Continuous dependence for NLS in fractional order spaces

www.elsevier.com/locate/anihpc

Continuous dependence for NLS in fractional order spaces

Thierry Cazenave

, Daoyuan Fang

, Zheng Han

Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China Received 15 June 2010; received in revised form 23 November 2010; accepted 29 November 2010

Available online 1 December 2010

Abstract

For the nonlinear Schrödinger equation iut+u+λ|u|^αu=0 inR^N, local existence of solutions inH^s is well known in theH^s-subcritical and critical cases 0< α4/(N−2s), where 0< s <min{N/2,1}. However, even though the solution is constructed by a fixed-point technique, continuous dependence inH^s does not follow from the contraction mapping argument.

In this paper, we show that the solution depends continuously on the initial value in the sense that the local flow is continuous H^s→H^s. If, in addition,α1 then the flow is locally Lipschitz.

©2010 Elsevier Masson SAS. All rights reserved.

MSC:primary 35Q55 ; secondary 35B30, 46E35

Keywords:Schrödinger’s equation; Initial value problem; Continuous dependence; Fractional order Sobolev spaces; Besov spaces

1. Introduction

In this paper, we study the continuity of the solution mapϕ→ufor the nonlinear Schrödinger equation iu_t+u+g(u)=0,

u(0)=ϕ, (NLS)

inH^s(R^N), whereN1 and 0< s <min

1,N

2

. (1.1)

We assume that the nonlinearitygsatisfies

g∈C¹(C,C), g(0)=0, (1.2)

* Corresponding author.

E-mail addresses:thierry.cazenave@upmc.fr (T. Cazenave), dyf@zju.edu.cn (D. Fang), hanzheng5400@yahoo.com.cn (Z. Han).

1 Current address: Université Pierre et Marie Curie & CNRS, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France.

2 Research supported in part by Zhejiang University’s Pao Yu-Kong International Fund.

3 Research supported by NSFC 10871175, 10931007, and Zhejiang NSFC Z6100217.

0294-1449/$ – see front matter ©2010 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2010.11.005

and g(u)A+B|u|^α, (1.3)

for allu∈C, where 0< α 4

N−2s. (1.4)

Under these assumptions, for every initial valueϕ∈H^s(R^N), there exists a local solution u∈C([0, T], H^s(R^N)) of (NLS), which is unique under an appropriate auxiliary condition. More precisely, the following result is well known. (Here and in the rest of the paper, a pair (q, r)is called admissible if 2r <2N/(N −2)(2r <∞if N=1,2) and 2/q=N (1/2−1/r).)

Theorem 1.1.(See [8,13,7].) Assume(1.1)–(1.4)and let(γ , ρ)be the admissible pair defined by ρ=N (α+2)

N+sα , γ= 4(α+2)

α(N−2s). (1.5)

Given ϕ ∈ H^s(R^N), there exist 0 < T_max∞ and a unique, maximal solution u ∈C([0, Tmax), H^s(R^N))∩ L^γ_loc([0, Tmax), B_ρ,2^s (R^N))of(NLS), in the sense that

u(t )=e^it ϕ+i t

0

e^i(t−s)g u(s)

ds, (1.6)

for all 0t < Tmax, where(e^it )t∈R is the Schrödinger group. Moreover,u∈L^q_loc([0, Tmax), B_r,2^s (R^N))for every admissible pair(q, r)andudepends continuously onϕin the following sense. There exists a timeT ∈(0, Tmax)such that if ϕ_n→ϕ inH^s(R^N)and ifu_n denotes the solution of (NLS) with the initial value ϕ_n, thenT_max(ϕ_n) > T for all sufficiently large n and u_n is bounded in L^q((0, T ), B_r,2^s (R^N)) for any admissible pair (q, r). Moreover, u_n→u inL^q((0, T ), B_r,2^s−ε(R^N)) asn→ ∞for allε >0 and all admissible pairs(q, r). In particular,u_n→u inC([0, T], H^s−ε(R^N))for allε >0.

Theorem 1.1 goes back to [8,13]. The precise statement which we give here is taken from [7, Theorems 4.9.1 and 4.9.7]. The admissible pair (γ , ρ) corresponds to one particular choice of an auxiliary space, which ensures that Eq. (NLS) makes sense and that the solution is unique. See [8,13,7] for details. Under certain conditions on α and s, an auxiliary space is not necessary: the equation makes sense by Sobolev’s embedding and uniqueness in C([0, T], H^s(R^N))holds “unconditionally”, see [13,11,19,26].

