ON FRACTIONAL NONLINEAR SCHR¨ODINGER EQUATION WITH COMBINED POWER-TYPE NONLINEARITIES

. We undertake a comprehensive study for the fractional nonlinear Schr¨odinger equation where < < α < α < s d − s . Firstly, we establish the local and global well-posedness results for non-radial and radial H s initial data, radial ˙ H s c ∩ ˙ H s initial data, where s c = d 2 − 2 sα 2 . Secondly, we study the asymptotic behavior of global radial H s solutions. Of particular interest is the L 2 -critical case and the results in this case are conditional on a conjectured global existence and spacetime estimate for the L 2 -critical fractional nonlinear Schr¨odinger equation. Thirdly, we obtain suﬃcient conditions about existence of blow-up radial ˙ H s c ∩ ˙ H s solutions, and derive the sharp threshold mass of blow-up and global existence for this equation with L 2 -critical and L 2 subcritical nonlinearities. Finally, we obtain the dynamical behaviour of blow- up solutions in both L 2 -critical and L 2 -supercritical cases, including mass-concentration and limiting proﬁle.


(Communicated by Pierre Germain)
Abstract. We undertake a comprehensive study for the fractional nonlinear Schrödinger equation i∂tu − (−∆) s u = µ 1 |u| α 1 u + µ 2 |u| α 2 u, u(0) = u 0 , where d 2d−1 ≤ s < 1, 0 < α 1 < α 2 < 4s d−2s . Firstly, we establish the local and global well-posedness results for non-radial and radial H s initial data, radialḢ sc ∩Ḣ s initial data, where sc = d 2 − 2s α 2 . Secondly, we study the asymptotic behavior of global radial H s solutions. Of particular interest is the L 2 -critical case and the results in this case are conditional on a conjectured global existence and spacetime estimate for the L 2 -critical fractional nonlinear Schrödinger equation. Thirdly, we obtain sufficient conditions about existence of blow-up radialḢ sc ∩Ḣ s solutions, and derive the sharp threshold mass of blow-up and global existence for this equation with L 2 -critical and L 2subcritical nonlinearities. Finally, we obtain the dynamical behaviour of blowup solutions in both L 2 -critical and L 2 -supercritical cases, including massconcentration and limiting profile.
1. Introduction. In recent years, there has been a great deal of interest in using fractional Laplacians to model physical phenomena. By extending the Feynman path integral from the Brownian-like to the Lévy-like quantum mechanical paths, Laskin in [37,38] used the theory of functionals over functional measure generated by the Lévy stochastic process to deduce the following nonlinear fractional Schrödinger equation where 0 < s < 1 and f (u) is the nonlinearity. The fractional nonlinear Schrödinger equation also appears in the continuum limit of discrete models with long-range interactions (see e.g. [36]) and in the description of Bonson stars as well as in water wave dynamics (see e.g. [28]). The fractional differential operator (−∆) s is defined by (−∆) s u = F −1 [|ξ| 2s F(u)], where F and F −1 are the Fourier transform and inverse Fourier transform, respectively. Recently, equation (1.1) has attracted more and more attentions in both the physics and mathematics fields, see [2,3,7,9,12,13,14,15,16,19,20,21,26,23,24,25,30,32,35,47,48,49]. For the Hartree-type nonlinearity f (u) = ±(|x| −γ * |u| 2 )u, Cho et al. in [8] proved existence and uniqueness of local and global solutions of (1.1). In the focusing case, i.e. there is a minus sign in front of the nonlinearity, the existence of blow-up solutions was shown by Cho et al. in [10]. The dynamical properties of blow-up solutions have been investigated in [9,12,48]. Zhang and Zhu in [47] studied the stability and instability of standing waves. Guo and Zhu in [32] established the sharp threshold of blow-up and scattering in the mass-supercritical and energy-subcritical case. The global existence in the focusing energy-critical case was shown by Cho et al. in [13]. For the local nonlinearity f (u) = ±|u| α u, the well-posedness and ill-posedness in the Sobolev space H s have been investigated in [11,34,19]. In [3], Boulenger et al. have obtained general criteria for blow-up of radial solutions of (1.1) with the focusing nonlinearity f (u) = −|u| α u and α ≥ 4s d in R d , d ≥ 2. Dynamics of blow-up solutions were studied recently by the first author in [20,21]. The sharp threshold of blow-up and scattering in the mass-supercritical and energy-subcritical case was established in [43,33]. Guo et al. in [30] shown the global existence and scattering in the energy-critical case. The orbital stability of standing waves for other kinds of fractional Schrödinger equations has been studied in [2,24,25,49,42].
