A sharp scattering condition for focusing mass-subcritical nonlinear Schr\"odinger equation

This article is concerned with time global behavior of solutions to focusing mass-subcritical nonlinear Schr\"odinger equation of power type with data in a critical homogeneous weighted $L^2$ space. We give a sharp sufficient condition for scattering by proving existence of a threshold solution which does not scatter at least for one time direction and of which initial data attains minimum value of a norm of the weighted $L^2$ space in all initial value of non-scattering solution. Unlike in the mass-critical or -supercritical case, ground state is not a threshold. This is an extension of previous author's result to the case where the exponent of nonlinearity is below so-called Strauss number. A main new ingredient is a stability estimate in a Lorenz-modified-Bezov type spacetime norm.


Introduction
In this article, we continue our study in [24] on time global behavior of solutions to the following focusing mass-subcritical nonlinear Schrödinger equation (NLS) i∂ t u + ∆u = −|u| p−1 u, (t, x) ∈ R 1+N with initial condition where N 1, the exponent of the nonlinearity is in the range (1.1) max 1 + 2 N , 1 + 4 N + 2 < p < 1 + 4 N , and s c : Here, the function space in which initial data lie is homogeneous weighted L 2 space with norm f FḢ sc = |x| sc f L 2 , where F stands for usual Fourier transform in R N and a homogeneous Sobolev spaceḢ s (R N ) is defined aṡ for 0 < s < N/2. Remark that 0 < s c < min(1, N/2) as long as (1.1). The equation (NLS) has the following scaling; if u(t, x) is a solution to (NLS) then for all λ > 0 (1.2) u [λ] (t, x) = λ 2 p−2 u(λ 2 t, λx) 1 is also a solution to (NLS). The initial space FḢ sc is scaling critical in such a sense that u 0{λ} FḢ sc = u 0 FḢ sc holds for any λ > 0, where is known to be useful to obtain a priori decay estimate of solution, while it is conserved only in the mass-critical case p = 1 + 4 N . In our setting u 0 ∈ FḢ sc , none of the above quantities make sense in general.
Our aim here is analysis of time global behavior of solutions to (NLS)-(IC) in mass-subcritical range p < 1 + 4 N . Nakanishi and Ozawa show in [27] that, under an assumption on p weaker than (1.1), if an initial data belongs to FḢ sc ∩ L 2 and if it is small with respect to the FḢ sc norm, then the solution to (NLS)-(IC) exists globally in time and behaves like a linear solution near t = ±∞ (see also [6,15]). This behavior, which is referred to as scattering, occurs when the linear dispersive effect becomes dominant for large time. Remark that smallness of datum gives that of corresponding solution, which is closely related to weakness of nonlinear effect relative to linear effect. On the other hand, there exists a non-scattering solution. A typical example is a standing-wave solution e it φ(x), where φ is a solution to the elliptic equation (1.6) − ∆φ + φ = |φ| p−1 φ.
There exists a unique positive radial solution Q(x) to (1.6), provided 1 < p < 1 + 4 N −2 (1 < p < ∞ if N = 1, 2), see [5] and references therein. It is called a ground state and is characterized as the function minimizing the energy E(u) among all nontrivial H 1 functions under mass constraint. Another behavior is blowup (in finite time). Fibich shows in [11] that a self-similar blowup solution exists in the mass-subcritical case 1 < p < 1 + 4/N .
Recently, remarkable progresses are made on classification of initial space according to global behavior of corresponding solution. In the energycritical case, global behavior ofḢ 1 -solutions of energy less than that of W = (1 + |x| 2 N (N −2) ) − N−2 2 , which is a solution to an elliptic equation ∆W + |W | 4 d−2 W = 0, is precisely analyzed. It is shown that initial space consists of two disjoint sets, say S and B, and that if an initial data belongs to S then a corresponding solution of (NLS) scatters for both time directions, while if data belongs to B (and if some further assumption is satisfied) then a solution blows up in finite time. This is first given by Kenig and Merle [19] for 3 N 5 under radial assumption, and is extended by Killip and Visan [22] to non-radial N 5 case. In [1,17], a similar classification is given in H 1 framework in the energy-subcritical and mass-supercritical case 1 + 4 N < p < 1 + 4 N −2 (1 + 4 N < p < ∞ if N = 1, 2). See also [9,10]. In [25], Nakanishi and Schlag consider (NLS) with p = N = 3 and radial H 1 data of energy at most slightly above that of the ground state Q, and show that initial space splits into nine nonempty disjoint sets which are characterized by distinct behavior of solution. For the mass-critical case, Dodson shows in [8] that if an L 2 -solution u(t) satisfies u(0) L 2 < Q L 2 then the solution exists globally in time and scatters in L 2 for both positive and negative time (see also [37]).
As for this subject, in contrast, the mass-subcritical case p < 1 + 4/N is less understood. In the previous paper [24], the case p St < p < 1 + 4/N and u 0 ∈ FH 1 := {f ∈ S ′ | F −1 f ∈ H 1 (R N )} is treated, where (1.7) p St := 1 + 2 − N + √ N 2 + 12N + 4 2N is a Strauss exponent, and a sharp sufficient condition for scattering is given by demonstrating that there exists a special solution u c (t) such that u c (t) does not scatter either for positive or negative time, and that if u 0 ∈ FH 1 satisfies ℓ F H 1 (u 0 ) < ℓ F H 1 ( u c (0)) then the corresponding solution u(t) to (NLS)-(IC) scatters for both positive and negative time, where ℓ F H 1 is defined as (1.8) ℓ It is obvious that ℓ F H 1 (·) is invariant under (1.3). This criterion is similar to that in the mass critical case. However, it is also proven that standing wave solutions are not the threshold u c (t) any more.
In this article, we improve this result in the following two directions. Firstly, the restriction p > p St is removed and the possible range of p is extended as in (1.1). Secondly, we take an initial data from FḢ sc which is wider than FH 1 .