We note that the continuous dependence statement in Theorem 1.1 is weaker than the expected one (i.e. with ε=0). Indeed, Theorem 1.1 is proved by applying a fixed point argument to Eq. (1.6), so one would expect that the dependence of the solution on the initial value is locally Lipschitz. However, the metric space in which one applies Banach’s fixed point theorem involves Sobolev (or Besov) norms of orders, while the distance only involves Lebesgue norms. (The reason for that choice of the distance is that the nonlinearity need not be locally Lipschitz for Sobolev or Besov norms of positive order.) Thus the flow is locally Lipschitz for Lebesgue norms, and continuous dependence in the sense of Theorem 1.1 follows by interpolation inequalities. See [8,13] for details.

In this paper, we show that continuous dependence holds inH^s in the standard sense under the assumptions of Theorem 1.1 (with an extra condition in the critical caseα=_N−⁴_2s). More precisely, our main result is the following.

Theorem 1.2.Assume(1.1)–(1.4)and suppose further that A=0 ifα= 4

N−2s, (1.7)

where A is the constant in(1.3). The solution of (NLS)given by Theorem 1.1depends continuously on ϕ in the following sense.

(i) The mappingϕ→T_max(ϕ)is lower semicontinuousH^s(R^N)→(0,∞].

(ii) If ϕ_n →ϕ in H^s(R^N) and if u_n (respectively, u) denotes the solution of (NLS) with the initial value ϕ_n (respectively,ϕ), thenu_n→uinL^q((0, T ), B_r,2^s (R^N))for all0< T < T_max(ϕ)and all admissible pairs(q, r).

In particular,u_n→uinC([0, T], H^s(R^N)).

Theorem 1.2 applies in particular to the model caseg(u)=λ|u|^αu withλ∈Cand 0< α4/(N−2s). If, in addition, α1 then the dependence is in fact locally Lipschitz. We summarize the corresponding results in the following corollary.

Corollary 1.3.Assume(1.1)and letg(u)=λ|u|^αuwithλ∈Cand0< α_N−⁴_2s. It follows that the solution of(NLS) given by Theorem1.1depends continuously on the initial value in the sense of Theorem1.2. If, in addition, α1 then the dependence is locally Lipschitz. More precisely, letϕ∈H^s(R^N)and letu be the corresponding solution of (NLS). Given0< T < T_max(ϕ)there existsδ >0such that ifψ∈H^s(R^N)satisfies ϕ−ψ _H^s δandvis the corresponding solution of (NLS), then for every admissible pair(q, r)

u−v _Lq((0,T ),B_r,2^s )C ϕ−ψ _Hs, (1.8)

whereCdepends onϕ, T , q, r. In particular, u−v _L∞((0,T ),H^s)C ϕ−ψ _H^s.

We are not aware of any previous continuous dependence result for (NLS) inH^s with nonintegers. For integers, the known results in the model case g(u)=λ|u|^αu, λ∈C are the following. Continuous dependence inL²(R^N) follows from [24] in the subcritical caseN α <4 and from [8] in the critical caseN α=4. Continuous dependence in H¹(R^N)is proved in [12] in the subcritical case(N−2)α <4 and in [8,16,22,17] in the critical case(N−2)α=4.

Continuous dependence inH²(R^N)follows from [12] in the subcritical case(N−4)α <4.

Our proof of Theorem 1.2 is based on the method used by Kato [12] to prove continuous dependence inH¹(R^N).

We briefly recall Kato’s argument. Convergence in Lebesgue spaces holds by the contraction mapping estimates, so the tricky part is the gradient estimate. By applying Strichartz estimates, this amounts in controlling∇[g(u_n)−g(u)] in someL^pspace. However,∇[g(u_n)−g(u)] =g(u_n)[∇u_n−∇u]+[g(u_n)−g(u)]∇u. The termg(u_n)[∇u_n−∇u] is easily absorbed by the left-hand side of the inequality, and the key observation is that the remaining term[g(u_n)− g(u)]∇uis of lower order, in the sense that the convergence ofu_ntouin appropriateL^pspaces implies that this last term converges to 0. We prove Theorem 1.2 by applying the same idea. The key argument of the proof is an estimate which shows thatg(u)−g(v)is bounded in Besov spaces by a Lipschitz term (i.e. with a factoru−v) plus some lower order term. (See Lemmas 2.1 and 2.2 below.) In the critical caseα=4/(N−2s), a further argument is required in order to show thatun→uin the appropriateL^pspace (estimate (4.14)). For this, following Tao and Visan [22], we use a Strichartz-type estimate for a non-admissible pair, and this is where we use the assumption (1.7). Local Lipschitz continuity in Corollary 1.3 follows from a different, much simpler argument: in this case, the nonlinearity is locally Lipschitz in the appropriate Besov spaces.

