Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomogeneous nonlinear Schr\"odinger equation

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Scattering theory in a weighted L

²

space for a class of the defocusing inhomogeneous nonlinear Schr’́odinger

equation

van Duong Dinh

To cite this version:

van Duong Dinh. Scattering theory in a weightedL²space for a class of the defocusing inhomogeneous nonlinear Schr’́odinger equation. 2017. �hal-01610390�

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THE DEFOCUSING INHOMOGENEOUS NONLINEAR SCHR ¨ODINGER EQUATION

VAN DUONG DINH

Abstract. In this paper, we consider the inhomogeneous nonlinear Schr¨odinger equation (INLS), namely

i∂tu+ ∆u+µ|x|^−b|u|^αu= 0, u(0) =u0∈H¹,

withb, α > 0. We firstly recall a recent result on the local well-posedness for the (INLS) of Guzman [15], and improve this result in the two and three spatial dimensional cases. We next study the decay of global solutions for the defocusing (INLS), i.e. µ=−1 when 0< α < α^? whereα^?=^4−2b_d−2 ford≥3, andα^?=∞ford= 1,2 by assuming that the initial data belongs to the weighted L² space Σ = {u ∈ H¹(R^d) :|x|u ∈ L²(R^d)}. We finally combine the local theory and the decaying property to show the scattering in Σ for the defocusing (INLS) in the caseα?< α < α^?, whereα?=^4−2b_d .

1. Introduction

Consider the inhomogeneous nonlinear Schr¨odinger equation, namely i∂tu+ ∆u+µ|x|^−b|u|^αu = 0,

u(0) = u0, (INLS)

where u : R×R^d → C, u₀ : R^d → C, µ = ±1 and α, b > 0. The terms µ = 1 and µ = −1 correspond to the focusing and defocusing cases respectively. The case b = 0 is the well-known nonlinear Schr¨odinger equation which has been studied extensively over the last three decades.

The inhomogeneous nonlinear Schr¨odinger equation arises naturally in nonlinear optics for the propagation of laser beams, and it is of a form

i∂tu+ ∆u+K(x)|u|^αu= 0. (1.1)

The (INLS) is a particular case of (1.1) withK(x) =|x|^−b. The equation (1.1) has been attracted a lot of interest in a past several years. Berg´e in [1] studied formally the stability condition for soliton solutions of (1.1). Towers-Malomed in [21] observed by means of variational approximation and direct simulations that a certain type of time-dependent nonlinear medium gives rise to completely stabe beams. Merle in [17] and Rapha¨el-Szeftel in [18] studied the problem of existence and nonexistence of minimal mass blowup solutions for (1.1) with k₁ < K(x) < k₂ and k₁, k₂ > 0.

Fibich-Wang in [10] investigated the stability of solitary waves for (1.1) with K(x) := K(|x|) where >0 is small andK ∈C⁴(R^d)∩L^∞(R^d). The caseK(x) = |x|^b withb >0 is studied by many authors (see e.g. [4,5,16,25] and references therein).

2010Mathematics Subject Classification. 35P25, 35Q55.

Key words and phrases. Inhomogeneous nonlinear Schr¨odinger equation; Local well-posedness; Decay solutions;

Virial identity; Scattering; WeightedL²space.

1

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In order to review the known results for the (INLS), we recall some facts for this equation. We firstly note that the (INLS) is invariant under the scaling,

uλ(t, x) :=λ^2−b^α u(λ²t, λx), λ >0.

An easy computation shows

kuλ(0)kH˙^γ(R^d)=λ^γ+^2−b^α ⁻^d²ku0kH˙^γ(R^d). Thus, the critical Sobolev exponent is given by

γ_c :=d

2−2−b

α . (1.2)

Moreover, the (INLS) has the following conserved quantities:

M(u(t)) :=ku(t)k²_L2(R^d)=M(u0), (1.3) E(u(t)) :=1

2k∇u(t)k²_L2(R^d)−µG(t) =E(u₀), (1.4) where

G(t) := 1 α+ 2

Z

|x|^−b|u(t, x)|^α+2dx. (1.5) The well-posedness for the (INLS) was firstly studied by Genoud-Stuart in [11, Appendix] (see also [13]). The proof is based on the abstract theory developed by Cazenave [2] which does not use Strichartz estimates. Precisely, the authors showed that the focusing (INLS) with 0 < b <

min{2, d}is well posed inH¹(R^d):

