Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 41, pp. 1–8.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.41

UNCONDITIONAL WELL-POSEDNESS FOR THE NONLINEAR

SCHRÖDINGER EQUATION IN BESSEL POTENTIAL SPACES

RYOSUKE HYAKUNA

Abstract. The Cauchy problem for the nonlinear Schrödinger equation is called uncondition-

ally well posed in a data space E if it is well posed in the usual sense and the solution is unique in
the space C([0, T ];E). In this paper, this notion of the unconditional well-posedness is redefined

so that it covers Lp-based Sobolev spaces as data space E and it is equivalent to the usual one
when E is an L2-based Sobolev space Hs. Based on this definition, it is shown that the Cauchy

problem for the 1D cubic NLS is unconditionally well posed in Bessel potential spaces Hs
p for

4/3 < p ≤ 2 under certain assumptions on s.

1. Introduction

We consider the Cauchy problem for the one dimensional cubic nonlinear Schrödinger equation

iut + uxx + |u|2u = 0, u(0) = ϕ ∈ E, (1.1)

where E is a Banach space of complex valued functions on R. Recall that Cauchy problem (1.1)
is called locally well posed in the data space E if, for any ϕ ∈ E there are a T = T (ϕ) > 0 and

a unique solution u of (1.1) in the space C([0, T ];E) ∩ YT ≜ ZT and the map ϕ 7→ u is locally
Lipschitz from E to ZT , where YT is a space of functions on [0, T ] × R. For example, (1.1) is
locally (and globally) well posed in Hs for s ≥ 0. The space YT is called an auxiliary space and
in some cases the well-posedness can be shown without using this space. In fact, if s > 1/2 the
solution can be obtained directly in a closed subset of L∞([0, T ];Hs) via the standard fixed point
argument and the local well-posedness holds with ZT = C([0, T ];Hs). This is possible because the
space Hs forms an algebra if s > 1/2. On the other hand, when s < 1/2, the solution of (1.1) is
usually constructed in the space L∞([0, T ];Hs)∩Lq([0, T ];Hs

r ) for a suitable choice of (q, r) with
the aid of the so-called Strichartz estimate (see e.g. [2]). In this case, YT = Lq([0, T ];Hs

r ), and the
uniquness is shown in this space, not in C([0, T ];Hs). Therefore, it is natural to ask if the well-
posedness holds with ZT = C([0, T ];E) for s < 1/2. This problem of whether or not the auxiliary
condition is removable was first proposed by Kato [11]. He said (1.1) is unconditionally well posed
in E if it is well posed with ZT = C([0, T ];E), or equivalently, the uniqueness of solutions holds in
the space C([0, T ];E) in addition to its well-posedness. In [11] he showed that the trivial sufficient
condition s > 1/2 for unconditional well-posedness in Hs can be pushed down to s = 1/6. It is
known that this is the minimal Sobolev regularity for the unconditional well-posedness of the 1D
cubic NLS (1.1). Thus, the fact that u ∈ Lq([0, T ];Hs

r ) is simply an additional regularity property
of the solution which is removable in the uniqueness assertion for s ≥ 1/6, while it is required to
ensure the well-posedness in Hs for s < 1/6. Since the pioneering work [11], the problem of the
unconditional well-posedness for nonlinear Schrödinger equations and other nonlinear dispersive
equations has been extensively studied. We refer to [13] and references therein for earliear results
in this direction.

2020 Mathematics Subject Classification. 35Q55.
Key words and phrases. Nonlinear Schrödinger equation; unconditional well-posedness; Lp-space;

Bessel potential space.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted August 24, 2024. Published April 16, 2025.