In Theorem 1.2 we assumes <min{1, N/2}. The assumptions < N/2 is natural, but the restrictions <1 is tech- nical. WhenN3, local existence inH^s(R^N)is known to hold for 0< s < N/2, in particular wheng(u)=λ|u|^αu, under some extra assumption onαandsthat ensures thatgis sufficiently smooth. See [8,13,18]. The limitations <1 first appears in our Besov space estimates. These could possibly be extended tos >1. It also appears in a more subtle way. For example, the introduction of certain exponents (4.21) in the critical case explicitly requiress <1. It is not impossible that the idea in [22] of using intermediate Besov spaces of lower order can be applied whens >1.

We next mention a few open questions. As observed above, the fixed point argument used in [8,13,7] to prove Theorem 1.1 does not show that the flow is locally Lipschitz inH^s(R^N). On the other hand, it does not show either that it is not locally Lipschitz. This raises the following question: under the assumptions (1.1)–(1.4), is the flow of (NLS) locally Lipschitz inH^s? We suspect that the answer might be negative. Note that in the pure power case g(u)=λ|u|^αu, Remark 2.3 implies that the nonlinearity is locally Hölder continuous in the appropriate Besov spaces of orders. Therefore it is natural to ask if the flow also is locally Hölder continuous inH^s. Finally, we note that in the critical caseα=_N−⁴_2s Theorem 1.2 imposes the restrictionA=0 in (1.3). Can this restriction be removed?

The rest of this paper is organized as follows. In Section 2 we establish estimates ofg(u)−g(v)in Besov spaces.

We complete the proof Theorem 1.2 in Section 3 in the subcritical caseα <_N−⁴_2s and in Section 4 in the critical case α=_N−⁴_2s. Section 5 is devoted to the proof of Corollary 1.3.

Notation.Given 1p∞, we denote byp its conjugate given by 1/p=1−1/pand we consider the standard (complex-valued) Lebesgue spacesL^p(R^N). Given 1p, q∞ands∈Rwe consider the usual (complex-valued) Sobolev and Besov spaces H^s(R^N)andB_p,q^s (R^N)and their homogeneous versions H˙^s(R^N)andB˙_p,q^s (R^N). See for example [1,23] for the definitions of these spaces and the corresponding Sobolev’s embeddings. We denote by (e^it )t∈R the Schrödinger group and we will use without further reference the standard Strichartz estimates, see [21,27,14,15].

2. Superposition operators in Besov spaces

The study of superposition operators in Besov spaces has a long history and necessary conditions (sometimes necessary and sufficient conditions) ong are known so thatu→g(u)is a bounded map of certain Besov spaces.

(See e.g. [20] and the references therein.) On the other hand, few works study the continuity of such maps, among which [2–6], but none of these works applies to such cases asg(u)= |u|^αu, which is a typical nonlinearity for (NLS).

Besides, for our proof of Theorem 1.2 we do not need only continuity but instead a rather specific property, namely that g(u)−g(v)can be estimated in a certain Besov space of orders by a Lipschitz term (i.e. involvingu−v in some other Besov space of orders) plus a lower order term. We establish such estimates in the two following lemmas.

The first one concerns functionsgsuch that|g(u)|C|u|^α, and the second functionsgthat are globally Lipschitz.

(A functiongsatisfying (1.2)–(1.3) can be decomposed as the sum of two such functions.) The proofs of these lemmas are based on the choice of an equivalent norm onB˙_p,q^s defined in terms of finite differences, and on rather elementary calculations.