The Strauss exponent p St , which is a positive root of N p 2 −(N +2)p−2 = 0, appears at least in the following two contexts. One is time global behavior of defocussing case where the right hand side of (NLS) has the opposite sign, +|u| p−1 u. It is known that asymptotic completeness holds in FH 1 for p p St . More precisely, if p p St then a priori decay estimate of FH 1 -solution, which follows from a priori estimate on the pseudo conformal energy (1.5), is sufficient to show that every FH 1 -solution scatters both for positive and negative times (see [6,27,31]). Another is validity of a stability type estimate called "long-time perturbation". When p > p St , we are able to obtain a stability type estimate with a Lebesgue-Lebesgue type space-time norm L ρ t L q x (see [18,24]). The restriction p > p St in the previous study [24] comes from the latter point. To remove this, we establish a new stability estimate with respect to a Lorentz-modified-Bezov type space. It is known that spatial decay of initial data yields time decay of solution to free Schrödinger equation (see [5,15,27]). As long as the linear dispersive effect is dominant, nonlinear solution inherits this property. The Lorentz-modified-Besov space, introduce in [27], enables us to take time decay effect due to fractionally weighted data into account efficiently and to treat fractional weights in a way very similar to that for fractional derivatives. To obtain the stability estimate, we need the restriction p > 1+ 4 N +2 for N 3 (see Remark 3.6). The other restriction p > 1 + 2 N is not a technical one. It is known that if p 1 + 2 N then any non-trivial L 2 -solution does not scatter (see [3,29]) . We would emphasize that, in our setting, finite-mass assumption u 0 ∈ L 2 is removed. The finite mass property is closely related to blowup phenomenon. In the mass-subcritical case 1 < p < 1 + 4/N , a solution with finite mass always exists globally in time due to the mass conservation law (see [32]), while blowup phenomenon can occur without the finite mass assumption as shown in [11]. Note that, however, it is not clear so far whether or not our assumption u 0 ∈ FḢ sc admits a finite-time blowup solution. It is worth mentioning that local well-posedness in FḢ sc is not found in the previous results [6,15,24,27]. The difference is that continuous dependence property was based on the finite mass assumption u 0 ∈ L 2 . Our stability estimate is a key for removing this assumption. For N 3 and p such that 0 < s c < 1/2, local well-posedness holds true inḢ −sc ⊃ FḢ sc under radial assumption (see [16]) 1.1. Main results. Before presenting our main results, we introduce several notations. The notion of an FḢ sc -solution to (NLS)-(IC) is given in Definition 3.1 below. Here, we merely note that an FḢ sc -solution u(t) is such that t → U (−t)u(t) is continuous R ⊃ I → FḢ sc , where U (t) := e it∆ is the free Schrödinger group and I is an time interval. In what follows, we say u(t) scatters as t → ∞ (resp. t → −∞) if I contains (τ, ∞) (resp. (−∞, τ )) for some τ ∈ R and the limit lim t→∞ U (−t)u(t) (resp. lim t→−∞ U (−t)u(t)) exists in FḢ sc sense, unless otherwise stated. Let S ± := u 0 ∈ FḢ sc solution u(t) to (NLS) with u(0) = u 0 scatters as t → ±∞. .
We also set a scattering set S = S + ∩ S − . Let us define Then, we have ℓ c ∈ (0, ∞). Indeed, by small data scattering, there exists a constant η > 0 such that ℓ c η (see Remark 3.12). On the other hand, ℓ c ℓ(Q) follows from the fact that the ground state solution e it Q(x) is a non-scattering solution. Remark that Q ∈ FḢ sc , that is, ℓ(Q) < ∞ since Q(x) is smooth and decays exponentially as |x| → ∞.
Our main theorems are as follows.
Remark 1.3. By definition of ℓ c , we obtain the following criterion for scattering; if u 0 ∈ FḢ sc is such that u 0 FḢ sc < u 0,c FḢ sc then u 0 ∈ S. Notice that Theorem 1.1 implies that this criterion is sharp. Namely, the above criterion actually fails when u 0 FḢ sc = u 0,c FḢ sc .
Remark 1.4. In Theorem 1.1, the assumption that the equation is focusing is used only for justifying ℓ c < ∞. Namely, the conclusion of Theorem 1.1 holds true also for the defocussing equation, provided we assume ℓ c < ∞. As for the short-range defocussing equations, a conjecture is that asymptotic completeness holds, that is, ℓ c = +∞ or equivalently S = FḢ sc . This conjecture is true when p St p < 1 + 4/N and initial space is FH 1 (see [6,27,31]). In the case 1 + 2/N < p < p St , it is shown in [33] that if initial data is taken from H 1 ∩ FH 1 then a solution globally in time and scatters in L 2 for both time directions. Theorem 1.1 reduces the conjecture to nonexistence of a critical element. If existence of the critical element u 0,c , the conclusion of Theorem 1.1, yields any contradiction, then we obtain the conjecture.
Remark 1.5. In the mass-critical case, the ground state Q(x) plays the role of u 0,c ( [8]). Theorem 1.2 says that the situation is completely different in the mass-subcritical case. It seems reasonable because the ground state solution e it Q(x) is orbitally stable in H 1 in the mass subcritical case (see [5] and references therein). Remark 1.6. Rigorously speaking, the statement in Theorems 1.1 and 1.2 is not a direct extension of that in [24]. It is because the minimizing problem with respect to · FḢ sc is different from that with respect to ℓ F H 1 (·). One sees this from the fact that there exist functions ψ 1 , ψ 2 ∈ FH 1 such that ψ 1 FḢ sc < ψ 2 FḢ sc and ℓ F H 1 (ψ 1 ) > ℓ F H 1 (ψ 2 ) (such example will be given in Remark 4.2). Nevertheless, the result of [24] can be also extended to the case (1.1) as mentioned in Remark 7.5. In other words, our argument enables us to consider the both minimizing problems. In this sense, Theorems 1.1 and 1.2 are "extensions" of results of [24].
If an initial data has a rapid quadratic oscillation, then a solution is global and scatters. This kind of result is known for p > p St and u 0 ∈ FH 1 (see [6,24]). As a byproduct of our main theorems, we improve this result as follows.
Theorem 1.7. Suppose (1.1). For any ψ ∈ FḢ sc , there exists b 0 > 0 such that if |b| > b 0 then e ib|x| 2 ψ ∈ S. Remark 1.8. Theorem 1.7 holds true also for ψ ∈ FH 1 : An FH 1 -solution of (NLS) with data e ib|x| 2 ψ scatters in FH 1 -sense for both time directions, provided |b| is sufficiently large (see Remark 6.2). This is a direct extension of the result of [6,24] to the case (1.1). Theorem 1.7 yields an unboundedness property of the scattering set S. Corollary 1.9. Suppose (1.1). It holds that Unboundedness of S itself is trivial in view of Galilean transform. Indeed, if u 0 ∈ S ∩ L 2 loc then, for any The unboundedness of S then follows from the fact that u 0 (· − x 0 ) FḢ sc → ∞ as |x 0 | → ∞. However, the unboundedness above follows by a completely different cause, a rapid oscillation of initial data. Theorem 1.7 also suggest that some quantity other than ℓ(u 0 ) should be taken into account for complete classification of behavior of solutions.