1



2 R. HYAKUNA EJDE-2025/41

The aim of this article is to extend the unconditional well-posedness results for the L2-based
Sobolev spaces Hs to the Bessel potential spaces Hs

p . One might think discussing such a problem
is meaningless, since if p ̸= 2 the Schrödinger equations are not well posed in Hs

p even in the linear
case (see [1, 7]). In particular, we cannot expect the persistence property u(t) ∈ Hs

p for data
u(0) ∈ Hs

p unless p = 2 let alone the uniqueness issue in the space C([0, T ];Hs
p). Nevertheless, we

stress that the problem of the unconditional well-posedness can be generalized to the Lp-setting
in a natural manner. Our idea here is motivated by Zhou. In [16] he considered the “twisted”

variable v(t) ≜ e−it∂2
xu(t) and rewrite the integral equation corresponding to (1.1) as

v(t) = ϕ+ i

∫ t

0

e−iτ∂2
x
[
(eiτ∂

2
xv(τ))(eiτ∂

2
xv(τ))(eiτ∂

2
xv(τ))

]
dτ. (1.2)

Note that (1.2) is well-known as the integral equation and v as a state both in the interaction
picture. Then he showed that a local solution v of (1.2) exists in the space C([0, T ] ; Lp) for
ϕ ∈ Lp, 1 < p < 2. This result implies the existence of a solution of the original Cauchy problem

(1.1) such that e−it∂2
xu(t) ∈ C([0, T ] ;Lp). Especially, note that if p = 2, this is equivalent to the

usual persistence property of the solution, that is, u ∈ C([0, T ];L2). Thus, a suitable space to
discuss the problem of the unconditional uniqueness in the Lp-space is

{u : e−it∂2
xu(t) ∈ C([0, T ];Lp)}. (1.3)

In this article, we formulate the notion of the unconditional well-posedness and uniqueness in the
Lp, and more generally, the Bessel potential spaces Hs

p based on this idea. Then, we show that
(1.1) is unconditionally locally well posed in Hs

p under some assumptions on the Sobolev regularity.

Finally, for the unconditional well-posedenss in other non-L2-based spaces, we refer to [14], where
the authors discuss the unconditional uniqueness of the periodic NLS in the Fourier-Lebesgue
spaces FLp as data space. Note also that for the Fourier-Lebesgue spaces, one does not need to
consider the twisted variable as in the case of Hs

p , since the solution is expected to have the usual
persistence property u ∈ C([0, T ];FLp). See [6, 10].

Notation. For 1 ≤ p ≤ ∞ and I ⊂ R, ∥f∥Lp(I) denotes the usual Lp-norm. p′ is the conjugate

exponent of p: 1/p + 1/p′ = 1. The Fourier transform of ϕ is denoted by ϕ̂. We denote by C a
positive constant which may vary from line to line. The Bessel potential spaces are defined by

Hs
p(R) ≜ {ϕ ∈ S ′(R) : ⟨D⟩sϕ ∈ Lp(R)},

equipped with the norm

∥ϕ∥Hs
p(R) ≜ ∥⟨D⟩sϕ∥Lp(R) ,

where ⟨̂D⟩sϕ(ξ) = (1 + |ξ|2)s/2ϕ̂(ξ). In particular, we write Hs = Hs
2 as usual. For simplicity we

often write Lp, Hs
p to denote Lp(R), Hs

p(R) respectively. For a function u : I × R → C, we set

∥u∥Lq(I;Hs
p)

≜
(∫

I

∥u(t, ·)∥qHs
p
dt
)1/q

.

We define function spaces C(I;E) and L∞(I;E) to introduce a concept of well-posedness of
(1.1) in Lp.

Definition 1.1. Let E ⊂ S ′(R) and I ⊂ R. The space C(I;E) is defined by

C(I;E) ≜ {u = eit∂
2
xv(t) : v ∈ C(I;E)}

equipped with the norm

∥u∥C(I;E) ≜ sup
t∈I

∥e−it∂2
xu∥E .

Similarly, the space L∞(I;E) is defined by

L∞(I;E) ≜ {u = eit∂
2
xv(t) : v ∈ L∞(I;E)}

equipped with the norm

∥u∥L∞(I;E) ≜ ∥e−it∂2
xu∥L∞(I;E).



EJDE-2025/41 UNCONDITIONAL WELL-POSEDNESS FOR NLS IN BESSEL POTENTIAL SPACES 3

We first give the precise definition of “twisted” local well-posedness for (1.1) which was intro-
duced by Zhou.

For a normed space E, BE(r) denotes the closed ball with the radius r > 0, namely BE(r) ≜
{u ∈ E : ∥u∥E ≤ r}.