Lemma 2.1.Let0< s <1,α >0,1q <∞and1p < r∞, and let0< σ <∞be defined by α

σ = 1 p−1

r. (2.1)

We use the convention that u _L^σ =(

R^N|u|^σ)^σ1 even ifσ <1. Letg∈C¹(C,C)satisfy

g(u)C|u|^α, (2.2)

for allu∈C. It follows that there exists a constantCsuch that ifu, v∈ ˙B_r,q^s (R^N)and u _L^σ, v _L^σ <∞, then g(v)−g(u) _˙

B^s_p,qC v ^α_Lσ v−u _B˙s

r,q+K(u, v), (2.3)

whereK(u, v)satisfies K(u, v)C

u ^α_Lσ+ v ^α_Lσ

u _B˙s

p,q, (2.4)

and

K(u, u_n)

n→∞

−−−−→0, (2.5)

if(un)n1⊂ ˙B_r,q^s (R^N)andu∈ ˙B_r,q^s (R^N)are such that u L^σ<∞and un−u L^σ →0asn→ ∞.

Proof. We denote byτ_ythe translation operator defined fory∈R^Nbyτ_yu(·)=u(· −y)and we recall that (see e.g.

Section 5.2.3, Theorem 2, p. 242 in [23]) u _B˙s

p,q ≈

R^N

τ_yu−u ^q

L^p(R^N)|y|^−N−sqdy ¹

q

. (2.6)

Givenz1, z2∈C, we have g(z1)−g(z2)=(z1−z2)

1 0

∂_zg

z2+θ (z1−z2)

dθ+(z1−z2) 1

0

∂_zg

z2+θ (z1−z2) dθ,

which we write, in short, g(z₁)−g(z₂)=(z₁−z₂)

1 0

g

z₂+θ (z₁−z₂)

dθ. (2.7)

We set

A(u, v, y)(·)=τ_y

g(v)−g(u)

−

g(v)−g(u)

, (2.8)

so that by (2.6) g(v)−g(u) _˙

B_p,q^s C

R^N

A(u, v, y) ^q

L^p|y|^−N−sqdy ¹

q

. (2.9)

We deduce from (2.8) and (2.7) that A(u, v, y)=

g(τ_yv)−g(v)

−

g(τ_yu)−g(u)

=(τ_yv−v) 1

0

g

v+θ (τ_yv−v)

dθ−(τ_yu−u) 1 0

g

u+θ (τ_yu−u) dθ;

and so

A(u, v, y)=

τ_y(v−u)−(v−u)¹

0

g

v+θ (τ_yv−v) dθ

+(τ_yu−u) 1 0

g

v+θ (τ_yv−v)

−g

u+θ (τ_yu−u) dθ

=:A₁(u, v, y)+A₂(u, v, y). (2.10)

It follows from (2.2) that A1(u, v, y)C

|v|^α+ |τyv|^ατ_y(v−u)−(v−u). We deduce by Hölder’s inequality (note thatσ/α1) that

A1(u, v, y)

L^pC v ^α_Lσ τy(v−u)−(v−u)

L^r, from which it follows by applying (2.6) that

R^N

A₁(u, v, y) ^q

L^p|y|^−N−sqdy ¹

q C v ^α_Lσ v−u _B˙s

r,q. (2.11)

We set

K(u, v)=

R^N

A₂(u, v, y) ^q

L^p|y|^−N−sqdy ¹

q

, (2.12)

and formula (2.3) follows from (2.9), (2.10), (2.11) and (2.12). Next, it follows from assumption (2.2) that A₂(u, v, y)C

|u|^α+ |τ_yu|^α+ |v|^α+ |τ_yv|^α

|τ_yu−u|. (2.13)

Applying Hölder’s inequality, we deduce as above that

R^N

A₂(u, v, y) ^q

L^p|y|^−N−sqdy _q¹

C

u ^α_Lσ+ v ^α_Lσ

u _B˙s

r,q, (2.14)

which proves (2.4). It remains to show (2.5). Let(u_n)_n1andube as in the statement, and assume by contradiction that there is a subsequence, still denoted by(u_n)_n1, andε >0 such that

K(u, u_n)ε. (2.15)

We set

Bn(y, θ )(·)= |τyu−u|g

un+θ (τyun−un)

−g

u+θ (τyu−u), (2.16)

and we deduce from (2.2) that B_n(y, θ )C

|u|^α+ |τ_yu|^α+ |u_n|^α+ |τ_yu_n|^α

|τ_yu−u|. (2.17)

It follows from (2.10) that A₂(u, u_n, y) ^p

L^p 1 0

R^N

B_n(y, θ )^pdx dθ. (2.18)