The rest of the paper is organized as follows. We first define function spaces which we work with and collect several preliminary facts and estimates in Section 2. We then begin our study with local well-posedness in FḢ sc (Theorem 3.7) and a stability type estimate, long time perturbation, in a Lorentz-modified-Bezov space (Theorem 3.13) in Section 3. A necessary and sufficient condition for scattering is also given in this section (Proposition 3.9). Section 4 is devoted to proof of concentration compactness for sequences bounded in FḢ sc (Proposition 4.3). Smallness due to rapid oscillation of initial data is considered there (Proposition 4.1). We then prove our main theorems in Sections 5, 6, and 7.

Preliminaries
2.1. Function spaces. For our analysis, we prepare several notations and function spaces.
We first define a function spaceẊ s q (t) defined by the following norm: for s, t ∈ R and 1 q ∞, where U (t) = e it∆ . For t = 0, let M (t) be a t-dependent multiplication operator defined by M (t) := e i |x| 2 4t × and let D(t) be a dilation operator defined by D(t)f := (2πit) −N/2 f x 2t . One deduces from the well-known factorization U (t) = M (t)D(t)FM (t) that the identity holds for a suitable function φ. Hence, we have the following equivalent norm for t = 0: The spaceẊ s q (t) is regarded as a modified Sobolev space in this sense. Let a sequence {ϕ j } j∈Z be a Littlewood-Paley decomposition, that is, , and j∈Z ϕ j (ξ) = 1 for ξ = 0. We use a function space defined by the following norm for s, t ∈ R, and 1 q, r ∞. By (2.2), we have the following equivalent norm is a norm of usual homogeneous Besov space. We refer the spaceṀ s q,r (t) to as a modified Besov space. The following fundamental properties ofẊ s q (t) andṀ s p,q (t) are summarized as follows.
Then, we have This is a special case of the following interpolation type inequality.
Proposition 2.5. Let 1 < ρ 1 , ρ 2 < ∞, s 1 , s 2 ∈ R, and 1 q 1 , q 2 , r 1 , Proof. Since ρ 1 , ρ 2 < ∞, we can take δ, M > 0 so that Then, it suffices to approximate v(t) ∈Ṁ s 1 q 1 ,r 1 (t) ∩Ṁ s 2 q 2 ,r 2 (t) by a function which has a compact support, for each fixed t ∈ [−M, −δ] ∪ [δ, M ]. By the equivalent representation (2.5), approximation of w := M (−t)v(t) in B s 1 q 1 ,r 1 ∩Ḃ s 2 q 2 ,r 2 is sufficient. This is an immediate consequence of the atomic decomposition. Indeed, for w ∈Ḃ s 1 q 1 ,r 1 , we have the following decomposition; where Q's are dyadic cubes, s Q 's are constants satisfying and a Q 's are the (s 1 , q 1 )-atoms. Remark that we can construct {s Q } and {a Q } independently of s, q, and r (see [13,Theorem 2.6]). Therefore, w ∈ B s 2 q 2 ,r 2 implies that the above decomposition holds inḂ s 2 q 2 ,r 2 with the same a Q and s Q . Now, a linear combination of sufficiently large (but finite) number of s Q a Q is a desired approximation of w since theḂ s 1 q 1 ,r 1 norm is equivalent to the infimum of the left hand side of (2.7) in all decompositions of the form (2.6) and since q 1 , q 2 , r 1 , r 2 < ∞.
An improved version of inhomogeneous Strichartz' estimate (the second estimate) for the Lorentz-modified-Sobolev space will be required in our analysis. Before stating it, we introduce an immediate consequence of Strichartz' estimate and Lemma 2.3.
Proposition 2.8. For pair (ρ, q) satisfying 0 < δ(q)−(s c −s) < min(1, N/2) and 2 ρ − δ(q) + s = s c for some s s c , it holds that Proof. Take ρ and q so that One verifies that which implies that ( ρ, q) is an admissible pair. Hence, U (t)f L ρ,2 (R,Ṁ sc q,2 ) C f FḢ sc by the Strichartz estimate. Hence the desired result is an immediate consequence of Lemma 2.3, Remark 2.9. Notice that a pair (ρ, q) is not always acceptable in the preceding proposition. Indeed, when N 3 the case ∞ > q 2 * , which is equivalent to δ(q) N/2, is allowed.
We now state the inhomogeneous Strichartz' estimate for the Lorentzmodified-Sobolev space with non-admissible pairs. See [12,18,23,36] for this kind of extension with respect to L ρ t L q x type norm.
then it holds that ) .
Proof. The estimate is known if either 1/q 1 + 1/q 2 = 1 or 2/ρ i − δ(q i ) = 0. Let us first consider the case where I j denotes the characteristic function of [2 j , 2 j+1 ). We begin with the well-known estimate . Apply this estimate to obtain for any fixed t ∈ R. Therefore, we have On the other hand, if 2/r = δ(q) ∈ (0, min(1, N/2)), where we have applied (2.8) and used the inclusion L r,2 ⊂ L r and L r ′ ⊂ L r ′ ,2 . From this inequality, we have .
Interpolating (2.9) and (2.10), one sees that As shown in [27], it follows that the operator is bounded as a map as long as 0 β < α < γ, where ℓ s,2 is a weighted ℓ 2 space defined by the norm a j ℓ s,2 := 2 js a j ℓ 2 .
We are now in a position to finish the proof. For λ ∈ (0, ∞) \ {1}, by the discrete J-functional method, we have Now, we fix ρ ∈ (1, ρ 1 ) and chose λ = 2 −1/ρ and ϕ j = K j f to obtain By means of (2.11) and (2.12), we conclude that as long as Similarly, we have a inhomogeneous estimate non-admissible for the Lorentzmodified-Sobolev space.