Definition 1.2. Let E be a normed space of functions on R. Cauchy problem (1.1) is locally well
posed in E if, for any M > 0 there exist a TM > 0 and a space YTM

of functions on [0, TM ] × R
such that: for any ϕ ∈ BE(M) there is a unique solution in u ∈ ZTM

≜ C([0, TM ];E) ∩ YTM
.

Moreover, the map ϕ 7→ u is Lipschitz from BE(M) to ZTM
.

Now we give the definition of the unconditional well-poseness for (1.1).

Definition 1.3. Let E be the same as Definition 1.2. Cauchy problem (1.1) is unconditionally
locally well posed in E if it is locally well posed in the sense of Definition 1.2 with ZTM

=
C([0, TM ];E), namely the uniqueness holds in the space C([0, TM ];E).

Remark 1.4. Note that in the case of E = Hs, (1.1) is locally well posed in the sense of Definition
1.2 if and only if it is locally well posed in the usual sense, since C([0, T ];Hs) = C([0, T ];Hs).
Similarly, it is unconditionally locally well posed in the sense of Definition 1.3 if and only if it is
unconditionally locally well posed in the usual sense.

Our results in this paper are as follows. We begin with the “conditional” well-posedness result.

Proposition 1.5. Let 4/3 < p ≤ 2 and 0 < s < 3/2 − 2/p. Then (1.1) is locally well posed in
Hs

p(R) in the sense of Definition 1.2 with YTM
= Lq([0, TM ];Hs

r ) for suitable q, r.

The conditional well-posedness results in Hs
p can be proved arguing similarly as in the proof

of [9, Theorem 1.1]. We prove this proposition at the end of this paper. The main results of
this paper is the unconditional well-posedness for the Cauchy problem under additional regularity
assumptions on the data.

Theorem 1.6. Let 4/3 < p ≤ 2 and s < 3/2 − 2/p. Then (1.1) is unconditionally locally well
posed in Hs

p(R) in the sense of Definition 1.3 if:

(1) 4/3 < p ≤ 3/2 and s > 0;
(2) 3/2 < p ≤ 2 and s ≥ 2/3− 1/p.

Remark 1.7. Let sc(p) ≜ max(0, 2/3− 1/p) for 1 ≤ p ≤ 2. Then Theorem 1.6 asserts that (1.1)
is unconditionally locally well posed in Hs

p for s > sc(p) when 4/3 < p ≤ 3/2 and for s ≥ sc(p)
when 3/2 < p ≤ 2. The exponent sc(p) is considered as a natural threshold for the unconditional
well-posedness in the following sense. In the case of the cubic NLS, one need |u|2u ∈ L1

loc so that
the nonlinear part makes sense in the distributional framework(see e.g. [13]). Thus we require

u ∈ L3
loc. Now we assume e−it∂2

xu(t) ∈ Hs
p . Then, by the well-known time decay property of the

evolution group eit∂
2
x , we have u(t) ∈ Hs

p′ , t ̸= 0. When 3/2 < p ≤ 2, we see that u(t) ∈ L3 if

s ≥ sc(p) = 2/3− 1/p by Sobolev’s embedding. When 1 ≤ p ≤ 3/2, we have u(t) ∈ Hs
p′ ⊂ Lp′

for

s ≥ sc(p) = 0 and thus we see that u(t) ∈ L3
loc since 3 ≤ p′ ≤ ∞.

2. Proof of main results

2.1. Key lemma and proposition. Our unconditional well-posedness results are proved by a
basic embedding theorem for L∞(I;Hs

p) and Strichartz type estimates.

Lemma 2.1. Let I ⊂ R be a finite interval. Let 2 ≤ q, r ≤ ∞, 1 ≤ p ≤ 2, s ≥ 0 satisfy

s ≥ 1− 1

p
− 1

r
, q

(1
p
− 1

2

)
< 1 (with the convention that ∞ · 0 = 0).

Then

∥f∥Lq(I;Lr(R)) ≤ CI∥f∥L∞(I;Hs
p(R)).