Note that|un−u|^σ→0 inL¹(R^N). In particular, by possibly extracting a subsequence, we see thatun→uasn→ ∞ a.e. and that there exists a functionw∈L¹(R^N)such that|un−u|^σ wa.e. It follows that|un|^σC(|u|^σ+w)∈ L¹(R^N). Moreover, by Young’s inequality

B_n(y, θ )^pC

|u|^σ+ |τ_yu|^σ+ |u_n|^σ+ |τ_yu_n|^σ+ |τ_yu−u|^r

. (2.19)

Since by (2.6)τ_yu−u∈L^r(R^N)for a.a.y∈R^N, we see that for a.a.y∈R^N,B_n(y, θ )^pis dominated by anL¹func- tion. Furthermore,gis continuous andu_n→ua.e., so thatB_n(y, θ )→0 a.e. asn→ ∞. We deduce by dominated convergence and (2.18) that

A2(u, un, y) ^p

L^p−−−−→n→∞ 0, (2.20)

for almost ally∈R^N. Next, it follows from (2.13) and Hölder’s inequality that A2(u, un, y)

L^pC

u ^α_Lσ+ un α L^σ

τyu−u L^r C τyu−u L^r.

Since

τ_yu−u ^q_Lr|y|^−N−sq∈L¹ R^N

,

by (2.6), we see that A₂(u, u_n, y) ^q_Lp|y|^−N−sq is dominated by anL¹ function. Applying (2.20), we deduce by dominated convergence thatK(u, u_n)→0 asn→ ∞. This contradicts (2.15) and completes the proof. 2

Lemma 2.2.Let0< s <1and1p∞,1q <∞. Letg∈C¹(C,C)withgbounded. It follows that there exists a constantCsuch that ifu, v∈ ˙B_p,q^s (R^N), then

g(v)−g(u) _˙

B^s_p,qC v−u _B˙s

p,q +K(u, v), (2.21)

whereK(u, v)satisfies K(u, v)C u _B˙s

p,q, (2.22)

and

K(u, un)

n→∞

−−−−→0, (2.23)

if(un)n1⊂ ˙B_p,q^s (R^N)andu∈ ˙B_p,q^s (R^N)are such thatun→uinL^μ(R^N)for some1μ∞.

Proof. The proof of Lemma 2.1 is easily adapted. 2

Remark 2.3.In the particular case wheng(u)= |u|^αuwithα >0, the estimate (2.3) can be refined. More precisely, g(v)−g(u) _˙

B_p,q^s C v ^α_Lσ v−u _B˙s

r,q+C u _B˙s

r,q u−v ^α_Lσ, (2.24)

if 0< α1 and g(v)−g(u) _˙

B_p,q^s C v ^α_Lσ v−u _B˙s

r,q+C u _B˙s r,q

u ^α_L⁻σ¹+ v ^α_L⁻σ¹

u−v _L^σ, (2.25)

ifα1. Indeed, note that∂zg(z)=(1+^α₂)|z|^α and∂zg(z)=^α₂|z|^α−2z². We claim that, |z₁|^α− |z₂|^α

α(|z₁|^α−1+ |z₂|^α−1)|z₁−z₂| ifα1,

|z₁−z₂|^α if 0< α1, (2.26)

and

|z₁|^α−2z²₁− |z₂|^α−2z²₂

C(|z₁|^α−1+ |z₂|^α−1)|z₁−z₂| ifα1,

C|z₁−z₂|^α if 0< α1. (2.27)

Assuming (2.26)–(2.27), estimates (2.24) and (2.25) follow from the argument used at the beginning of the proof of Lemma 2.1. Indeed, the left-hand side of (2.14) (whereA2is defined by (2.10)) is easily estimated by applying (2.26)–

(2.27) and Hölder’s inequality. It remains to prove the claim (2.26)–(2.27). The first three estimates are quite standard, and we only prove the last one. We assume, without loss of generality, that 0<|z2||z1|. Note first that

|z1|^α−2z²₁− |z2|^α−2z²₂= |z1|^α

z²₁

|z₁|²− z₂²

|z₂|²

+ z²₂

|z₂|²

|z1|^α− |z2|^α |z₁|^α

z²₁

|z1|²− z₂²

|z2|²

+ |z₁−z₂|^α.