• locally if 0< α < α^?,

• globally for any initial data if 0< α < α?,

• globally for small initial data ifα_?≤α < α^?. Here α? andα^? are defined by

α?:=4−2b

d , α^?:=

( _4−2b

d−2 ifd≥3,

∞ ifd= 1,2. (1.6)

In the case α=α? (L²-critical), Genoud in [12] showed that the focusing (INLS) with 0< b <

min{2, d}is globally well-posed inH¹(R^d) assumingu0∈H¹(R^d) and ku0k_L2(R^d)<kQk_L2(R^d),

where Q is the unique nonnegative, radially symmetric, decreasing solution of the ground state equation

∆Q−Q+|x|^−b|Q|^4−2b^d Q= 0. (1.7)

Also, Combet-Genoud in [6] established the classification of minimal mass blow-up solutions for the focusingL²-critical (INLS).

In the case α_? < α < α^?, Farah in [7] showed that the focusing (INLS) with 0< b <min{2, d}

is globally well-posedness inH¹(R^d) assumingu₀∈H¹(R^d) and

E(u₀)^γ^cM(u₀)^1−γ^c< E(Q)^γ^cM(Q)^1−γ^c, (1.8) k∇u₀k^γ_L^c₂₍

R^d)ku₀k^1−γ_L₂₍^c

R^d)<k∇Qk^γ_L^c₂₍

R^d)kQk^1−γ_L₂₍^c

R^d), (1.9)

where Q is the unique nonnegative, radially symmetric, decreasing solution of the ground state equation

∆Q−Q+|x|^−b|Q|^αQ= 0. (1.10)

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Note that the existence and uniqueness of nonnegative, radially symmetric, decreasing solutions to (1.7) and (1.10) were proved by Toland [23] and Yanagida [24] (see also Genoud-Stuart [11]).

Their results hold under the assumption 0 < b < min{2, d} and 0 < α < α^?. Farah in [7] also proved that ifu₀∈Σ satisfies (1.8) and

k∇u0k^γ_L^c₂₍

R^d)ku0k^1−γ_L₂₍^c

R^d)>k∇Qk^γ_L^c₂₍

R^d)kQk^1−γ_L₂₍^c

R^d), (1.11)

then the blow-up in H¹(R^d) must occur. Afterwards, Farah-Guzman in [8, 9] proved that the above global solution is scattering under the radial condition of the initial data.

Recently, Guzman in [15] used Strichartz estimates and the contraction mapping argument to establish the well-posedness for the (INLS) in Sobolev space. Precisely, he showed that:

• if 0 < α < α_? and 0 < b <min{2, d}, then the (INLS) is locally well-posed in L²(R^d).

Thus, it is globally well-posed inL²(R^d) by mass conservation.

• if 0< α <α,e 0< b <eband max{0, γc}< γ≤mind

2,1 where αe:=

( _4−2b

d−2γ ifγ < ^d₂,

∞ ifγ= ^d₂, and eb:=

( _d

3 ifd= 1,2,3,

2 ifd≥4, (1.12)

then the (INLS) is locally well-posedness in H^γ(R^d).

• ifα_?< α <α, 0e < b <ebandγ_c < γ≤mind

2,1 , then the (INLS) is globally well-posed in H^γ(R^d) for small initial data.

In particular, he proved the following local well-posedness in the energy space for the (INLS).

Theorem 1.1 ([15]). Let d≥2,0< b <eband0< α < α^?, where eb:=

( _d

3 if d= 2,3, 2 if d≥4.

Then the(INLS)is locally well-posed inH¹(R^d). Moreover, the solutions satisfyu∈L^p_loc(R, L^q(R^d)) for any Schr¨odinger admissible pair(p, q).