Proposition 2.12. Let (ρ i , q i ) (i = 1, 2) be two acceptable pairs. If s 0 The proof is done in the essentially same way. We omit details.
2.3. Specific choice of function spaces. We introduce several explicit spaces which is used throughout this paper. The choice depends on whether p 2 or not. Recall that we are concerned with the case (1.1). Hence, p 2 occurs only when N 3. Similarly, p < 2 implies N 3. We introduce pairs And, for an interval I ⊂ R, we let (2.14) We omit (I) when it is clear from the context. When p 2, W 1 (I) = W 2 (I) = W (I).
When p < 2, we use two more function spaces. Let s 0 = s 0 (p, N ) be a real number given by and define two more spaces; . and the embedding W 2 (I) ֒→ X(I) ֒→ L(I) hold.

2.4.
Estimates on the nonlinearity. In this subsection, we collect several estimate on the nonlinearity.
Lemma 2.15. Let I ⊂ R be an interval. There exists a constant C such that (2.20) |u| p−1 u F (I) C u p−1 L(I) u W 1 (I) whenever the right hand side makes sense. Lemma 2.16. Let I ⊂ R be an interval. There exists a constant C such that if p 2, whenever the right hand side makes sense.
By (2.15), the L(I) norms can be replaced by S(I) norm or W 2 (I) norm in the above lemmas. To prove these lemmas, we introduce difference representation of Besov norm (see [4]).
Lemma 2.17. Let s > 0 and q, r ∈ [1, ∞]. Let M be an integer larger than s. Let δ = δ a be a difference operator, i.e. δf (x) = f (x + a) − f (x). Then, we have the following equivalent expression of the Besov norm: By an elementary computation, We deduce from the second relation of (2.16) that . Then, taking sup |a| 2 −j , multiplying by 2 scj , and then taking ℓ 2 -norm in j, we see from Lemma 2.17 that The lemma now follows from (2.5) and the generalized Hölder inequality in time.
Proof of Lemma 2.16. We keep the notation g 1 and g 2 . Let us first consider the case p < 2. It is clear that g 1 and g 2 are Hölder continuous of order Since g j is Hölder continuous of order p − 1, we obtain Arguing as in the previous lemma, we obtain (2.21).
On the other hand, when p 2 there exists a constant C such that for any z 1 , z 2 ∈ C. Hence, we obtain from which (2.22) follows.
The following is the key estimates of the analysis for p < 2.
Lemma 2.18. Let I ⊂ R be an interval. There exists a constant C such that whenever the right hand side makes sense.
To prove this lemma, we need following two lemmas.
whenever the right hand makes sense.
Proof of Lemma 2.18. As in the proof of (2.23), it holds that Indeed, one can see this by replacing u(x + a) and u(x) by u 1 and u 2 , respectively. Hence, by Lemma 2.19, where 1/q 0 = 1/q(Y ) − 1/q(L). Lemma 2.20 then yields .
Remark 2.21. The lower bound p > 1 + 4 N +2 of our main theorem comes from this lemma. Indeed, it is necessary that the exponent s 0 , which denotes the order of weight of X(I) and Y (I), obeys are acceptable pairs which fulfill the assumption of Proposition 2.12; and that the relations (2.16) and (2.19) and the embedding W 2 (I) ֒→ X(I) ֒→ L(I) hold true. On the other hand, since |z| p−1 z is Hölder continuous of order p, to estimate the left hand side of (2.24), the exponent s 0 must be smaller than p − 1. Further, in view of (2.25), we need the relation s 0 < s c (p − 1). These restrictions yield the bound s c < 1, which is nothing but p > 1 + 4 N +2 . 2.5. An estimate on Lorentz space. In this subsection, we prove the following.
For the proof of this proposition, we need the following.
Lemma 2.23. Let I be an interval. Let 1 < ρ < ∞ and 1 < r < ∞. Let f ∈ L ρ,r (I). Let {I j } k j=1 be a subdivision of I. Then, there exists a constant C > 0 independent of k and f such that Introduce a linear operator T by where 1 I j is a characteristic function of I j . We now claim that T is a bounded Indeed, by the Hôlder inequality, . By the real interpolation method, we conclude that , which yields the desired estimate.
Proof of Proposition 2.22. Suppose M > δ, otherwise there is nothing to prove. Take t 1 ∈ I so that t 1 > t 0 and f L ρ,r (I 1 ) = δ. Similarly, as long as f L ρ,r (t j−1 ,sup I) > δ, we define t j ∈ I so that t j > t j−1 and f L ρ,r (I j ) = δ. Let us now show that, under this procedure, the assumption fails in finite steps, that is, there exists k 0 such that Indeed, take k 1 and suppose that we are able to take t j so that f L ρ,r (I j ) = δ for 1 j k. Then, one verifies from Lemma 2.23 that which implies k obeys the estimate Thus, such k is bounded from above and so there exists k 0 satisfying (2.26). Put t k 0 = sup I.
2.6. Some other estimates. We finish this section with several useful estimates.
Proof. Take f ∈Ḃ s q,r . Let M be an integer such that s < M s + 1. By Lemma 2.17, for k ∈ [0, M ]. One then sees that Since s > 0 and M − s > 0 and since s k < M − k, taking ℓ r norm with respect to j, we obtain The embeddinġ follows by assumption r 2 and by definitions of s k and q k . Thus, which completes the proof.
Proof. By the preceding lemma, for any t = 0, Take L ρ(W 1 ),2 -norm to obtain the desired result.

Local well-posedness and Perturbation arguments
The aim of this section is to establish local well-posedness of (NLS)-(IC) in FḢ sc and some related results. Let us first make the notion of a solution clear.
Definition 3.1. Let I ⊂ R be a nonempty time interval and let t 0 ∈ I. We say a function u: respectively, for all compact K ⊂ I, and u obeys the Duhamel formula in the FḢ sc sense for all t 1 ∈ I. We refer to the interval I as the lifespan of u. We say that u is a global solution if I = R. We say a solution u of (NLS) satisfies (IC) if I contains 0 and u(0) = u 0 .
Remark 3.2. The second term of the right hand of (3.1) makes sense. Indeed, if u ∈ W (K) then |u| p−1 u ∈ F (K) by Lemma 2.15 and embedding (2.15). Then, Strichartz' estimate implies that the term belongs to C 0 holds in theẊ sc 2 (t) sense or in theṀ sc 2,2 (t) sense for t ∈ I. 3.1. Local existence and uniqueness. In this subsection, we show a local existence theorem under an additional condition u 0 ∈ L 2 . This assumption will be removed after we establish a short-time perturbation.