4 R. HYAKUNA EJDE-2025/41

Proof. Let 2 ≤ r ≤ ∞. We first fix t ∈ I. Then, by Sobolev’s embedding and the well-known

decay property of the free group eit∂
2
x , we have

∥eit∂
2
xf(t)∥Lr ≤ C∥eit∂

2
xf(t)∥

Ḣ
s(r)

p′
≤ C(4π|t|)−(1/p−1/2)∥f(t)∥

Ḣ
s(r)
p

,

where s(r) = 1− 1/p− 1/r. Replacing f with e−it∂2
xf , we obtain

∥f(t)∥Lr ≤ C(4π|t|)−( 1
p−

1
2 )∥e−it∂2

xf(t)∥Hs
p
, (2.1)

for any s ≥ s(r) and t ∈ I. The desired embedding is obtained after taking ∥ · ∥Lq(I)-norm of both
sides and applying Hölder’s inequality in the right hand side. □

The next key estimates are the inhomogeneous Strichartz inequalities for not necessarily ad-
missible pairs.

Proposition 2.2 (See [5, 12, 15]). Let 2 ≤ q, r < ∞, 1 ≤ ρ ≤ 2, 1 < γ ≤ 2 be such that

2 +
2

q
+

1

r
=

2

γ
+

1

ρ
,

1

q
+

1

r
<

1

2
,

3

2
− 1

ρ
<

1

γ
< 1.

Then ∥∥∫ t

0

ei(t−τ)∂2
xF (τ)dτ

∥∥
Lq(I;Lr(R)) ≤ C∥F∥Lγ(I;Lρ(R)). (2.2)

Proof of Theorem 1.6. We prove the unconditional well-posedness results. We assume that the
conditional local well-posedness result (Proposition 1.5) holds, which will be proved in the last
section. Then it is enough to show the uniqueness of the solution in the space C([0, T ];Hs

p) to
conclude the unconditional well-posedness. We first consider case (i) and we let 4/3 < p ≤ 3/2
and s > 0. We may assume s is sufficiently small, since the uniqueness of the solution in a function
space also implies the uniqueness in any smaller spaces. Let δ > 0 and we set

1

q
=

1

p
− 1

2
+

δ

2
,

1

r
= 1− 1

p
− δ,

1

ρ
= 3− 3

p
− 3δ(=

3

r
),

1

γ
=

2

p
− 1

2
+

3

2
δ.

Then the pair (q, r) satisfies the assumption of Lemma 2.1 with s = δ and the quadruple (q, r, γ, ρ)
satisfies the assumption of Proposition 2.2 if δ is sufficiently small. Now we consider two solutions
u, v ∈ C([0, T ];Hδ

p(R)) of (1.1) with u(0) = v(0) ∈ Hδ
p(R). We want to show that u(t) = v(t) for

all t ∈ [0, T ]. By Lemma 2.1 we see that u, v ∈ Lq([0, T ];Lr). Let 0 < T0 ≤ T . We estimate the
difference ∥u− v∥Lq([0,T0];Lr). By Duhamel’s formula and Proposition 2.2,

∥u− v∥Lq([0,T0];Lr) =
∥∥∫ t

0

ei(t−τ)∂2
x(|u|2u− |v|2v)dτ

∥∥
Lq([0,T0];Lr)

≤ C∥|u|2u− |v|2v∥Lγ([0,T0];Lρ).

By Hölder’s inequality in the time variable and the triangle inequality, the norm in the right hand
side is estimated by

∥u2(ū− v̄)∥Lγ([0,T0];Lρ) + ∥uv̄(u− v)∥Lγ([0,T ];Lρ) + ∥vv̄(u− v)∥Lγ([0,T0];Lρ)

≤ CT
1− 1

p

0

(
∥u∥2Lq([0,T ];Lr) + ∥u∥Lq([0,T ];Lr)∥v∥Lq([0,T ];Lr) + ∥v∥2Lq([0,T ];Lr)

)
∥u− v∥Lq([0,T0];Lr).

We put ηT ≜ max
(
∥u∥Lq([0,T ];Lr), ∥v∥Lq([0,T ];Lr)

)
. Then we have

∥u− v∥Lq([0,T0];Lr) ≤ 3CT
1− 1

p

0 η2T ∥u− v∥Lq([0,T0];Lr). (2.3)

Now we choose T0 so that

3CT
1− 1

p

0 η2T ≤ 1

2
.

