we immediately conclude that q0(1) ≡ q(2)0 =: q0 from (5.13).

Next, we carry out the induction step. We begin with the assumption of (5.14) q(1)` ≡ q`(2) =: q` for all ` = 0, · · · , m − 1,

where m ∈ {1, · · · , L} and will show

(5.15) qm(1) ≡ qm(2).

Applying the same argument as in the proof of (5.9), we can show that (5.10) remains valid, i.e.

(5.16) |qm(1)− q(2)m |1+αm1 u0 = 0 in Ω.

Similarly, (5.16) implies

q(1)m (x) = qm(2)(x) a.e. in Ω.

 Applying the similar argument as above, we now want to prove the unique determination of a Hartree-type nonlinearity in Theorem 2.5.

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Proof of Theorem 2.5. By the assumptions of

Λq(1)(g) = Λq(2)(g),

u(1) = u(2) in Ωe, and using the unique continuation property of the fractional Laplacian (Lemma 5.1), we have that

u(1) ≡ u(2) =: u in Rn. In other words,

q1(1)(x) − q1(2)(x)

 Z

Rn

|x − y|−d|u(y)|2dy



u(x) = 0.

Since g 6≡ 0, R

Rn|x − y|−d|u(y)|2dy > 0, and hence

(5.17) q1(1)(x) − q1(2)(x)u(x) = 0 in Ω.

Note that u ∈ L(Rn) is compactly supported. It is then clear that u satisfies (5.12).

Therefore we can apply Lemma 5.4 here. We then prove q(1)1 ≡ q1(2) in Ω

as above. 

6. Well-posedness of time-evolution semi-linear equations

In this section, we will establish the well-posedness of the initial boundary value for the time-evolution semi-linear fractional Schrödinger equation. The approach is similar to that in the time-independent case. By Assumption 2.7, we define the bounded linear operator S : L(ΩT) → L(0, T ; Hs(Rn)) ∩ L(ΩT) by SF := v, where v is the unique solution of





P(∂t) + (−∆)sv + q0v = F in ΩT, Q(1)(∂t), · · · , Q(k)(∂t)v = 0 on {0} × Ω,

v = 0 in (Ωe)T,

where q0 ∈ L(ΩT), and there exists a constant C such that (6.1) kSF kL(0,T ;Hs(Rn))∩L(ΩT) ≤ CkF kL(ΩT).

We are now ready to prove the well-posedness of the initial boundary value problem (2.9).

Proof of Theorem 2.12. If u is a solution of (2.9), then v := u − u0 satisfies

(6.2)





P(∂t) + (−∆)sv + q0v = −F (v) in ΩT, Q(1)(∂t), · · · , Q(k)(∂t)v = 0 on {0} × Ω,

v = 0 in (Ωe)T,

where

F (v) := (q(·, |v + u0|) − q0)(v + u0).

Let kgk = δ ≤ δ0, where δ0 > 0 is a small constant to be determined later. We then define the Banach space

Xδ := v ∈ L(0, T ; Hs(Rn)) ∩ L(ΩT) kvkL(0,T ;Hs(Rn))∩L(ΩT) ≤ δ . We first show that

(6.3) SF (v) ∈ Xδ for all v ∈ Xδ.

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