If one considers initial data inḢ sc ∩Ḣ s , then the equation only has energy conservation. The conservation of mass is no longer available in this setting. One of motivations for considering (1.2) is the lack of scaling invariance. It is wellknown that there is a natural scaling invariance associated to the single nonlinear Schrödinger equation (1.5) More precisely, the scaling u λ (t, x) := λ 2s α u(λ 2s t, λx), λ > 0 leaves (1.5) invariant, that is, if u is a solution of (1.5), then u λ is also a solution of (1.5). In our consideration with combined nonlinearities α 1 < α 2 , there is no scaling that leaves (1.2) invariant. However, one can use scaling and homogeneity to normalize both µ 1 and µ 2 to have magnitude one without difficulty. When s = 1, Tao et al. in [44] undertook a comprehensive study for the following nonlinear Schrödinger equation with combined power-type nonlinearities i∂ t u + ∆u = µ 1 |u| α1 u + µ 2 |u| α2 u, where 0 < α 1 < α 2 ≤ 4 d−2 , d ≥ 3 and µ 1 , µ 2 are non-zero real numbers. More precisely, they addressed questions related to local and global well-posedness, finite time blow-up in weighted space Σ := H 1 ∩ L 2 (|x| 2 dx), and asymptotic behaviour (scattering) in both energy space H 1 and weighted space Σ. The scattering versus blow-up for some particular cases of (1.6) was studied in [40,46,6,41]. Recently, in [22], the second author proved the existence of blow-up solutions and found the sharp threshold of blow-up and global existence for (1.6) with 0 < α 1 < 4 d , α 2 = 4 d , µ 1 > 0 and µ 2 < 0, which is a complement to the result in [44].
For the fractional nonlinear Schrödinger equation (1.2), the second author in [23] established some sufficient conditions about the existence of blow-up solutions, sharp thresholds of blow-up and global existence and dynamical properties of blowup solutions in the L 2 -critical case, i.e. 0 < α 1 < 4s d , α 2 = 4s d , µ 1 > 0 and µ 2 < 0, including mass-concentration, blow-up rates, and limiting profile.
In this paper, we will systematically study the Cauchy problem (1.2). We are interested in local and global well-posedness, asymptotic behavior(scattering), the existence of finite time blow-up solutions and dynamical properties of blow-up solutions in the L 2 -critical and L 2 -supercritical cases, including mass-concentration and limiting profile. We also mention that in this paper, we do not consider the energy-critical case, i.e. α 2 = 4s d−2s . The reasons for that are the lack of dispersive estimates for the fractional Schrödinger operator e −it(−∆) s as well as the lack of a good global theory for the single energy-critical equation. We hope to consider this interesting problem in a future work.
Firstly, applying a fixed point argument and Strichartz estimates, we establish the local well-posedness results of (1.2) for non-radial and radial H s initial data, radialḢ sc ∩Ḣ s initial data. These results are complements to the ones in [23,49]. Note that non-radial Strichartz estimates for the fractional Schrödinger equation are well-known to have a loss of derivatives. This is the main reason why we mainly consider radial data in this paper. We refer the reader to Section 3 for more details.

VAN DUONG DINH AND BINHUA FENG
Secondly, using some elementary inequalities, we establish an a priori estimate on the kinetic energy, namely where E and M are the conserved energy and mass respectively. With this a priori bound, the blow-up alternative yields the global existence of H s solutions to (1.2) in two cases: d−2s , µ 1 ∈ R and µ 2 > 0. We refer the reader to Section 6 for more details. We also give some criteria for the global existence of radialḢ sc ∩Ḣ s solutions to (1.2) at the end of Section 8.
Thirdly, we will study the asymptotic behavior of global radial H s solutions to (1.2) with 4s d ≤ α 1 < α 2 < 4s d−2s and µ 2 > 0. In the case α 1 = 4s d , the L 2 -stability is exploited. To do so, by a similar idea as in [44], we need to assume a good global theory for the single defocusing L 2 -critical FNLS, namely More precisely, we need the following assumption.
Unlike the nonlinear Schrödinger equation s = 1, equation (1.2) does not enjoy the a priori interaction Morawetz estimate. However, thanks to the radial assumption, we are able to derive a priori radial Morawetz estimates in the defocusing case, i.e. µ 1 , µ 2 > 0 and 4s d ≤ α 1 < α 2 < 4s d−2s . This estimate allows us to control the solution in the L m -norm with α 1 + 1 + 2 d−2s ≤ m ≤ α 2 + 1 + 2 d−2s . Using this global L m -norm bound, we can derive the global Strichartz bound of solutions. With the help of this global Strichartz bound, the scattering follows easily. We refer the reader to Section 7 for more details.