Remark 3.4. For any u 0 ∈Ṁ sc 2,2 (t 0 ), there exists an interval I ∋ t 0 in which a corresponding solution u(t) exists. Indeed, thanks to Strichartz' estimate, we have U (t − t 0 )u 0 W (R) < +∞ for any u 0 ∈Ṁ sc 2,2 (t 0 ). Hence, we are always able to choose an interval I ∋ t 0 so that U (t − t 0 )u 0 W (I) η 0 . Notice that the interval I depends on the profile of the data, not simply on the size of the data.
for m 1. Let us first establish a uniform a priori bound on u (m) . Strichartz' estimates (Proposition 2.7), Lemma 2.15, and (2.15) give us Hence, if η 0 is sufficiently small, then we can show by induction that Again by Strichartz' estimate, We next estimate the difference u (m+1) − u (m) . We set By Strichartz' estimate and (3.6), Further, by the assumption u 0 ∈ L 2 , we have u (1) as m → ∞. It therefore holds that as m → 0, which shows that u(t) is a solution of (NLS). By a priori estimates (3.6) and (3.7), one has desired bounds on u(t). Uniqueness follows from a standard argument.

3.2.
Short-time perturbation. We next establish a stability estimate what is called a short-time perturbation. This is one of the key estimate of our argument. The local well-posedness, shown in the forthcoming subsection, relies on this estimate. The result depends on whether p 2 or not. When p 2 we are able to obtain Lipschitz continuity. However, it becomes merely Hölder if p < 2. We follow an argument by [30], in which the energy critical equation is considered. In [30], in order to establish a stability estimate for p < 2, they introduce an "intermediate space" and an exotic Strichartz estimate on it. In our case, X(I) is the intermediate space. Heart of the matter is that we take not a Lorentz-modified-Besov type space but a Lorentz-modified-Sobolev space. Then, the non-admissible Strichartz estimate (Proposition 2.12) plays the role of the exotic Strichartz estimate. This stability estimate will be upgraded in Theorem 3.13. However, the main point lies in this proposition.
for some 0 < ε ε 0 then a solution u(t) of (NLS) with initial data u(t 0 ) at t = t 0 obeys Proof. Put w := u − u. Then, w satisfies We first consider the case p 2. By the Strichartz estimate, (2.22), (2.15), and assumption, one sees that If ε 0 is small, we obtain w W (I) Cε + C w p W (I) , which implies w W (I) Cε for ε ε 0 if ε 0 is small. Then, another use of Strichartz' estimate yields We finally note that which completes the proof.
We next consider the case p < 2. By the embedding W 2 (I) ֒→ X(I) ֒→ L(I), Strichartz' estimate (Proposition 2.7), the non-admissible Strichartz estimate (Proposition 2.10), Lemma 2.18, and the assumption, one sees that Let us now estimate u W (I) . By the assumption, we have Cε 0 + Cε p 0 . If ε 0 is sufficiently smaller than η 0 , which is a constant given in Theorem 3.3, then the life span of u contains I and we have (3.10) u W (I) Cε 0 .
Substituting this estimate to (3.9), we obtain w X(I) Cε+Cε p−1 0 w X(I) , which implies if ε 0 is small. Again by Strichartz' estimate, We now use (2.21) to the second term of the right hand side to yield Then, by W 2 (I) ֒→ X(I) ֒→ L(I) and by (3.10) and (3.11), Hence, if ε 0 is small, then we have The other estimates follow as in the p 2 case.
Remark 3.6. The lower bound p > 1 + 4 N +2 comes from this proposition. The point of the above argument for p < 2 is that we first establish (3.11) to deal with w p−1 L(I) in the right hand side of (3.12). To obtain (3.11), it is essential to employ Lemma 2.18 which requires the lower bound p >  Proof. Let us first extend the existence result to the case u 0 ∈Ṁ sc 2,2 (t 0 ) \ L 2 . Let η 0 be the constant given in Theorem 3.3, and assume for some η 1 2 η 0 . Take a sequence of functions {u 0,n } ∞ n=1 ⊂Ṁ sc 2,2 (t 0 ) ∩ L 2 such that u 0,n → u 0 inṀ sc 2,2 (t 0 ) as n → ∞. Since for all n N . One then deduces from Theorem 3.3 that there exists a sequence of solutions {u n } ∞ n=N with the bounds u n W (I) Cη, One also obtains a standard blowup criterion. It is worth mentioning that that if u 0 ∈ L 2 then blowup never occurs due to the mass conservation. Proof. Assume T + < ∞ and u S([t 0 ,T + )) < +∞ for contradiction. Then, for any δ > 0 there exists t 1 ∈ (t 0 , T + ) such that u S([t 1 ,T + )) δ. Then, for τ ∈ [t 1 , T + ), . We now take δ (and t 1 ) so that where η 0 is given in Theorem 3.3. One can take t 3 > T + so that By means of Theorem 3.3, this implies u is extended beyond t = T + , which is a contradiction.
3.4. Scattering criterion. In this subsection, we give a necessary and sufficient condition for scattering. It is well-known that a space-time bound of a solution is a sufficient condition for scattering. Here, we will prove that this space-time bound is also a necessary condition. We restrict ourselves to nonnegative time only, but it is obvious that a similar result holds true for negative time.
Proposition 3.9. Let u 0 ∈ FḢ sc and let u(t) be a corresponding unique solution to (NLS) given in Theorem 3.7. Let (T − , T + ) ∋ 0 be the maximal interval of u(t). Then, the following three conditions are equivalent.
Let us show "(3)⇒(2)". Since u S(R + ) < ∞, for any ε > 0, there exists T 0 such that u S([T 0 ,∞)) ε. Now, u(t) satisfies One sees from Strichartz' estimate that Remark 3.10. The above criterion is valid also for FH 1 -solutions. More precisely, if u 0 ∈ FH 1 then a corresponding FH 1 -solution u(t) of (NLS)-(IC), scatters (in FH 1 ) for positive time if and only if u S(R + ) < ∞ (or the other properties in the proposition). The outline of proof is as follows. If a solution scatters in FH 1 then it does so also in the FḢ sc sense. Hence the first property of Proposition 3.9 is satisfied. On the other hand, if the third property of Proposition 3.9 are satisfied, then a persistence-of-regularity type argument shows that it scatters in FH 1 sense. The argument is very similar to that in the above proof.