Next,

|z₁|^α z²₁

|z1|²− z²₂

|z2|²

2|z₁|^α z₁

|z1|− z₂

|z2|

=2|z1|^α−1

z1−z2+ z₂

|z₂|

|z2| − |z1|4|z1|^α−1|z1−z2|.

If|z₁−z₂||z₁|, then (sinceα1)|z₁|^α−1|z₁−z₂||z₁−z₂|^α. If|z₁−z₂||z₁|, then (recall that|z₁−z₂|

|z₁| + |z₂|2|z₁|) we see that|z₁|^α−1|z₁−z₂|2|z₁|^α2|z₁−z₂|^α. This shows the second estimate in (2.27).

3. Proof of Theorem 1.2 in the subcritical caseα <_N−⁴_2s

Throughout this section, we assumeα < _N−⁴_2s and we use the admissible pair(γ , ρ)defined by (1.5). We need only to show that, givenϕ∈H^s(R^N), there existsT =T ( ϕ H^s) >0 such that if ϕ H^s < M thenTmax(ϕ) > T and such that ifϕn→ϕinH^s(R^N), thenTmax(ϕn) > T for all sufficiently largenand the corresponding solutionun

of (NLS) satisfiesun→uinL^q((0, T ), B_r,2^s (R^N))for all admissible pairs(q, r). SinceT =T ( ϕ H^s), properties (i) and (ii) easily follow by a standard iteration argument.

We note that by Theorem 1.1 there existsT =T ( ϕ _H^s) >0 such that if ϕ _H^s < MthenT_max(ϕ) > T. Moreover, by possibly choosingT smaller (but still depending on ϕ _H^s), ifϕ_n→ϕ inH^s(R^N), then T_max(ϕ_n) > T for all sufficiently largenand the corresponding solutionu_nof (NLS) satisfies

sup

n1

un L^q((0,T ),B^s_r,2)<∞, (3.1)

and

u_{n n}−−−−→_→∞ u inL^q

(0, T ), B_r,2^s−ε R^N

∩C

[0, T], H^s−ε R^N

, (3.2)

for allε >0 and all admissible pairs(q, r). It is not specified in Theorem 1.1 that, in the subcritical case,T can be bounded from below in terms of ϕ _H^s but this is immediate from the proof.

We set

σ=N (α+2)

N−2s > ρ, (3.3)

so that by Sobolev’s embedding B˙_ρ,2^s

R^N →L^σ

R^N

, (3.4)

and we claim that u_{n n}

−−−−→→∞ u inL^γ−ε

(0, T ), L^σ R^N

, (3.5)

for all 0< εγ−1. This is a consequence of (3.2). (In fact, if we could letε=0 in (3.2), then we would obtain by (3.4) convergence inL^γ((0, T ), L^σ(R^N)).) Indeed, givenη >0 and small, we haveB^s−

ηN ρ(ρ+η)

ρ+η,2 (R^N) →L^σ(R^N).

Ifηis sufficiently small,ρ+η <2N/(N−2)⁺so that there existsγ_ηsuch that(γ_η, ρ+η)is an admissible pair, and we deduce from (3.2) thatu_n→uinL^γη((0, T ), L^σ(R^N)). Sinceγ_η→γ asη→0, we conclude that (3.5) holds.

We decomposegin the formg=g₁+g₂whereg₁, g₁∈C¹(C,C),g₁(0)=g₂(0)=0 and

g₁(u)C, (3.6)

g₂(u)C|u|^α, (3.7)

for allu∈C. By Strichartz estimates in Besov spaces (see Theorem 2.2 in [8]), given any admissible pair(q, r)there exists a constantCsuch that

u_n−u _Lq((0,T ),B˙_r,2^s )C ϕ_n−ϕ _H˙s

+C g₁(u_n)−g₁(u)

L¹((0,T ),B˙_2,2^s )+C g₂(u_n)−g₂(u)

L^γ((0,T ),B˙_ρ,2^s ). (3.8) We estimate the last two terms in (3.8) by applying Lemmas 2.2 and 2.1, respectively. We first apply Lemma 2.2 tog₁ withq=p=2 and we obtain

g₁(u_n)−g₁(u) _˙

B^s_2,2C u_n−u _B˙s

2,2+K₁(u, u_n). (3.9)