Fourthly, we will investigate sufficient conditions about the existence of blow-up radialḢ sc ∩Ḣ s solutions for (1.2) by using the method of Boulenger et al. [3]. When µ 1 > 0, 0 < α 1 < α 2 = 4s d and µ 2 < 0, Feng in [23] found the sharp threshold mass Q L 2 of blow-up and global existence for (1.2), where Q is the ground state solution of (2.11) with α = 4s d . However, there is an error in the proof of the existence of blow-up solutions given in [23]. In addition, in this case, it follows from Lemma 4.4 that Therefore, by using the argument of Boulenger et al [3], we can prove the existence of blow-up solutions by choosing the initial data u 0 such that E(u 0 ) < 0. But when µ 1 < 0 and 0 < α 1 < 4s Because u(t) α1+2 L α 1 +2 is a positive uncertain function, which may be bounded or not relative to t. Hence, it is hard to choose E(u 0 ) to ensure the existence of blowup solutions. We develop a new argument by contradiction to solve this problem. In addition, our method can be easily applied to prove the existence of blow-up solutions for (1.6) with µ 1 < 0, µ 2 < 0 and 0 < α 1 < α 2 = 4 d , which is an open problem left by Tao et al. in [44]. As far as we know, this result has not been proved yet. Therefore, this type of result for (1.2) is new even if s = 1.
Finally, we obtain the dynamical behaviour of blow-up solutions to (1.2) in both L 2 -critical and L 2 -supercritical cases, including mass-concentration and limiting profile. In the L 2 -critical case, i.e. 0 < α 1 < α 2 = 4s d , the second author in [23] studied dynamical properties of finite time blow-up solutions with µ 1 > 0 and µ 2 < 0. In this paper, we extend the result of [23] to µ 1 ∈ R. In the L 2 -supercritical case, i.e. 4s d < α 2 < 4s d−2s and 0 < α 1 < α 2 , since the uniqueness of solutions to elliptic equations (2.14) and (2.17) are not known yet, we need to introduce notions of Sobolev and Lebesgue ground states in order to describe the dynamical behaviour of the blow-up solutions to (1.2) in the homogenoues setting.
This paper is organized as follows. In Section 2, we present some preliminaries including Strichartz estimates, profile decompositions and compactness lemmas. In Section 3, we establish the local well-posedness results of (1.2) for non-radial and radial H s initial data as well as radialḢ sc ∩Ḣ s initial data. In Section 4, we recall and establish some new virial estimates related to (1.2) for both radial H s data and radialḢ sc ∩Ḣ s data. In Section 5, we study the stability of the L 2 -critical fractional nonlinear Schrödinger equation. In Section 6, we establish the global well-posedness of radial H s solutions to (1.2). In Section 7, we show the asymptotic behavior in the energy space for global radial H s solutions to (1.2). In Section 8, we will establish some sufficient conditions about the existence of blow-up solutions for (1.2), and then obtain some sharp thresholds of blow-up and global existence. In Section 9, we study the dynamical behaviour of the blow-up solutions to (1.2) in both L 2 -critical and L 2 -supercritical cases, including mass-concentration and limiting profile.
with the usual modification when either p or q are infinity. In the case p = q, we shall use L p (J × R d ) instead of L p (J, L p ). We next recall Strichartz estimates for the unitary group e −it(−∆) s . It is well-known that e −it(−∆) s enjoys several types of Strichartz estimates, in particular non-radial and radial Strichartz estimates which are recalled as follows.
• Non-radial Strichartz estimates [16,19]: for d ≥ 1, s ∈ (0, 1)\{1/2}, where (p, q) and (a, b) are Schrödinger admissible pairs, i.e. and p , we see that s p,q > 0 for any Schrödinger admissible pairs except (p, q) = (∞, 2). This shows that non-radial Strichartz estimates have a loss of derivatives. This loss of derivatives makes the study of local well-posedness with non-radial data more difficult. We refer to Section 3 for more details.
• Radial Strichartz estimates [31,35,14]: for d ≥ 2, s ∈ (0, 1)\{1/2}, the estimates (2.1) and (2.2) hold true provided (p, q) and (a, b) satisfy the radial Schrödinger admssible condition The last condition in (2.5) allows us to choose (p, q) so that s p,q = 0. Pluging it into 2 p + 2d−1 q ≤ 2d−1 2 , we have the following radial Strichartz estimates: for where u 0 and f are radially symmetric and (p, q), (a, b) are fractional admissible (2.8) These Strichartz estimates without loss of derivatives allow us to give a better local well-posedness result. This is the reason why we mainly consider radially symmetric initial data throughout this paper.

Profile decompositions.