A consequence of Proposition 3.9 is the following small data scattering. Lemma 3.11 (Small data scattering). Let u 0 ∈ FḢ sc and let u(t) be a corresponding unique solution given in Theorem 3.7. Then, we have the following.
This lemma is proven in [27] with an additional assumption u 0 ∈ L 2 . Another version is given later (Proposition 6.1) by using a long-time perturbation, Theorem 3.13.
Proof. If η 1 is sufficiently small, it follows from (3.2) that u W (R) Cη 1 . Hence, we obtain the former part by Proposition 3.9. The latter statement is an immediate consequence of the former and Strichartz' estimate.
Remark 3.12. The second property of Lemma 3.11 implies that the infimum ℓ c given in (1.9) is greater than or equal to η 2 > 0.
3.5. Long time perturbation. The following are our second stability estimate.

Smallness via quadratic oscillation.
In this subsection, we show that rapid oscillation of initial data gives smallness of a solution to linear Schrödinger equation. This is one of the key estimate of profile decomposition. It will be also a key tool for the proof of Theorem 1.7. This kind of smallness is given for FH 1 data in [6,24]. = 0.
Proof. By the symmetry (U (t)e ib|x| 2 ψ)(x) = (U (−t)e −ib|x| 2 ψ)(x), it suffices to show the case b → ∞. Further, if we show the limit for some specific acceptable pair (ρ, q) with 2 ρ − δ(q) + s = s c for some s ∈ [0, s c ) then it holds for all other pairs, thanks to Lemma 2.4 and Proposition 2.8. We therefore fix such a pair (ρ, q) and consider the limit b → ∞. Since (ρ, q) is acceptable, we have 0 < δ(q) < 1 and 1 < ρδ(q). Further, by assumption s < s c , we have ρδ(q) < 2. Let ϕ j ∈ C ∞ 0 (R N ) be a cut-off function in the definition ofṀ s q,r (t). We deduce from Stichartz' estimate that, for any ε > 0, there exists a number J = J(u 0 , ε) > 0 such that Since j∈Z ϕ j = 1 on {x = 0}, it suffices to show (4.1) for |j| J ϕ j u 0 . For this, we prove (4.1) for each ϕ j u 0 . Hence, we may suppose by scaling that supp u 0 ⊂ {1/2 |x| 2}. It holds that Let us restrict our attention to the case j = 0 since the estimates for ϕ 1 u 0 and ϕ −1 u 0 are essentially the same. It is obvious by assumption that ϕ 0 u 0 = u 0 . The pseudo-conformal transform gives us As a result, proof of (4.1) boils down to showing Let us prove (4.2). Take a > 0. Let β = β(a) > 0 to be chose later. We divide R into the following six intervals , 0 , Notice that I j = ∅ for j = 1, 2, . . . , 6 for sufficiently large b as long as β is chosen independently of b. By the L p -L q estimate, we have It follows from the assumptions supp u 0 ⊂ {1/2 |x| 2} and 0 < δ(q) < 1 that u 0 L q ′ C u 0 FḢ sc . This yields G b (t) C|t| −δ(q) u 0 FḢ sc . Then, where 1 I = 1 I (t) is a characteristic function of I. Now, recall an equivalent representation of L ρ,2 norm where f * is a non-increasing rearrangement of f (see [4]). It then holds that Hence, plugging this to (4.3), we have The same argument shows On the other hand, by the generalized Hölder inequality, .
The same argument yields .
It then follows from these three estimates that .

Concentration compactness.
In this subsection, we show that a bounded sequence of functions in FḢ sc is decomposed into a sum of several profiles. Compared with a sequence bounded in ℓ F H 1 (·), which is an FH 1 -bounded sequence up to a scaling normalization, the feature of FḢ scbounded sequence is that it admits a sum of functions with extremely distinct scales. Indeed, take 0 ≡ ψ ∈ S with supp ψ ⊂ {1 |x| 2} and put where we use the notation (1.3). Then, ψ n FḢ sc = 2 ψ FḢ sc < ∞ but ℓ F H 1 (ψ n ) → ∞ as n → ∞. Thus, in order to extract some specific profiles from an FḢ sc bounded sequence, we have to take this kind of scale decomposition into account. To do so, we employ results in [2], as in [21].
The result is the following.   as n → ∞. Furthermore, for any 1 < ρ, q < ∞ with δ(q)−s c ∈ (0, min(1, N/2)) and 2/ρ − δ(q) = s c , it holds that  However, just in order to simplify our argument, we work with (4.11) since it is sufficient for later use.
Scale decomposition. We first show that an FḢ sc -bounded sequence is decomposed into several portions which have distinct scales.
Definition 4.5. Let {f n } be a bounded sequence in L 2 . Let h = {h n } be a sequence of positive numbers.
For a given sequence {f n } bounded in FḢ sc , {F(|x| sc f n )} is a bounded sequence of L 2 . Applying decomposition of L 2 -bounded sequence in [2] to this sequence, we obtain the following. The error estimate (4.12) also yields smallness of linear evolution of the error term. To see this, we recall a refinement of the Sobolev embedding. Lemma 4.7 ([2,14]). Let 1 < p < q < ∞ and s > 0 satisfy Then, there exists a constant C = C(N, p, q, s) such that This lemma yields the following.  Proof. First, we consider the special case δ(q) − s c = 0, that is, In this case, ρ = ρ 0 := s −1 c . By means of Lemma 4.7, it holds for t = 0 that .
The general case follows by interpolation lemma (Lemma 2.4) since U (t)R l n is bounded in L ρ,∞ (R, L q ) for any (ρ, q) satisfying the assumption in light of Proposition 2.8.
