INVERSE PROBLEMS FOR SOME FRACTIONAL EQUATIONS WITH GENERAL NON-LINEARITY

(1)

GENERAL NON-LINEARITY

PU-ZHAO KOW AND JENN-NAN WANG

Abstract. Inspired by some interesting equations modeling anomalous diffusion and nonlinear 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 complement 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 potentials, Global uniqueness of inverse problems.

1

(2)

with 0 < s < 1. Here the fractional Laplacian (−∆)^s : H^s(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 C_Kerr > 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|) − q₀k_L^∞_(Ω) ≤ C_Kerrω(|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 C_Kerr 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 < · · · < α_L < ∞ and the functions q₀, q₁, · · · , q_L∈ L^∞(Ω) with q₀ ≥ 0. We remark that q₁, · · · , q_L∈ L^∞(Ω) can be complex-valued. Using Lemma1.3, it can be easily verified that (1.3) satisfies Assumption 1.1 with

C_kerr = 2^1+α^L^−min{1,α¹^}

L

X

`=1

kq_`k_L^∞_(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.

(3)

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 H^γ(Rⁿ) be the standard L²-based fractional Sobolev space. Let U be an open set in Rⁿ and denote ˜H^γ(U ) the closure of C_c^∞(U ) in H^γ(Rⁿ), where H^γ(U ) is the restriction of H^γ(Rⁿ) in U equipped with the standard quotient norm. Moreover, if U is a bounded Lipschitz domain, then we have the following identifications:

( ˜H^γ(U ))⁰ = H^−γ(U ), (H^γ(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 ∈ C_c^∞(Ωe), using [GSU20, Lemma 2.3] and [LL19, Proposition 3.3], there exists a unique solution u0 ∈ L^∞(Ω)∩

H^s(Rⁿ) of (2.1)

((−∆)^su₀+ q₀(x)u₀ = 0 in Ω,

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

(2.2) ku0kL^∞(Ω)∩H^s(Rⁿ) ≤ CkgkL^∞(Ωe)∩H^s(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 Ω_e,

with g ∈ C_c^∞(Ω_e). 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 kgk_L^∞_(Ω_e_)∩H^s_(Rⁿ₎ ≤ δ₀, then there is a unique solution u ∈ L^∞(Ω) ∩ H^s(Rⁿ) to (2.3). In addition, the following estimate holds:

(2.4) ku − u₀k_L∞(Ω)∩ ˜H^s(Ω) ≤ Cω(Cδ)δ,

where δ = kgk_L^∞_(Ω_e_)∩H^s_(Rⁿ₎ and u₀ ∈ L^∞(Ω) ∩ H^s(Rⁿ) is the unique solution of (2.1).

(4)

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

(2.5) kuk_L^∞_(Ω)∩H^s_(Rⁿ₎ ≤ Ckgk_L^∞_(Ω_e_)∩H^s_(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 ⊂ Ω_e, let q satisfy Assumption 1.1 and u_q ∈ L^∞(Ω) ∩ H^s(Rⁿ) be the unique solution to (2.3) provided g ∈ C_c^∞(V ) and kgk_L^∞_{(V )∩H}^s_(Rⁿ₎≤ δ₀. We now define the Dirichlet-to-Neumann map (DN-map):

Λ_q(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 Λ_q 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) Λ_q(1)(g) = Λ_q(2)(g)

for all g ∈ C_c^∞(V ) with kgk_L^∞_{(V )∩H}^s_(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 ∈ C_c^∞(V ) such that g 6≡ 0. If

(2.8) Λ_q(1)(g) = Λ_q(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

(5)

with q₁^(j) ∈ L^∞(Ω). Let u^(j) ∈ L^∞(Ω) ∩ H^s(Rⁿ) be the unique solution of (2.3) with u^(j) = g in Ω_e, where 0 6≡ g ∈ C_c^∞(V ). If

Λ_q(1)(g) = Λ_q(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 C_Kerr > 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 (Ω_e)_T := (0, T ) × Ω_e

with g ∈ C_c^∞((Ω_e)_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 Ω_T, Q⁽¹⁾(∂_t), · · · , Q^(k)(∂_t)u₀ = 0 on {0} × Ω,

u₀ = g in (Ω_e)_T.

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

Assumption 2.7. Let q₀ ∈ L^∞(Ω_T). For each F ∈ L^∞(Ω_T) and g ∈ C_c^∞((Ω_e)_T), there exists a unique solution w of

(2.12)







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

w = g in (Ω_e)_T,

and satisfying

kwk_L^∞_{(0,T ;H}^s_(Rⁿ_))∩L^∞_(Ω_T₎ ≤ C kF k_L^∞_(Ω_T₎+ kgk∗

with some suitable norm k · k∗.