Lemma 2.1 (H s profile decomposition [48,20]). Let d ≥ 2 and 0 < s < 1. Let (v n ) n≥1 be a bounded sequence in H s . Then there exist a subsequence still denoted by (v n ) n≥1 , a family (x j n ) j≥1 of sequences in R d and a sequence (V j ) j≥1 of H s functions such that • for every k = j, |x k n − x j n | → ∞, as n → ∞; • for every l ≥ 1 and every as n → ∞.

Sharp Gagliardo-Nirenberg inequality.
A first application of the profile decomposition is the following sharp Gagliardo-Nirenberg inequalities. 3,49]). Let d ≥ 2, 0 < s < 1 and 0 < α < 4s d−2s . Then for any u ∈ H s , where the optimal constant C opt is given by Here Q is the unique (up to symmetries) positive radial solution to the elliptic equation (2.11) Moreover, the following Pohozaev's identities hold true: (2.12) Remark 1. The uniqueness of positive radial solution to (2.11) was shown recently in [26,27]. Note that the estimate (2.10) still holds true in one dimension (see e.g. [26]).
• Then for any u ∈Ḣ γc ∩Ḣ s , where the optimal constant A opt is given by with W a solution to the elliptic equation (2.14) Moreover, the following Pohozaev's identities hold true: • Then for any u ∈ L βc ∩Ḣ s , where the optimal constant B opt is given by with R a solution to the elliptic equation Moreover, the following Pohozaev's identities hold true: Since the uniqueness of solutions to (2.14) and (2.17) are still unknown. To study dynamical properties of blow-up solutions in the homogeneous setting, we need to introduce the notions of ground states. Denote 1. We call Sobolev ground states the maximizers of G which are solutions to (2.14). We denote the set of Sobolev ground states by G. 2. We call Lebesgue ground states the maximizers of K which are solutions to (2.17). We denote the set of Lebesgue ground states by K.
It follows from the definition of ground states that if g and k are Sobolev ground state and Lebesgue ground state respectively, then This implies that all Sobolev ground states have the sameḢ γc -norm and all Lebesgue ground states have the same L βc -norm. We thus denote S gs := g Ḣγc , ∀g ∈ G, (2.19) L gs := k L βc , ∀k ∈ K. (2.20) The sharp Gagliardo-Nirenberg inequalities (2.13) and (2.16) can be written as where Q is the unique (up to symmetries) positive radial solution to the elliptic equation (2.11) with α = 4s d . Remark 2. The lower bound (2.23) is optimal. Indeed, taking v n = Q where Q is given in 2.3, we get the equality.
Lemma 2.7 (Compactness lemma II [21]). Let d ≥ 2 and 0 < s < 1, 4s d < α < 4s and γ c , β c be as in (2.9). Let (v n ) n≥1 be a bounded sequence inḢ γc ∩Ḣ s such that • Then there exists a sequence (y n ) n≥1 in R d such that up to a subsequence, v n (· + y n ) P weakly inḢ γc ∩Ḣ s , for some P ∈Ḣ γc ∩Ḣ s satisfying • Then there exists a sequence (z n ) n≥1 in R d such that up to a subsequence, 3. Local well-posedness. In this section, we establish the local well-posedness for (1.2) in the case of non-radial, radial H s initial data as well as radialḢ sc ∩Ḣ s initial data. The proofs are based on Strichartz estimates and the standard fixed point argument, so we will omit them and only give some comments.
3.1. Non-radial H s initial data.
Then for all u 0 ∈ H s , there exist T ∈ (0, +∞] and a unique solution to (1.2) Moreover, the following properties hold: • The solution enjoys conservation of mass and energy, i.e. M (u(t)) = M (u 0 ) and E(u(t)) = E(u 0 ) for all t ∈ [0, T ).
We refer the reader to [20] (or [19]) for the proof of this result. Note that in the case of non-radial H s initial data, Strichartz estimates have a loss of derivatives. However, the loss of derivatives can be compensated by using the Sobolev embedding.
Remark 4. It follows from (3.1) and s ∈ (0, 1)\{1/2} that the local well-posedness for non-radial H s initial data is available only for Then for any u 0 ∈ H s radial, there exist T ∈ (0, +∞] and a unique solution to . Moreover, the following properties hold: • The solution enjoys conservation of mass and energy, i.e. M (u(t)) = M (u 0 ) and E(u(t)) = E(u 0 ) for all t ∈ [0, T ).
We again refer the reader to [20] for the proof of this result. In this case, Strichartz estimates have no loss of derivatives. We thus get a better local wellposedness result compared to the one in Proposition 1.
≤ α 1 < α 2 and s c be as in (1.3). Let (p j , q j ) be as in (3.3). Then for any u 0 ∈Ḣ sc ∩Ḣ s radial, there exist T > 0 and a unique solution u to (1.2) satisfying Moreover, the following properties hold: Proof. The proof is similar to the one of Proposition 2 (see also [21]). We thus omit the details. Note that the condition , hence the energy functional is welldefined. Note also that the conservation of mass is no longer available in this setting.