Next, we apply Lemma 2.1 tog2withq=2,r=ρandp=ρ, and we obtain g₂(u_n)−g₂(u) _˙

B^s_ρ,2C u_n ^α_Lσ u_n−u _B˙s

ρ,2+K₂(u, u_n), (3.10)

whereσ is defined by (3.3). Applying Hölder’s inequality in time, we deduce from (3.9) that g₁(u_n)−g₁(u)

L¹((0,T ),B˙_2,2^s )CT u_n−u _{L∞((0,T ),B˙}s

2,2)+ K₁(u, u_n)

L¹(0,T ), (3.11)

and from (3.10) that g₂(u_n)−g₂(u)

L^γ((0,T ),B˙^s_ρ,2)CT^{4−α(N4−2s)} u_n ^α_Lγ((0,T ),L^σ) u_n−u _Lγ((0,T ),B˙_ρ,2^s )

+ K₂(u, u_n)

L^γ(0,T ). (3.12)

Note that u_n ^α_Lγ((0,T ),L^σ)is bounded by (3.1) and (3.4). Thus we see that, by possibly choosingT smaller (but still depending on ϕ _H^s), we can absorb the first term in the right-hand side of (3.11) by the left-hand side of (3.8) (with the choice(q, r)=(∞,2)), and similarly for (3.12) (with the choice(q, r)=(γ , ρ)). By doing so, we deduce from (3.8) that

u_n−u _Lγ((0,T ),B˙^s_ρ,2)+ u_n−u _{L∞((0,T ),H˙}s)

C ϕ_n−ϕ _H˙s +C K₁(u, u_n)

L¹(0,T )+C K₂(u, u_n)

L^γ(0,T ). (3.13)

Next, we note that by (2.22)K₁(u, u_n) u(t ) _B˙s

2,2 for allt∈(0, T )and thatu∈L^∞((0, T ), B_2,2^s (R^N)). Moreover, un→uinC([0, T], L²(R^N)), so thatun(t )→u(t )inL²(R^N)for allt∈(0, T ). Applying (2.23) (withμ=2), we deduce thatK1(un, u)→0 for allt∈(0, T ). By dominated convergence, we conclude that

K₁(u, u_n)

L¹(0,T ) n−−−−→→∞ 0. (3.14)

We now show that K₂(u, u_n)

L^γ(0,T ) n−−−−→→∞ 0. (3.15)

Indeed, suppose by contradiction that there exist a subsequence, still denoted by(u_n)_n1, andδ >0 such that K2(u, un)

L^γ(0,T )δ. (3.16)

We note that by (2.4) and Young’s inequality K2(u_n, u)^γC

u_n ^αγ_Lσ+ u ^αγ_Lσ

u ^γ_˙

B^s_ρ,2

C

u_n

αγ γ−2

L^σ + u

αγ γ−2

L^σ + u ^γ_˙

B_ρ,2^s

. (3.17)

Sinceαγ /(γ−2) < γ, we deduce from (3.5) that, after possibly extracting a subsequence,K2(un, u)^γ is dominated by anL¹function. It also follows from (3.5) that, after possibly extracting a subsequence,un(t )→u(t )inL^σ(R^N) for a.a.t∈(0, T )so that, applying (2.5),

K₂(u, u_n)

n→∞

−−−−→0, (3.18)

for a.a.t∈(0, T ). By dominated convergence, K2(u, un) _Lγ(0,T )→0. This contradicts (3.16), thus proving (3.15).

Finally, it follows from (3.13), (3.14) and (3.15) that

u_n−u _Lγ((0,T ),B˙_ρ,2^s )+ u_n−u _{L∞((0,T ),H˙}s) n−−−−→→∞ 0. (3.19) Given any admissible pair (q, r), we deduce from (3.8), (3.11), (3.12), (3.14), (3.15) and (3.19) that un − u _Lq((0,T ),B˙_r,2^s )→0 asn→ ∞. This completes the proof.

4. Proof of Theorem 1.2 in the critical caseα=_N−⁴_2s

Throughout this section, we assumeα=_N−⁴_2s and we use the admissible pair(γ , ρ)defined by (1.5). We recall that by assumption (1.7),

g(u)C|u|^α. (4.1)

The argument of the preceding section fails essentially at one point. More precisely, hereαγ /(γ −2)=γ, so that the convergence (3.5) does not imply as in Section 3 (see (3.17)) thatK2(un, u)^γis dominated by anL¹function. To overcome this difficulty, we show in Lemma 4.1 below that (3.5) holds withε=0. This is the key difference with the subcritical case, and the rest of the proof is very similar.