(6)

Remark 2.8. In view of Assumption 2.7, the unique solution u₀ of (2.11) satisfies (2.13) ku₀k_L^∞_{(0,T ;H}^s_(Rⁿ_))∩L^∞_(Ω_T₎ ≤ 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 (Ω_e)_T.

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

(2.14) kgk∗ = kgk_L^∞_{(0,T ;H}^s_(Rⁿ_))∩L^∞_((Ω_e₎_T₎+ k(−∆)^sgk_L²_(Ω_T₎.

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 Ω_T,

w = ∂_tw = 0 on {0} × Ω,

w = g in (Ω_e)_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 H^s ⊂ 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) = ∂_t^β with 1 < β < 2 be the Caputo derivative. Then (2.12) becomes







∂_t^β + (−∆)^sw + q₀(·)w = F in Ω_T,

w = ∂_tw = 0 on {0} × Ω,

w = g in (Ω_e)_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∗ = kgk_L^∞_{(0,T ;H}^s_(Rⁿ₎₎+ k(−∆)^sgk_L²_(Ω_T₎

for n = 1 and ¹₂ < s < 1 (since the proof utilized the continuous embedding of H^s ⊂ 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 ; H^s(Rⁿ)) ∩ L^∞(Ω_T) of (2.9) having the following estimate

(2.15) ku − u₀k_L∞(0,T ; ˜H^s(Ω))∩L^∞(ΩT) ≤ Cω(Ckgk∗)kgk∗, where u₀ is the unique solution of (2.11).

(7)

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

(2.16) kuk_L^∞_{(0,T ;H}^s_(Rⁿ_))∩L^∞_(Ω_T₎≤ Ckgk∗.

2.2.2. Inverse problems. Fixing any open sets V, W ⊂ Ω_e 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:

Λ_q(j)(g) := (−∆)^su^(j)|_W_T

for all g ∈ C_c^∞(V_T) 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 < · · · < α_L < ∞ and the functions q^(j)₀ , q₁^(j), · · · , q_L^(j) ∈ C⁰(Ω). Suppose that Assumption 2.7 holds. Let g ∈ C_c^∞(VT) be given satisfying g(t, x) 6≡ 0, then

Λ_q(1)(g) = Λ_q(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 ∈ C_c^∞(V_T), let P_q₀g := w, where w is the unique solution of (2.12) with F ≡ 0. We assume the set

{ Pq0g|_Ω_T g ∈ C_c^∞(V_T) } is dense in L²(Ω_T).

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) ∈ Ω_T and z ∈ C,

where 0 < α₁ < · · · < α_L < ∞ and q^(j)₀ , q₁^(j), · · · , q_L^(j)∈ L^∞(Ω_T). Suppose that Assumption2.7 and Assumption 2.16 hold. If

Λ_q(1)(g) = Λ_q(2)(g)

for all g ∈ C_c^∞(V_T) with kgk∗ ≤ δ₀, where δ₀ is defined in Theorem 2.12, then q⁽¹⁾ ≡ q⁽²⁾.

(8)

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 C^s( ¯Ω).

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

∞

X

k=1

q_k(x)z^k

k!, q₁ ≥ 0 and all q_k ∈ C^s( ¯Ω).

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

Q(x, z) =

L

X

k=0

q_k(x)z^k

and q_k ∈ C^s( ¯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.

(9)

• [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 nonlinear potentials considered in that paper mimick assumption (3.1).

Finally, we want to point out that the global uniqueness of determining the Hartree potential (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) ∩ ˜H^s(Ω) by SF := v, where v is the unique solution of

((−∆)^sv + q0v = F in Ω,

v = 0 in Ω_e,

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

(4.1) kSF k_L∞(Ω)∩ ˜H^s(Ω) ≤ CkF k_L^∞_(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 ∈ C_c^∞(Ω_e) satisfying kgk_L^∞_(Ω_e_)∩H^s_(Rⁿ₎ = δ ≤ δ₀, where δ₀ > 0 is a small constant to be determined later. We consider the Banach space

Xδ :=

n

v ∈ L^∞(Ω) ∩ ˜H^s(Ω) kvk_L∞(Ω)∩ ˜H^s(Ω) ≤ δ o .

(10)

We first show that

(4.3) SF (v) ∈ X_δ for all v ∈ X_δ.

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

(4.4)

kSF (v)k_L∞(Ω)∩ ˜H^s(Ω) ≤ CkF (v)k_L^∞_(Ω)

≤ Cω(kv + u₀k_L^∞_(Ω))kv + u₀k_L^∞_(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 X_δ.