VAN DUONG DINH AND BINHUA FENG
4. Virial estimates. In this section, we recall virial estimates related to (1.2). Let us start with the following estimate.
for some C > 0 depending only on ∇ϕ W 1,∞ and d.
To study the time evolution of M ϕ (u(t)), we need the following auxiliary function where c s := sin πs π .
Using Plancherel's and Fubini's theorem, it follows that If we make formal substitution and take the unbouned function ∇ϕ(x) = 2x, then we have ∂ 2 jk ϕ = 2δ jk and ∆ 2 ϕ = 0. Using (4.4), we find formally the virial identity Now let ϕ : R d → R be as above. We assume in addition that ϕ is radially symmetric and satisfies ϕ(r) := r 2 for r ≤ 1, const. for r ≥ 10, and ϕ (r) ≤ 2 for r ≥ 0.
Here the precise constant is not important. For R > 0 given, we define the rescaled function ϕ R : It is easy to see that Moreover, By a similar argument as Lemma 2.2 in [3], we have the following virial estimate for the time evolution of M ϕ R (u(t)).
We also have the following refined version of Lemma 4.3 in the L 2 -critical case.
Proof. The proof is essentially given in [3, Lemma 2.3]. For the reader's convenience, we provide some details. By Lemma 4.2, By Lemma A.2 of [3] and the conservation of mass, we have We next use (4.4) to write We also have We thus get Since supp(ψ 2,R ) ⊂ {|x| > R}, we use the radial Sobolev embedding (see e.g. [15]): and the conservation of mass to estimate Here we use the Young inequality ab ηa p + η −q/p b q with 1/p + 1/q = 1 to have the last inequality. Note that the assumption 0 < α 1 < α 2 = 4s d and d > 2s ensures that α 1 , α 2 < 2. By the same argument as in (2.24) of [3], we obtain Therefore, for some constant C > 0. The same estimate holds true if α 1 is replaced by α 2 . Combining (4.10) and (4.12), we complete the proof.
Recall that s c is given in (1.3). Let ϕ R be as in (4.7). We define the localized virial action
Proof. We apply Lemma 4.2 to have Since ϕ R (x) = |x| 2 for |x| ≤ R, we use (4.5) to write Using the fact and (4.4) and (4.8), we estimate By Lemma A.2 of [3], the choice of ϕ R , we have from (4.14) that Collecting above estimates and using 2d − ∆ϕ R L ∞ 1, we get Let us now control u(t) αj +2 L α j +2 (|x|>R) , j = 1, 2. For H s solutions, one can take advantage of the mass conservation law. In this setting, the conservation of mass is no longer available. To overcome this difficulty, we use the technique of [39] (see also [21]). Consider for A > 0 the annulus C := {A < |x| ≤ 2A}. Using the radial 4582 VAN DUONG DINH AND BINHUA FENG Sobolev embedding (4.11) and (4.14), we have In the case α = α 2 , we see that Thanks to the assumption α 2 < 4s, we can choose 1 2 < β 2 < s so that β2α2 s < 2. The Young inequality then implies for any η > 0, . This shows for any η > 0, there exists C 2 (η) > 0 such that In the case α = α 1 , we have Applying the Young inequality, we get for any η > 0, We need to showθ 1 > 0. Since α 1 < α 2 < 4s, taking β 1 = 1 2 + for some > 0 small enough, we see thatθ 1 > 0 provided that It is easy to check that the assumption (4.16) ensures that the above inequality holds true. This shows that for any η > 0, there exists C 1 (η) > 0 such that for someθ 1 > 0. In both cases, we have shown for some ϑ > 0. We now write and apply (4.21) with A = 2 j R to get Denote also is a fractional admissible pair.
Consider the defocusing L 2 -critical fractional nonlinear Schrödinger equation To study the stability of (5.2), we assume that Assumption 1.1 holds true.
Proposition 4 (L 2 -critical stability). Let d ≥ 2 and d 2d−1 ≤ s < 1. Let J ⊂ R + be a compact interval and letṽ be a radial approximate solution to (5.2) in the sense that for some function e. Assume that for some constants M, L > 0. Let t 0 ∈ J and let v(t 0 ) be radially symmetric and close toṽ(t 0 ) in the sense that for some M > 0. Assume in addition that for some η > 0 small enough to be chosen shortly. By Strichartz estimates, v(t 0 ) Ḣγ for all k = 0, · · · , K − 1. Summing over all subintervals J k , we prove (5.13).  2) if one of the following conditions holds true: • 0 < α 1 < α 2 < 4s d and µ 1 , µ 2 ∈ R; • 0 < α 1 < α 2 < 4s d−2s and µ 1 ∈ R, µ 2 > 0. Moreover, for all compact intervals J ⊂ R + , the global solution satisfies the spacetime bound Here for simplicity, we only state the global well-posedness for radial H s data. However, it still holds true for non-radial H s data provided the local theory is available (see e.g. Proposition 1).