We now go into details and we first recall some facts concerning the local theory. (See e.g. [8].) We define G(u)(t )=

t

0

e^i(t−s)g u(s)

ds. (4.2)

If(γ , ρ)is the admissible pair defined by (1.5), then g(u)

L^γ((0,T ),B_ρ,2^s )C u ^α_L⁺γ((0,T ),B^{1 s}_ρ,2), and g(v)−g(u)

L^γ((0,T ),L^ρ)C

u ^α_Lγ((0,T ),B_ρ,2^s )+ u ^α_Lγ((0,T ),B_ρ,2^s )

u−v _L^γ_{((0,T ),L}^ρ₎,

where the constant C is independent of T >0. By Strichartz estimates in Besov spaces, it follows that the prob- lem (1.6) can be solved by a fixed point argument in the setE= {u∈L^γ((0, T ), B_ρ,2^s (R^N)); u _Lγ((0,T ),B^s_ρ,2)η} equipped with the distance d(u, v)= u−v _Lγ((0,T ),L^ρ)forη >0 sufficiently small. (Note that(E,d)is a complete metric space. Indeed,L^γ((0, T ), B_ρ,2^s )is reflexive, so its closed ball of radiusηis weakly compact.) By doing so, we see that there existsδ₀>0 (independent ofϕ∈H^s(R^N)andT >0) such that if

e^i·ϕ

L^γ((0,T ),B_ρ,2^s )< δδ₀, (4.3)

thenT_max(ϕ) > T and the corresponding solution of (1.6) satisfies

u _Lγ((0,T ),B_ρ,2^s )2δ. (4.4)

Moreover, given any admissible pair(q, r), there exists a constantC(q, r)such that

u _Lq((0,T ),B_r,2^s )δC(q, r). (4.5)

In addition, ifϕ, ψboth satisfy (4.3) andu, vare the corresponding solutions of (NLS), then

u−v L^γ((0,T ),L^ρ)C ϕ−ψ _L2. (4.6)

We also recall the Strichartz estimate in Besov spaces e^i·ϕ

L^q((0,T ),B_r,2^s )C(q, r) ϕ _H^s, (4.7)

for every admissible pair(q, r). In particular, e^i·ψ

L^γ((0,T ),B_ρ,2^s ) e^i·ϕ

L^γ((0,T ),B_ρ,2^s )+C ψ−ϕ H^s, (4.8)

for everyϕ, ψ∈H^s(R^N).

We now fixϕ∈H^s(R^N)satisfying (4.3) and we consider(ϕ_n)_n1⊂H^s(R^N)such that ϕ_n−ϕ _H^s

n→∞

−−−−→0. (4.9)

We denote byuand(u_n)_n1the corresponding solutions of (NLS).

We show that, by lettingδ₀in (4.3) possibly smaller (but still independent ofT andϕ), un−u _L^q_{((0,T ),B}^s

r,2) n→∞−−−−→0, (4.10)

for all admissible pairs(q, r). Indeed, arguing as in the subcritical case, we see that (see the proof of (3.13)) u_n−u _Lγ((0,T ),B˙^s_ρ,2)C ϕ_n−ϕ _H˙s

+C u_n ^α_Lγ((0,T ),L^σ) u_n−u _Lγ((0,T ),B˙^s_ρ,2)+ K(u, u_n)

L^γ(0,T ), (4.11) whereK(u, u_n)is given by Lemma 2.1. We next observe that, using (4.4), (3.4) and possibly choosingδ₀ smaller, C u_n ^α_Lγ((0,T ),L^σ)1/2 fornlarge. We then deduce from (4.11) that

u_n−u _Lγ((0,T ),B˙^s_ρ,2)C ϕ_n−ϕ _H˙s+C K(u, u_n)

L^γ(0,T ). (4.12)

We claim that K(u, un)

L^γ(0,T ) n→∞−−−−→0. (4.13)

For proving (4.13), we use the following lemma, whose proof is postponed until the end of this section.

Lemma 4.1.Under the above assumptions, and forδ₀in(4.3)possibly smaller(but still independent ofT andϕ)it follows that

un−u L^γ((0,T ),L^σ) n−−−−→→∞ 0, (4.14)

whereσ is defined by(3.3).