Then ∥u−v∥Lq([0,T0];Lr) = 0 and thus u(t) = v(t) for all t ∈ [0, T0]. In a similar manner, we can see
that u(t) = v(t) for all t ∈ [T0, 2T0], since the time interval T0 can be determined depending only
on ηT . Repeating this argument, we finally get u = v on [0, T ], which concludes the uniqueness in
the space C([0, T ];Hδ

p).



EJDE-2025/41 UNCONDITIONAL WELL-POSEDNESS FOR NLS IN BESSEL POTENTIAL SPACES 5

The uniqueness assertion for case (ii) of Theorem 1.3 can be treated in the same manner. We
let 3/2 < p ≤ 2 and s = sc(p) = 2/3− 1/p. For a sufficiently small δ > 0 we set

1

q
=

1

6
− δ,

1

r
=

1

3
,

1

ρ
= 1(=

3

r
),

1

γ
=

5

6
− δ.

Then it is easy to check that q, r, γ, ρ satisfy the assumption of Proposition 2.2 if δ is small enough.
Moreover, s, q, r satisfy the condition of Lemma 2.1 if δ < 2/3− 1/p. Thus

L∞([0, T ];Hs
p) ↪→ Lq([0, T ];Lr).

Then we get the desired uniqueness in C([0, T ];Hs
p) by considering two solutions u, v ∈ C([0, T ];Hs

p)
with u(0) = v(0) and estimating the difference u− v in Lq([0, T ];Lr) as in case (i). □

3. Proof of the conditional well-posedness result

3.1. Strichartz estimates. In the last section we prove the conditional well-posedness for (1.1)
in Hs

p . We exploit the standard Strichartz technique here. This approach is well known in the

case where the initial data lie in the L2 and Hs spaces (s ≥ 0). It is shown in [9] that it can also
work well for the case where data are in Lp, 4/3 < p < 2. We begin with the estimates for the
homogeneous equation.

Proposition 3.1 ([12, Theorem 3.2]). Let 1 < p ≤ 2 and let q, r ∈ [2,∞] be such that

2

q
+

1

r
=

1

p
,

1

q
+

1

r
<

1

2
.

Then
∥eit∂

2
xϕ∥Lq(R;Lr(R)) ≤ C∥ϕ∥Lp(R). (3.1)

In particular, for any s ∈ R we have

∥eit∂
2
xϕ∥Lq(R;Hs

r (R)) ≤ C∥ϕ∥Hs
p(R). (3.2)

Inequalitites (3.1) and (3.2) are used for estimating the linear part of the corresponding integral
equation. Moreover, when 3/4 < p ≤ 2, stronger estimates are known:

Proposition 3.2 ([10, Theorem 5]). Let 4/3 < p ≤ 2 and let q, r ∈ [2,∞] be such that

2

q
+

1

r
=

1

p
, 0 <

1

q
< min

(1
2
− 1

r
,
1

4

)
.

Then
∥eit∂

2
xϕ∥Lq(R;Lr(R)) ≤ C∥ϕ̂∥Lp′ (R). (3.3)

The diagonal case q = r = 3p of (3.3) goes back to Fefferman [4] and is used to obtain the
well-posedness results for (1.1) in [6, 3, 10]. In this paper we essentially use (3.3) to deal with the
nonlinear term. Indeed, the following equivalent form of Proposition 3.2 is useful in the estimates
of the Duhamel term of the integral equation.

Corollary 3.3 ([9, Proposition 2.6]). Let 4/3 < p ≤ 2 and let σ, ρ be such that (q, r) ≜ (σ′, ρ′)
satisfies the assumption of Proposition 3.2. Then

∥
∫ t

0

ei(t−τ)∂2
xF (τ)dτ∥L∞([0,T ];Lp(R)) ≤ C∥τ

1
p−

1
2F (τ)∥Lσ([0,T ];Lρ(R)).