Proof. The proof of this result is based on the blow-up alternative which asserts that the time of existence depends only on the H s -norm of initial data, and an a priori estimate on the kinetic energy, namely where E and M are the conserved energy and mass respectively. To prove (6.2), we consider the following three cases: • When µ 1 < 0 and µ 2 > 0, we use the following inequality together with the conservation of mass and energy to have To see (6.3), we use the Young inequality to have for any η > 0, Multiplying both sides by µ1 α1+2 , we get We next choose η > 0 so that µ1η α1+2 = −µ2 α2+2 or η = − µ2(α1+2) µ1(α2+2) > 0 and obtain (6.3).

VAN DUONG DINH AND BINHUA FENG
We next apply the Young inequality Taking c > 0 sufficiently small, we absorb u(t) 2Ḣ s in the right hand side to the left hand side and get Combining three cases, we prove (6.2). The proof is complete.

7.
Scattering. In this section, we show the asymptotic behavior in the energy space H s for (1.2). More precisely, we prove the following.
The proof of this result is based on the combination of the L 2 -stability and the radial Morawetz inequality. We will give the proof of Theorem 7.1 at the end of this section.  Proof. We firstly note that under the assumptions of Lemma 7.2, the solution exists globally in time due to Section 6. Consider ϕ(x) = |x|. A direct computation shows In particular, ∇ϕ(x) = x |x| , ∆ϕ(x) = d−1 |x| . Moreover, where δ 0 is the Dirac delta function. Note also that since ϕ is a convex function, it is well-known that Applying formally Lemma 4.2 with ϕ(x) = |x|, we obtain Taking the time integration, we have Recall (see [ Let us start with the following useful estimate.
We next bound We next use the Hölder inequality and the Sobolev embedding to have We thus obtain This shows (7.4) with In order to perform the above estimates, we need to show a( ) and b( ) are both positive for > 0 small enough. Since → a( ) and → b( ) are continuous functions and the limits Taking > 0 small enough, we see that a( ), b( ) > 0. The proof is complete.
where the V -norm is given in (5.1).
Proof. By Hölder's inequality and fractional derivative estimates, we have .
Using the Sobolev embedding , we prove the desired estimate.
We are now able to show global Strichartz bounds (7.3) for solutions to (1.2). We will consider three cases.
(1) The case 4s d = α 1 < α 2 < 4s d−2s and µ 1 , µ 2 > 0. Without loss of generality, we assume that µ 1 = µ 2 = 1. In this case, we view (1.2) as a perturbation to the L 2 -critical fractional nonlinear Schrödinger equation. Note that in this case, we need to assume a satisfactory global theory for the L 2 -critical problem, more precisely Assumption 1.1.
We are only interested in those intervals J k = [t k , t k+1 ] which have a nonempty intersection with I n . Without loss of generality, we may assume that Indeed, by Strichartz estimates and (7.9), where p is the conjugate exponent of p = 2(d+2s) d . Note that (p, p) is fractional admissible and 1 p = 4s dp + 1 p .
Choosing δ > 0 small enough, we obtain In order to estimate F s -norm of u on I n × R d , we use the the stability technique as follows. We firstly compare u to v on the slab [t 0 , t 1 ] × R d by the L 2 -stability (see Proposition 4). We then use the result as an input to compare u to v on the slab [t 1 , t 2 ] × R d . By induction, we derive bounds on u from bounds on v on all slab J k × R d , k = 0, · · · , L. Adding these bounds, we get the desired estimate of u on I n × R d . For k = 0, we will check the hypotheses of Proposition 4. Note that u and |u| α2 u play the roles ofṽ and e in (5.3) respectively. The condition (5.4) is satisfied by the conservation of mass. The conditions (5.6) and (5.7) are obvious since u(t 0 ) = v(t 0 ). It remains to check (5.5) and (5.8). Using Duhamel's formula Strichartz estimates, (7.8) and (7.10) imply . By taking δ, η > 0 small enough, the continuity argument yields u Ẋ0 (J0) ≤ 4δ. (7.11) In particular, (5.5) holds. It remains to check (5.8). Estimating as in (7.8), we get This shows that (5.8) on J 0 by choosing δ, η > 0 small depending only on E and M . Applying Proposition 4, we obtain Moreover, we also have from Strichartz estimates, (7.4), (7.9) and (7.11) that (7.14) By taking δ, η > 0 small enough, we get For k = 1, we see that the condition (5.4) is again satisfied by the conservation of mass. By Strichartz estimates, (7.13) implies This shows (5.6) and (5.7). By Duhamel's formula, Strichartz estimates, (7.9), (7.10) and (7.16), we have . Taking δ, η > 0 small enough, the continuity argument implies u Ẋ0 (J1) ≤ 4δ. (7.17) This shows in particular that (5.5) holds. Using (7.17), a similar argument as in (7.9) gives Choosing δ, η > 0 small depending only on E and M , the condition (5.8) holds on J 1 . Applying Proposition 4, we get By the same argument as in (7.14), we also have u Ḟ s (J1) ≤ C(E).