Let v₁, v₂ ∈ X_δ 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)k_L∞(Ω)∩ ˜H^s(Ω)

≤ CkF (v₁) − F (v₂)k_L^∞_(Ω)

≤ C ω(kv₁+ u₀k_L^∞_(Ω)) + ω(kv₂ + u₀k_L^∞_(Ω))kv1− v₂k_L^∞_(Ω)

≤ Cω(Cδ)kv₁− v₂k_L∞(Ω)∩ ˜H^s(Ω).

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

2kv1− v2k_L∞(Ω)∩ ˜H^s(Ω)

and therefore (4.5).

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

(4.7) v = SF (v),

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

kvk_L∞(Ω)∩ ˜H^s(Ω)≤ Cω(kv + u₀k_L^∞_(Ω))kv + u₀k_L^∞_(Ω)

≤ Cω(Cδ)kvk_L^∞_(Ω)+ Cω(Cδ)ku₀k_L^∞_(Ω)

≤ Cω(Cδ)kvk_L∞(Ω)∩ ˜H^s(Ω)+ Cω(Cδ)δ.

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

(11)

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)|)|

≤ kq₁k_L^∞_(Ω) Z

Rⁿ

|x − y|^−d

|w₁(y)|²− |w₂(y)|² dy

≤ kq₁k_L^∞_(Ω) kw₁k_L^∞_(Rⁿ₎+ kw₂k_L^∞_(Rⁿ₎ Z

Rⁿ

|x − y|^−d

|w₁(y)| − |w₂(y)|

dy

≤ kq1k_L^∞_(Ω) kw1k_L^∞_(Rⁿ₎+ kw2k_L^∞_(Rⁿ₎ Z

Rⁿ

|x − y|^−d|w1(y) − w2(y)| dy

≤ kq₁k_L^∞_(Ω) kw₁k_L^∞_(Rⁿ₎+ kw₂k_L^∞_(Rⁿ₎kw₁− w₂k_L^∞_(Ω) Z

Ω

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

(4.8) kq(·, |w₁|) − q(·, |w₂|)k_L^∞_(Ω)

≤ Ckq₁k_L^∞_(Ω) kw₁k_L^∞_(Rⁿ₎+ kw₂k_L^∞_(Rⁿ₎kw₁− w₂k_L^∞_(Ω), where C = C(d, n, Ω). Similarly, if w_j has compact support, we can also estimate (4.9) kq(·, |w_j|) − q₀k_L^∞_(Ω) ≤ C(d, n, Ω, supp (w_j))kq₁k_L^∞_(Ω)kw_jk²_L∞(Rⁿ). Combining (4.8) and (4.9), we obtain

(4.10)

kq(·, |w₁|)w₁− q(·, |w₂|)w₂− q₀(·)(w₁− w₂)k_L^∞_(Ω)

≤ kw₁k_L^∞_(R³₎kq(·, |w₁|) − q(·, |w₂|)k_L^∞_(Ω) + kq(·, |w₂|) − q₀k_L^∞_(Ω)kw₁− w₂k_L^∞_(Ω)

≤ ˜C kw₁k²_L∞(R³)+ kw₂k²_L∞(R³)kw₁− w₂k_L^∞_(Ω)

for all w_j ∈ L^∞(R³) with compact support and satisfying supp (w₁− w₂) ⊂ Ω, where ˜C = Ckq₁k_L^∞_(Ω). Let wj = v_j + 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 ∈ H^γ(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 Ω_e, then the set

R :=

v|_Ω ((−∆)^s+ q₀)v = 0 in Ω v = g ∈ C_c^∞(V )

(12)

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 Λ_q(j)

0

: L^∞(V ) ∩ H^s(Rⁿ) → H^−s(W ):

Λ_q(j) 0

(g) := (−∆)^su^(j)₀ |_W for g ∈ C_c^∞(V ), where u^(j)₀ ∈ L^∞(Ω) ∩ H^s(Rⁿ) is the unique solution of (2.1) with (5.1) u⁽¹⁾₀ = u⁽²⁾₀ = g in Ω_e.

Likewise, let u^(j) ∈ L^∞(Ω) ∩ H^s(Rⁿ) be the unique solution of (2.3) with

(5.2) u⁽¹⁾ = u⁽²⁾ = g in Ω_e,

where g ∈ C_c^∞(V ) satisfies kgk_L^∞_(Ω_e_)∩H^s_(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 kΛ_q(j)(g) − Λ_q(j)

0

(g)k_H^−s_{(W )} ≤ k(−∆)^s(u − u^(j)₀ )k_H^−s_(Rⁿ₎

≤ Cku − u^(j)₀ k_H^s_(Rⁿ₎ = Cku − u^(j)₀ kH˜^s(Ω) ≤ Ckgk^1+α_L∞(V )∩H¹ ^s(Rⁿ)