Note that the result of Guzman [15] about the local well-posedness of (INLS) in H¹(R^d) is weaker than the one of Genoud-Stuart [11]. Precisely, it does not treat the cased= 1, and there is a restriction on the validity ofbwhend= 2 or 3. Although the result showed by Genoud-Stuart is strong, but one does not know whether the local solutions belong to L^p_loc(R, L^q(R^d)) for any Schr¨odinger admissible pair (p, q). This property plays an important role in proving the scattering for the defocusing (INLS). Our first result is the following local well-posedness in H¹(R^d) which improves Guzman’s result on the range ofbin the two and three spatial dimensions.

Theorem 1.2. Let

d≥4, 0< b <2, 0< α < α^?, or

d= 3, 0< b <1, 0< α < α^?, or

d= 3, 1≤b < 3

2, 0< α < 6−4b 2b−1, or

d= 2, 0< b <1, 0< α < α^?.

Then the(INLS)is locally well-posed inH¹(R^d). Moreover, the solutions satisfyu∈L^p_loc(R, L^q(R^d)) for any Schr¨odinger admissible pair(p, q).

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We will see in Section3that one can not expect a similar result as in Theorem1.1and Theorem 1.2holds in the one dimensional case by using Strichartz estimates. Thus the local well-posedness in the energy space for the (INLS) of Genoud-Stuart is the best known result.

The local well-posedness¹of Genoud-Stuart in [11,13] combines with the conservations of mass and energy immediately give the global well-posedness in H¹(R^d) for the defocusing (INLS), i.e.

µ=−1. To our knowledge, there are few results concerning long-time dynamics of the defocusing (INLS). Let us introduce the following weighted space

Σ :=H¹(R^d)∩L²(R^d,|x|²dx) ={u∈H¹(R^d) :|x|u∈L²(R^d)}, equipped with the norm

kuk_Σ:=kuk_H1(R^d)+kxuk_L2(R^d).

Our next result concerns with the decay of global solutions to the defocusing (INLS) by assuming the initial data in Σ.

Theorem 1.3. Let 0 < b < min{2, d}. Let u0 ∈ Σ and u∈ C(R, H¹(R^d)) be the unique global solution to the defocusing (INLS). Then, the following properties hold:

1. If α∈[α?, α^?), then for every







2≤q≤_d−2^2d ifd≥3, 2≤q <∞ ifd= 2, 2≤q≤ ∞ ifd= 1,

(1.13) there exists C >0 such that

ku(t)k_Lq(R^d)≤C|t|^−d(¹₂−¹_q), (1.14) for all t∈R\{0}.

2. If α∈(0, α?), then for every qgiven in (1.13), there existsC >0such that

ku(t)k_Lq(R^d)≤C|t|⁻^d(2b+dα)⁴ (¹2−¹_q), (1.15) for all t∈R\{0}.

This result extends the well-known result of the classical (i.e. b = 0) nonlinear Schr¨odinger equation (see e.g. [2, Theorem 7.3.1] and references cited therein).

We then use this decay and Strichartz estimates to show the scattering for global solutions to the defocusing (INLS). Due to the singularity of |x|^−b, the scattering result does not cover the same range of exponents bandαas in Theorem1.2. Precisely, we have the following:

Theorem 1.4. Let

d≥4, 0< b <2, α?≤α < α^?, or

d= 3, 0< b <1, 5−2b

3 < α <3−2b, or

d= 2, 0< b <1, α_?≤α < α^?.

Let u₀∈Σandube the unique global solution to the defocusing (INLS). Then there existsu^±₀ ∈Σ such that

t→±∞lim ku(t)−e^it∆u^±₀k_Σ= 0.

1The local well-posedness inH¹(R^d) of Genoud-Stuart is still valid for the defocusing case.

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In this theorem, we only consider the case α∈[α?, α^?). A similar result in the caseα∈(0, α^?) is possible, but it is complicated due to the rate of decays in (1.15). We will give some comments about this case in the end of Section 6.

This paper is organized as follows. In the next section, we introduce some notation and recall Strichartz estimates for the linear Schr¨odinger equation. In Section 3, we prove the local well- posedness given in Theorem 1.2. In Section 4, we derive the virial identity and show the pseudo- conformal conservation law related to the defocusing (INLS). We will give the proof of Theorem 1.3 in Section 5. Section 6 is devoted to the scattering result of Theorem1.4.