VAN DUONG DINH AND BINHUA FENG
By induction, taking δ, η > 0 smaller in each step, we obtain for each k = 0, · · · , L − 1. Adding these estimates over all subintervals J k which have a nonempty intersection with I n , we get Combining these bounds, we prove (7.7). The proof is complete.
We will use the bounds ofũ to derive the bounds of u on each spacetime slab J k × R d . For k = 0, we first deduce from the Duhamel formula and the fact u(0) =ũ(0) = u 0 that By Strichartz estimates and Lemma 7.4, we have Taking η > 0 small enough, the standard continuity argument yields Similarly, by Strichartz estimates and Lemma 7.4, .
Choosing M sufficiently small, the continuity argument shows

Moreover, Strichartz estimates and Lemma 7.4 again imply that
where δ > 0 is a small constant provided that η > 0 is taken small enough. For k = 1, we use Strichartz estimates together with (7.21) to get By Duhamel's formula, Strichartz estimates and the triangle inequality, we have By Duhamel's formula, we see that The continuity argument yields u Y s (J1) ≤ 2η, (7.23) provided that η and M are chosen sufficiently small. Similarly, .
Choosing M sufficiently small, we get We also have from Strichartz estimates, (7.22), (7.23) and (7.24) that provided that δ > 0 is chosen sufficiently small. The same argument applies for the next spacetime slab J 2 × R d . By induction, we obtain u Y s (J k ) ≤ 2η, ∀k = 0, · · · , K − 1. Summing these bounds over all subintervals J k , we get u Y s (R + ) ≤ C(E). By Strichartz estimates and Lemma 7.4, we have This shows (7.3).
7.3. Global Strichartz bounds imply scattering. In this subsection, we will use the global Strichartz bounds (7.3) to show the scattering. We first show that e it(−∆) s u(t) has a limit in H s as t → +∞. To see this, let 0 < t 1 < t 2 . By Strichartz estimates and Lemma 7.4,

VAN DUONG DINH AND BINHUA FENG
The global Strichartz bounds (7.3) implies that as t 1 , t 2 → +∞. This implies that the limit exists in H s . Moreover, By the same argument as above, we get This completes the proof of Theorem 7.1.
8. Blow-up criteria. In this section, we show some criteria for the existence of blow-up H s andḢ sc ∩Ḣ s solutions for (1.2).

H s blow-up criteria.
By the global well-posedness given in Section 6, the existence of blow-up H s solutions may only occur for µ 2 < 0 and 4s d ≤ α 2 < 4s d−2s . When 4s d < α 2 < 4s d−2s , the second author in [23] has established some sufficient conditions about existence of blow-up solutions, and derived some sharp thresholds of blow-up and global existence. Here, we investigate the sharp threshold mass of blow-up and global existence for (1.2) with L 2 -critical and L 2 -subcritical nonlinearities, i.e., 0 < α 1 < α 2 = 4s d . Theorem 8.1 (H s sharp global existence and blow-up criteria). Let d ≥ 2, d 2d−1 ≤ s < 1, µ 2 < 0, µ 1 ∈ R and 0 < α 1 < α 2 = 4s d . Let u 0 ∈ H s be radial and u ∈ C([0, T ), H s ) be the corresponding solution to (1.2). Let Q 2 is the unique (up to symmetries) positive radial solution to the elliptic equation (2.11) with α = α 2 . Then we have the following sharp criteria for global existence and blow-up of (1.2).