Decomposition of each scale. Proposition 4.6 gives us a procedure to decompose an FḢ sc -bounded sequence {f n } n into some pieces {g j n } n,j of which scales are pairwise orthogonal. Let us next decompose each pieces {g j n } n into sums of functions of the form e is j n |x| 2 e iy j n ·x ψ j . For this, we consider a 1-scaled sequence, where we denote by 1 a sequence of positive numbers of which all component is equal to one. For a sequence {φ n } n ⊂ FḢ sc , we set ν({φ n }) := φ ∈ FḢ sc ∃s n ∈ R, ∃y n ∈ R N s.t. e −isn|x| 2 e −iynx φ n ⇀ φ in FḢ sc as n → ∞, up to subsequence. and η({φ n }) := sup It is obvious by definition that Proposition 4.9. Let {φ n } n ⊂ FḢ sc be a bounded sequence such that {|x| sc φ n } is 1-scaled. Then, there exist a subsequence of {φ n } n , which is denoted again by {φ n } n , and sequences {ψ j } j ⊂ ν({φ n }), {w j n } j,n ⊂ FḢ sc {s j n } j,n ⊂ R, and {y j n } j,n ⊂ R N such that if j = k then (4.14) |s j n − s k n | + |y j n − y k n | → ∞ as n → ∞ and that, for all l 1, Proof. If η({φ n }) = 0 then the result follows by choosing ψ j ≡ 0 and w l n = φ n . Otherwise, there exists ψ 1 ∈ ν({φ n }) satisfying By the definition of ν, there also exist sequences {s 1 n } ⊂ R and {y 1 n } ⊂ R N such that e −is 1 n |x| 2 e −iy 1 n x φ n ⇀ ψ 1 in FḢ sc . Define a sequence {w 1 n } by w 1 n = φ n − e is 1 n |x| 2 e iy 1 n x ψ 1 . It then follows that FḢ sc + o(1). as n → ∞ (up to subsequence). We claim that |s 1 n − s 2 n | + |y 1 n − y 2 n | → ∞ as n → ∞. Indeed, otherwise we see from (4.16) and (4.17) that ψ 2 = 0, a contradiction.
By this procedure, we inductively define ψ j , {s j n }, {y j n }, and {w l n }. Then, for any l 1, It is possible to upgrade the smallness property (4.15) as follows.
Since {|x| sc w l n } is uniformly 1-scaled, we see from Lemma 2.3 and Strichartz' estimate that there exists ζ(R) : On the other hand, for R ≫ 1, where C is a positive constant and χ j ∈ C ∞ 0 is the function defined in the definition ofṀ s q,r (t). Let θ ∈ (1/2, 1) to be chosen later. Using Hölder's inequality twice, we see that for t = 0. Generalized Hödler's inequality gives us (4.20) Here, we have (4.21) where the supremum is taken over all sequences {t n } ⊂ R \ {0} and {x n } ⊂ R N . Further, it follows from the well-known integral representation of U (t) that where s n = −1/4t n and y n = x n /2t n . The limit supremum in the right hand side of (4.21) is hence bounded by Thus, (4.22) lim sup On the other hand, one deduces from generalized Hödler's inequality that (4.23) Notice that ρ is well-defined if θ is sufficiently close to one. Define ρ ± by with ε > 0 to be chosen later. ρ ± is well-defined if ε is sufficiently small. Then, by Lemma 2.3 and Strichartz' estimate, we have as long as δ(qθ) − s ± ∈ [0, min(1, N 2 )), where Notice that if θ is sufficiently close to one and ε is sufficiently small then |s ± − s c | + |δ(q) − δ(qθ)| ≪ 1.
Completion of the proof. We are now ready to prove Proposition 4.3.

Proof of Proposition 4.3.
In what follows, we denote various subsequences of {φ n } by {φ n }. By Proposition 4.6, there exist {R j n } ⊂ FḢ sc , {h j n } ⊂ R + , and {q j n } such that where |x| sc g j n is {h j n }-scaled and R J n is {h j n }-singular for all j ∈ [1, J]. Set a sequence {P j n } by g j n := (P j n ) {h j n } or, equivalently, by P j n := (g j n ) {1/h j n } . Then, one sees that |x| sc P j n is 1-scaled. Moreover, which in particular implies a uniform bound on {P j n }; Take an integer l 1. Using Lemma 4.8, one can choose an integer J 0 (l) 1 such that holds as long as J J 0 (l). We may assume J 0 (l + 1) J 0 (l) without loss of generality. Further, for each j ∈ [1, J 0 (l)], take a number K 0,j = K 0,j (l) so that (4.26) lim sup n→∞ w (j,K) n L ρ,2 (R,Ṁ 0 q,2 ) 2 −l J 0 (l) −1 as long as K K 0,j (l). This is possible because of Lemma 4.10. Replacing with larger one if necessary, we let K 0,j (l+1) K 0,j (l). We also set K 0,j (l) = 0 if j > J 0 (l). We define a set A l ⊂ Z 2 + by It is obvious that A l 1 ∩ A l 2 = ∅ if l 1 = l 2 . Introduce two subsets of A l by We enumerate indices (j, k) belonging to ∪ l 1 B l in the following manner.
. Then, take a bijection m : Z → ∪ l 1 B l so that For m ′ ∈ (m l , m l+1 ), we set as n → ∞. Hence, (4.9) holds. If j(m(m 1 )) = j(m(m 2 )) then |s m 1 n − s m 2 n | + |y m 1 n − y m 2 n | tends to infinity as n → ∞ by means of (4.14). Recall that s m i n converges to a number s m i ∈ R by the definition of B l . Therefore, we have as n → ∞. The limit (4.9) is true also in this case. The equality (4.10) is rather trivial by Propositions 4.6 and 4.9. We only note that (4.28) ( as n → ∞ because s m n converges to s m ∈ R. Let us show (4.11). Let m = m l . One verifies from Proposition 2.8 and (4.28) that (4.29) lim n→∞ U (t)I 1 L ρ,2 (R,Ṁ 0 q,2 ) = 0.
Next we consider I 2 . By Galilean transform, and so Since (j, k) ∈ ∪ l 1 C l implies s (j,k) equals to +∞ or −∞, it follows from Proposition 4.1 that which proves (4.11) since l > 0 is arbitrary.

Proof of Theorem 1.2
In what follows, we prove our main theorems. Let us begin with the proof of Theorem 1.2.