In particular, for any s ∈ R we have

∥
∫ t

0

ei(t−τ)∂2
xF (τ)dτ∥L∞([0,T ];Hs

p(R)) ≤ C∥τ
1
p−

1
2F (τ)∥Lσ([0,T ];Hs

ρ(R)). (3.4)

Proof of Proposition 1.5. We first introduce several particular exponents. We set

1

q
=

1

2p
− 1

8
− s

4
,

1

r
=

1 + 2s

4
,

1

ρ
=

3

4
− s

2

(
=

3

r
− 2s

)
,

1

γ
=

5

8
+

s

4
+

1

2p
,

1

σ
=

9

8
− 1

2p
+

s

4

(
=

1

q′
)
.



6 R. HYAKUNA EJDE-2025/41

Then it is easy to check that (q, r, γ, σ), (q, r), and (ρ, σ) satisfy the assmption of Proposition 2.2,
Proposition 3.1, and Corollary 3.3 if 4/3 < p ≤ 2 and 0 < s < 3/2 − 2/p. Let T > 0. We claim
that

∥
∫ t

0

ei(t−τ)∂2
x(u1u2u3)dτ∥Lq([0,T ];Hs

r (R)) ≤ CT 1+s− 1
p

3∏
j=1

∥uj∥Lq([0,T ];Hs
r (R)), (3.5)

and

∥
∫ t

0

ei(t−τ)∂2
x(u1u2u3)dτ∥L∞([0,T ];Hs

p(R)) ≤ CT 1+s− 1
p

3∏
j=1

∥uj∥Lq([0,T ];Hs
r (R)), (3.6)

for all uj ∈ Lq([0, T ];Hs
r (R)), j = 1, 2, 3, where C is independent of T . We first prove (3.5).

Applying (2.2) with F = ⟨D⟩s(u1u2u3), we have

∥
∫ t

0

ei(t−τ)∂2
x(u1u2u3)dτ∥Lq([0,T ];Hs

r )
≤ C∥u1u2u3∥Lγ([0,T ];Hs

ρ(R)).

We estimate the right hand side. By the fractional Leibniz rule (see e.g. [8])

∥u1u2u3∥Hs
ρ(R) ≤ C

(
∥u1∥Hs

r (R)∥u2u3∥Lκ(R) + ∥u1∥L2κ(R)∥u2u3∥Hs(R)
)
,

where
1

κ
=

1

ρ
− 1

r
=

1

2
− s.

For the first term in the right hand side we have by Schwartz’s inequality and Sobolev’s embedding

∥u1∥Hs
r (R)∥u2u3∥Lκ(R) ≤ ∥u1∥Hs

r (R)∥u2∥L2κ(R)∥u3∥L2κ(R) ≤ C

3∏
j=1

∥uj∥Hs
r (R).

Noting the relation
1

2
=

(1
4
− s

2

)
+

(1
4
+

s

2

)
=

1

2κ
+

1

r
,

the second term can be estimated as

∥u1∥L2κ(R)∥u2u3∥Hs(R) ≤ C∥u1∥L2κ(R)
(
∥u2∥Hs

r (R)∥u3∥L2κ(R) + ∥u2∥L2κ(R)∥u3∥Hs
r (R)

)
,

the right hand side of which can be controlled by ∥u1∥Hs
r
∥u2∥Hs

r
∥u3∥Hs

r
by Sobolev’s embedding.

Consequently,

∥u1u2u3∥Lγ([0,T ];Hs
ρ(R)) ≤ C∥

3∏
j=1

∥uj∥Hs
r (R)∥Lγ([0,T ]) ≤ CT 1+s− 1

p

3∏
j=1

∥uj∥Lq([0,T ];Hs
r (R))

by Hölder’s inequality in the time variable. This concludes the proof of (3.5). To prove (3.6) we
use (3.4) to obtain

∥
∫ t

0

ei(t−τ)∂2
x(u1u2u3)dτ∥L∞([0,T ];Hs

p)
≤ C∥τ

1
p−

1
2u1(τ)u2(τ)u3(τ)∥Lσ([0,T ];Hs

ρ(R)).