2. Preliminaries

In the sequel, the notationA.B denotes an estimate of the formA≤CB for some constant C >0. The constantC >0 may change from line to line.

2.1. Nonlinearity. Let F(x, z) :=|x|^−bf(z) withb >0 andf(z) :=|z|^αz. The complex deriva- tives off are

∂zf(z) = α+ 2

2 |z|^α, ∂zf(z) = α

2|z|^α−2z². We have for z, w∈C,

f(z)−f(w) = Z 1

0

∂zf(w+θ(z−w))(z−w) +∂_zf(w+θ(z−w))z−w dθ.

Thus,

|F(x, z)−F(x, w)|.|x|^−b(|z|^α+|w|^α)|z−w|. (2.1) To deal with the singularity |x|^−b, we have the following remark.

Remark 2.1 ([15]). LetB=B(0,1) ={x∈R^d:|x|<1} andB^c=R^d\B. Then k|x|^−bk_L^γ_x_(B)<∞, if d

γ > b, and

k|x|^−bk_L^γ_x_(Bc)<∞, if d γ < b.

2.2. Strichartz estimates. LetI⊂Randp, q∈[1,∞]. We define the mixed norm kuk_L^p

t(I,L^q_x):=Z

I

Z

R^d

|u(t, x)|^qdx¹_q¹_p

with a usual modification when either por q are infinity. When there is no risk of confusion, we may writeL^p_tL^q_x instead ofL^p_t(I, L^q_x). We also use L^p_t,xwhenp=q.

Definition 2.2. A pair(p, q) is said to beSchr¨odinger admissible, for short (p, q)∈S, if (p, q)∈[2,∞]², (p, q, d)6= (2,∞,2), 2

p+d q =d

2. We denote for any spacetime slabI×R^d,

kuk_S(L2,I):= sup

(p,q)∈S

kuk_L^p

t(I,L^q_x), kvk_S0(L²,I):= inf

(p,q)∈Skvk_Lp0

t (I,L^q_x⁰). (2.2) We next recall well-known Strichartz estimates for the linear Schr¨odinger equation. We refer the reader to [2,19] for more details.

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Proposition 2.3. Letube a solution to the linear Schr¨odinger equation, namely u(t) =e^it∆u0+

Z t 0

e^i(t−s)∆F(s)ds, for some data u0, F. Then,

kuk_S(L2,R).ku0k_L2

x+kFk_S⁰_(L2,R). (2.3) 3. Local existence

In this section, we give the proof of the local well-posedness given in Theorem 1.2. To prove this result, we need the following lemmas which give some estimates of the nonlinearity.

Lemma 3.1 ([15]). Let d ≥ 4 and 0 < b < 2 or d = 3 and 0 < b < 1. Let 0 < α < α^? and I= [0, T]. Then, there existθ₁, θ₂>0such that

k|x|^−b|u|^αvk_S0(L²,I). T^θ¹+T^θ²

k∇uk^α_S(L2,I)kvk_S(L2,I), (3.1) k∇(|x|^−b|u|^αu)kS⁰(L²,I). T^θ¹+T^θ²

k∇uk^α+1_S(L2,I). (3.2) The proof of this result is given in [15, Lemma 3.4]. For reader’s convenience and later use, we give some details.

Proof of Lemma 3.1. We bound

k|x|^−b|u|^αvk_S0(L²,I)≤ k|x|^−b|u|^αvk_S0(L²(B),I)+k|x|^−b|u|^αvk_S0(L²(B^c),I)=:A₁+A₂, k∇(|x|^−b|u|^αu)k_S⁰_(L2,I)≤ k∇(|x|^−b|u|^αu)k_S⁰_(L2(B),I)+k∇(|x|^−b|u|^αu)k_S⁰_(L2(B^c),I)=:B1+B2.