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

0

(g) − Λ_q(2) 0

(g)k_H^−s_{(W )}

≤

2

X

j=1

kΛ_q(j)(g) − Λ

q^(j)₀ (g)k_H^−s_{(W )}≤ Ckgk^1+α_L∞(V )∩H¹ ^s(Rⁿ). Observe that Λ_q^(j)

0

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

(5.4)

kΛ_q(1) 0

(g) − Λ_q(2) 0

(g)k_H^−s_{(W )}

= ⁻¹kΛ_q(1) 0

(g) − Λ_q(2) 0

(g)k_H^−s_{(W )}

≤ ⁻¹Ckgk^1+α_L∞(V )∩H¹ ^s(Rⁿ) = ^α¹Ckgk^1+α_L∞(V )∩H¹ ^s(Rⁿ). Taking → 0 in (5.4), we conclude that

Λ_q⁽¹⁾

0 (g) = Λ_q⁽²⁾

0 (g) for all g ∈ C_c^∞(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,

(13)

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

(5.7) q_m⁽¹⁾ ≡ q_m⁽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 − q_m⁽²⁾||u|^1+α^m =

(q_m⁽¹⁾− q⁽²⁾_m )|u|^α^mu

=

L

X

`=m+1

(q_`⁽¹⁾− q_`⁽²⁾)|u|^α^`u

≤ Ckgk^1+α_L∞(Ω^m+1e)∩H^s(Rⁿ), and thus

|q⁽¹⁾_m − q_m⁽²⁾|^1+αm¹ |u| ≤ Ckgk

1+αm+1 1+αm

L^∞(Ωe)∩H^s(Rⁿ). Consequently, from (2.4), it follows that

|q_m⁽¹⁾− q⁽²⁾_m |^1+αm¹ |u₀|

≤ |q⁽¹⁾_m − q_m⁽²⁾|^1+αm¹ |u₀− u| + |q_m⁽¹⁾− q_m⁽²⁾|^1+αm¹ |u|

≤ C kgk^1+α_L∞(Ω¹ e)∩H^s(Rⁿ)+ kgk

1+αm+1 1+αm

L^∞(Ωe)∩H^s(Rⁿ).

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

(5.9)

|q⁽¹⁾_m − q_m⁽²⁾|^1+αm¹ |u₀|

= ⁻¹|q⁽¹⁾_m − q_m⁽²⁾|^1+αm¹ |u₀|

≤ ⁻¹C kgk^1+α_L∞(Ω¹e)∩H^s(Rⁿ)+ kgk

1+αm+1 1+αm

L^∞(Ωe)∩H^s(Rⁿ)

≤ ^αC kgk^1+α_L∞(Ω¹e)∩H^s(Rⁿ)+ kgk

1+αm+1 1+αm

L^∞(Ωe)∩H^s(Rⁿ)

with

α = min

α₁,1 + αm+1

1 + α_m − 1

> 0.

Taking → 0 in (5.9) gives

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

⇐⇒ |q_m⁽¹⁾− q⁽²⁾_m |^1+αm¹ u₀ = 0 in Ω.

Then we have

Z

|q_m⁽¹⁾− q⁽²⁾_m |^1+αm¹ u0dx = 0 for all g ∈ C_c^∞(V ).

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

|q⁽¹⁾_m − q_m⁽²⁾|^1+αm¹ ≡ 0,

which implies (5.7).

(14)

We now prove Theorem 2.4 in which less data are used. To begin, we recall the following 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 ∈ H^s(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 Λ_q(1)

0

(g) = Λ_q(2) 0

(g)

for any fixed 0 6= g ∈ C_c^∞(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) q_m⁽¹⁾ ≡ q_m⁽²⁾.

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

(5.16) |q_m⁽¹⁾− q⁽²⁾_m |^1+αm¹ u0 = 0 in Ω.

Similarly, (5.16) implies

q⁽¹⁾_m (x) = q_m⁽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.

(15)

Proof of Theorem 2.5. By the assumptions of

Λ_q(1)(g) = Λ_q(2)(g),

u⁽¹⁾ = u⁽²⁾ in Ω_e, 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 ; H^s(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 (Ω_e)_T,

where q0 ∈ L^∞(ΩT), and there exists a constant C such that (6.1) kSF k_L^∞_{(0,T ;H}^s_(Rⁿ_))∩L^∞_(Ω_T₎ ≤ CkF k_L^∞_(Ω_T₎.

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 Ω_T, Q⁽¹⁾(∂_t), · · · , Q^(k)(∂_t)v = 0 on {0} × Ω,

v = 0 in (Ω_e)_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

X_δ := v ∈ L^∞(0, T ; H^s(Rⁿ)) ∩ L^∞(Ω_T) kvk_L^∞_{(0,T ;H}^s_(Rⁿ_))∩L^∞_(Ω_T₎ ≤ δ . We first show that

(6.3) SF (v) ∈ X_δ for all v ∈ X_δ.