As observed in the proof of (3.5), the norm ∥τ
1
p−

1
2u1(τ)u2(τ)u3(τ)∥Hs

ρ
can be controlled by

Cτ
1
p−

1
2 ∥u1(τ)∥Hs

r
∥u2(τ)∥Hs

r
∥u3(τ)∥Hs

r
. Thus the right hand side of the above inequality is bounded

from above by

C∥τ
1
p−

1
2

3∏
j=1

∥uj∥Lq([0,T ];Hs
r (R))∥Lσ([0,T ]) ≤ C∥τ

1
p−

1
2 ∥

L
σq

q−3σ ([0,T ])

3∏
j=1

∥uj∥Lq([0,T ];Hs
r (R))

≤ CT 1+s− 1
p

3∏
j=1

∥uj∥Lq([0,T ];Hs
r (R)),

where we have used Hölder’s inequality in the time variable. This proves (3.6).
Now we establish a local solution of the corresponding integral equation

u(t) = eit∂
2
xϕ+ i

∫ t

0

ei(t−τ)∂2
x(|u|2u)dτ.



EJDE-2025/41 UNCONDITIONAL WELL-POSEDNESS FOR NLS IN BESSEL POTENTIAL SPACES 7

We find a fixed point of the operator

Su ≜ eit∂
2
xϕ+ i

∫ t

0

ei(t−τ)∂2
x(|u|2u)dτ

in a suitable closed subset of L∞([0, T ];Hs
p) ∩ Lq([0, T ];Hs

r ). For T,R > 0 we define

VT,R ≜ {u ∈ L∞([0, T ];Hs
p) ∩ Lq([0, T ];Hs

r ) | ∥u∥L∞([0,T ];Hs
p)

+ ∥u∥Lq([0,T ];Hs
r )

≤ R}

equipped with the distance

d(u, v) ≜ ∥u− v∥L∞([0,T ];Hs
p)

+ ∥u− v∥Lq([0,T ];Hs
r )
.

By (3.2), (3.5), (3.6), for any u ∈ VT,R we have

∥Su∥L∞([0,T ];Hs
p)

+ ∥Su∥Lq([0,T ];Hs
r )

≤ C
(
∥ϕ∥Hs

p
+ T 1+s− 1

p ∥u∥3Lq([0,T ];Hs
r )

)
≤ C

(
∥ϕ∥Hs

p
+ T 1+s− 1

pR3
)
.

Now we take T,R so that

R = 2C∥ϕ∥Hs
p
, T 1+s− 1

p ≤ R−2

4C
.

Then the right-hand side is smaller than R/2+R/4 ≤ R and thus S : VT,R → VT,R is well defined.
Similarly, for u, v ∈ VT,R

d(Su, Sv) ≤ CT 1+s− 1
p (∥u∥2Lq([0,T ];Hs

r )
+ ∥u∥Lq([0,T ];Hs

r )
∥v∥Lq([0,T ];Hs

r )
+ ∥v∥2Lq([0,T ];Hs

r )
)

× ∥u− v∥Lq([0,T ];Hs
r )

≤ C
R−2

4C
3R2∥u− v∥Lq([0,T ];Hs

r )

≤ 3

4
d(u, v),

from which we see that S : VT,R → VT,R is a contraction mapping. Consequently, a local solution

v ∈ L∞([0, T ];Hs
p) ∩ Lq([0, T ];Hs

r ) with T ∼ ∥ϕ∥
− 2

1+s− 1
p

Hs
p

can be constructed by the standard

fixed point theorem. Finally a difference estimate similar to the proof of Theorem 1.3 gives the
uniqueness of the solution in the space Lq([0, T ];Hs

r ) and the continuous dependence on data. □

Acknowledgments. The author would like to express his gratitude to the anonymous referee
for his/her careful reading of the paper and valuable comments. This work was supported by the
JSPS KAKENHI Grant Number 20K03710.

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Ryosuke Hyakuna

Polytechnic University of Japan, Tokyo, Japan
Email address: 107r107r@gmail.com


	1. Introduction
	2. Proof of main results
	2.1. Key lemma and proposition

	3. Proof of the conditional well-posedness result
	3.1. Strichartz estimates
	Acknowledgments

	References