On B.By H¨older inequality and Remark2.1, A1≤ k|x|^−b|u|^αvk

L^p

0 1 t (I,L^q

0

x1(B)).k|x|^−bk_L^γ1

x (B)k|u|^αvk

L^p

0 1 t (I,L^υ_x¹)

.kuk^α_Lm1

t (I,Lⁿ_x¹)kvk_L^p1 t (I,L^q_x¹)

.T^θ¹k∇uk_L^p1

t (I,L^q_x¹)kvk_L^p1 t (I,L^q_x¹), provided that (p₁, q₁)∈S and

1 q₁⁰ = 1

γ1

+ 1 υ1

, d

γ1

> b, 1 υ1

= α n1

+ 1 q1

, 1

p⁰₁ = α m1

+ 1 p1

, θ₁= α m1

− α p1

,

and

q₁< d, 1 n1

= 1 q1

−1 d.

Here the last condition ensures the Sobolev embedding ˙W^1,q¹(R^d)⊂Lⁿ¹(R^d). We see that condition _γ^d

1 > bimplies d γ1

=d−d(α+ 2) q1

+α > b or q₁> d(α+ 2)

d+α−b. (3.3)

Let us choose

q1= d(α+ 2) d+α−b+,

for some 0< 1 to be chosen later. By taking >0 small enough, we see thatq1< d implies d > b+ 2 which is true since we are consideringd ≥4,0 < b < 2 or d= 3,0 < b < 1. On the other hand, using 0< α < α^? and choosing >0 sufficiently small, we see that 2< q₁< _d−2^2d . It remains to check θ1>0. This condition is equivalent to

α m₁ − α

p₁ = 1−α+ 2

p₁ >0 or p₁> α+ 2.

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Since (p1, q1)∈S, the above inequality implies d 2− d

q₁ = 2 p₁ < 2

α+ 2. A direct computation shows

d(α+ 2)[4−2b−(d−2)α] +(d+α−b)(4−d(α+ 2))>0

Since α∈(0, α^?), we see that 4−2b−(d−2)α >0. Thus, by taking >0 sufficiently small, the above inequality holds true. Therefore, we have for a sufficiently small value of ,

A1.T^θ¹k∇uk^α_S(L2,I)kukS(L²,I). (3.4) We next bound

B₁≤ k|x|^−b∇(|u|^αu)k_S0(L²(B),I)+k|x|^−b−1|u|^αuk_S0(L²(B),I)=:B₁₁+B₁₂. The term B11 is treated similarly as forA1 by using the fractional chain rule. We obtain

B11.T^θ¹k∇uk^α+1_S(L2,I), (3.5) provided >0 is taken small enough. Using Remark2.1, we estimate

B12≤ k|x|^−b−1|u|^αuk

L^p

0 1 t (I,L^q

0

x1(B)).k|x|^−b−1k_L^γ1

x (B)k|u|^αuk

L^p

0 1 t (I,L^υ_x¹)

.kuk^α_L^m1

t (I,Lⁿ_x¹)kuk_L^p1 t (I,Lⁿ_x¹)

.T^θ¹k∇uk^α+1_Lp1 t (I,L^q_x¹), provided that (p1, q1)∈S and

1 q₁⁰ = 1

γ1

+ 1 υ1

, d

γ1

> b+ 1, 1 υ1

=α+ 1 n1

, 1

p⁰₁ = α m1

+ 1 p1

, θ1= α m1

− α p1

,

and

q1< d, 1 n1

= 1 q1

−1 d. We see that

d

γ₁ =d−d(α+ 2)

q₁ +α+ 1> b+ 1 or q1> d(α+ 2) d+α−b.

The last condition is similar to (3.3). Thus, by choosing q₁ as above, we obtain for > 0 small enough,

B12.T^θ¹k∇uk^α+1_S(L2,I). (3.6) On B^c.Let us choose the following Schr¨odinger admissible pair

p2= 4(α+ 2)

(d−2)α, q2= d(α+ 2) d+α . Letm₂, n₂be such that

1 q⁰₂ = α

n2

+ 1 q2

, 1

p⁰₂ = α m2

+ 1 p2

. (3.7)

A direct computation shows θ2:= α

m2

− α p2

= 1−α+ 2 p2

= 1−(d−2)α 4 >0.

Note that in our consideration, we always have (d−2)α <4. Moreover, it is easy to check that 1

n₂ = 1 q₂ −1

d.