Proof of Theorem 1.2. Suppose u 0 ∈Ḣ 1 ∩ FḢ sc satisfies E[u 0 ] < 0. We first note that Hardy's inequality implies FḢ sc ֒→Ḣ −sc . Therefore, u 0 ∈ H 1 ∩Ḣ −sc ⊂ H 1 . It is known that (NLS)-(IC) is globally well-posed in H 1 and we see that the solution u belongs to C(R, H 1 ) and that the mass and the energy is conserved, for any t ∈ R. By uniqueness, this solution coincides with the one given by Theorem 3.7. By assumption, Since q(L) > p+1 > 2, by means of Hölder's inequality, there exists θ ∈ (0, 1) independent of t such that . By the mass conservation and (5.1), one sees that u(t) L q(L) is bounded by some positive constant from below. Hence, u S(R + ) = ∞. Thanks to Proposition 3.9, this implies u 0 ∈ S + . Furthermore, if 0 < d < 1 is sufficiently close to one then Then, du 0 ∈ S + as shown above, and so which completes the proof.
6. Proof of Theorem 1.7 For the proof of Theorem 1.7, let us establish a version of small data scattering. This follows from the long-time perturbation. Theorem 1.7 immediately follows from this proposition and Proposition 4.1. Proof. We apply Theorem 3.13 with t 0 = 0, I = R, and u(t) = U (t)u 0 . Then, e(t) = |U (t)u 0 | p−1 U (t)u 0 . By Strichartz' estimate, Hence, if we take η small, we have e F (R) ε 1 , where ε 1 = ε 1 (M ) is the number given in Theorem 3.13. Then, Theorem 3.13 yields which implies u W (R) < ∞. Thus, we conclude from Proposition 3.9 that u 0 ∈ S Remark 6.2. Since scattering in FḢ sc and scattering in FH 1 is equivalent for FH 1 -solutions (see Remark 3.10), if u 0 ∈ FH 1 is added to the assumption of the above proposition then the result holds true with replacing the meaning of scattering by FH 1 sense. This is the reason why Theorem 1.7 holds for FH 1 solutions.
To prove Lemmas 7.1 and 7.2, we first establish the following.
Take L ρ(F ),2 (R + )-norm of the both sides. By scaling and by the generalized Hölder inequality, we conclude that lim sup Since Ψ j L(R + ) < ∞ by embedding W 2 (R + ) ֒→ L(R + ), the proof is now reduced to showing that Then, together with (7.22), for any l > l 0 and n n 0 (l, l 0 , ε). One easily sees that there exists ε 0 > 0 such that if 0 < ε ε 0 then the inequality (7.23) implies which completes the proof.
Proof of Lemma 7.2. Set F (z) = |z| p−1 z. By definition of e l n , . By Lemma 7.3, the last term of the right hand side tends to zero as n → ∞ for all l. Moreover, the second and the third terms become small if we take l 0 sufficiently large in light of Lemma 7.1 and (2.21). Thus we shall estimate the first term. Without loss of generality, we may assume that supp Ψ j ⊂ [m, M ] × B 0 (R) holds for some m, M, R > 0 and for all j ∈ [1, l 0 ]. Set .
By the orthogonality condition, Σ j n (1 j l 0 ) are mutually disjoint for large n. For such n, we have Set χ j n = (χ j n ) p . Then, χ j n F ( v l 0 n + U (t)W l n ) = F (χ j n v l 0 n + χ j n U (t)W l n ) = F (Ψ j [1,ξ j n ] + χ j n U (t)W l n ) for any j ∈ [1, l 0 ], provided n is sufficiently large. Similarly, χ j n F ( v l 0 n ) = F (Ψ j [1,ξ j n ] ) for large n. Further, one easily verifies that 1 − l j=1 χ j n ≡ 0 on ∪ l 0 j=1 supp Ψ j [1,ξ j n ] (t, x). Therefore,   1 − and (1 − l j=1 χ j n )F ( v l 0 n ) ≡ 0. Thus, for large n, we have Since W l n is uniformly bounded in FḢ sc , we deduce from Strichartz' estimate and (7.6) that lim k→∞ lim sup The estimate of III is done in essentially the same way. We only note that changing of variable x − tξ j n = y and application of the Galilean transform give us is uniformly bounded. Let us proceed to the estimate of I. We consider only j = 1, i.e. we treat .
Arguing as in the proof of (2.21), we have → 0 as k → ∞. Putting this estimate and above estimates to (7.24), the estimate of I is completed.
For this θ, we set q 1 by the relation . Now, we use the following Lemma by [20]; Lemma 7.4. For any 1 < p, q, r < ∞ with 1 p = 1 q + 1 r and 0 < α, α 1 , α 2 < 1 with α = α 1 + α 2 , we have |∇| α (f g) − f |∇| α g − g|∇| α f L p C |∇| α 1 f L q |∇| α 2 g L r whenever the right hand side is bounded. . Therefore we conclude from (7.6) that Remark 7.5. The minimizing problem ℓ c,F H 1 := inf{ℓ F H 1 (u 0 ) | u 0 ∈ FH 1 \ S} can be treated in a similar way, where FH 1 is given in (1.8). Existence of the minimizer to this problem is shown in [24] under p St < p < 1 + 4/N . Here, we extend it as follows: Under the assumption (1.1), there exists u 0,c ∈ FH 1 such that u 0,c ∈ S + and ℓ F H 1 ( u 0,c ) = ℓ c,F H 1 . Further, a solution u c (t) to (NLS) with data u 0,c is not a standing wave. We give a sketch of proof. The strategy is the same as in the proof of Theorem 1.1 We first take a minimizing sequence for ℓ c,F H 1 (we replace S with S + without loss of generality). By scaling, we may further assume that each function has a unit mass and so that they are uniformly bounded in FH 1 . Then, apply the profile decomposition (Proposition 4.3) to the sequence. Uniform boundedness in FH 1 enables us to establish the Pythagoras decomposition (4.10) with FH 1 norm (see [24]). The rest of the argument is the same. Recall that scattering in FH 1 is equivalent to scattering in FḢ sc , as noted in Remark 3.10. We prove that only one profile is involved in the decomposition and that the profile must not belong to S + . Remark 7.6. A naive conjecture is that the above u 0,c is one of the function satisfying the properties of Theorem 1.1. However, it is not clear at least by the following two reasons. First is that we do not know whether u 0,c given in Theorem 1.1 belongs to FH 1 or not. Even when u 0,c ∈ FH 1 is true, minimality with respect to ℓ(·) and that to ℓ F H 1 (·) are different, which is the second reason. This fact is easily checked by the example given in Remark 4.2. Notice that u c (t) may blow up in finite time while u c (t) is global in time.