2. If u 0 (x) = cλ Here we extend it to µ 1 ∈ R and give a correct proof.
Proof of Theorem 8.1. (1) We consider separately two cases µ 1 > 0 and µ 1 < 0. In the first case, without loss of generality we take µ 1 = 1 and µ 2 = −1. By the sharp Gagliardo-Nirenberg inequality and the conservation of mass, we have By the assumption u 0 L 2 < Q L 2 and the conservation of energy, we see that u(t) Ḣs is bounded from above for all t ∈ [0, T ). The blow-up alternative implies the solution exists globally in time, i.e. T = +∞. In the second case, we assume µ 1 = µ 2 = −1. By the sharp Gagliardo-Nirenberg inequality and the conservation of mass, We next apply the Young inequality to have for any η > 0 small enough, Taking η > 0 small enough depending on u 0 L 2 (for instance η = c 2 u 0 dα 1 for some 0 < c 1. The conservation of energy and (8.2) imply This shows that u(t) Ḣs is bounded from above for all t ∈ [0, T ). Therefore the solution exists globally in time. This completes the proof of (1).
In the case µ 1 = 1, we obviously have for all t ∈ [0, T ). If T < +∞, then the proof is done. Otherwise, we can take T = +∞. By (8.4), we infer that for all t ≥ t 1 with some sufficiently large time t 1 > 0 and some constant c > 0 depending on s and E(u 0 ). Moreover, we have from Lemma 4.1 that Here we use the conservation of mass and the interpolation inequality |∇| for all t ≥ t * with some constants C > 0 and t * > 0 that depend only on u 0 , s and d.
In the case µ 1 = −1, we have from (8.4) that If T < +∞, then the proof is done. Otherwise, T = +∞ and we will show (8.1). Indeed, if (8.1) does not hold, then there exists C > 0 such that Interpolating between L 2 and L 2d d−2s and using the Sobolev embedding, we have .
Note that the constant C may change from lines to lines. By (8.3) and (8.7), we get .
If we choose λ > 0 such that for all t ∈ [0, +∞) with some constant υ > 0. Arguing as in the previous case, we find for all t ≥ t * with some constant C > 0 and t * > 0. This is a contradiction to (8.8).
The proof is complete.
Proof. Let us start with the following reduction: , we see that d dt M ϕ R (u(t)) ≤ −C for some C > 0. Integrating this bound, we have that M ϕ R (u(t)) < 0 for all t ≥ t 1 with some t 1 1 large enough. Taking integration over [t 1 , t] of (8.9), we obtain for all t ≥ t 1 . On the other hand, by (4.15) and the assumption (4.13), By (8.10) and (8.12), we see that We thus get from (8.11) and (8.13) that for all t ≥ t 1 . By nonlinear integral inequality, it yields that M ϕ R (u(t)) −C(ϕ R )| t − t * | 1−2s for some finite t * < +∞. This shows that M ϕ R (u(t)) → −∞ as t ↑ t * . Therefore the solution cannot exist for all time t ≥ 0 and consequencely we must have T < +∞. We now prove (8.9) and (8.10) under the hypotheses of Proposition 8.2. The second condition (8.10) follows easily from the fact E(u 0 ) < 0. In fact, suppose it is not true. Then there exists a sequence (t k ) k ⊂ [0, +∞) such that u(t k ) Ḣs → 0 as k → ∞. Thanks to the Gagliardo-Nirenberg inequality given in Lemma 2.4 and the assumption (4.13), we see that u(t k ) αj +2 L α j +2 → 0 as k → ∞. We thus get E(u(t k )) → 0, which is a contradiction to E(u(t k )) = E(u 0 ) < 0. Let us show (8.9).
We end this section by giving some criteria for global existence ofḢ sc ∩Ḣ s solutions to (1.2). where S gs is given in (2.19), then the solution exists globally in time, i.e. T = +∞.
Proof. We have In the case µ 1 > 0 and µ 2 < 0 (WLG we assume µ 1 = 1 and µ 2 = −1), we use the sharp Gagliardo-Nirenberg inequality (2.21) to have Thanks to the conservation of energy and the assumption (8.14), we obtain u(t) Ḣs < ∞ for all t ∈ [0, T ). The blow-up alternative implies the solution exists globally in time.
Let u 0 ∈Ḣ sc ∩Ḣ s be radial and the corresponding solution u to (1.2) defined on the maximal time interval [0, T ). If where S gs and L gs are given in (2.19) and (2.20) respectively, then the solution exists globally in time, i.e. T = +∞.
Proof. The proof is similar to the one of Lemma 8.3 by using (2.22). We omit the details.
Proof. The proof is again similar to the one of Lemma 8.3. We omit the details. 9. Blow-up dymanics.
9.1. Blow-up dynamics in the mass-critical case. In this subsection, we study dynamical properties of blow-up H s solutions for (1.2) with µ 1 ∈ R, µ 2 < 0 and 0 < α 1 < α 2 = 4s d . Note that in this setting, the existence of blow-up H s solutions has been established in Theorem 8.1. Using the compactness lemma given in Lemma 2.6, we obtain the following L 2 -concentration of blow-up H s solutions to (1.2).
d and u 0 ∈ H s be radial. Assume that the corresponding solution u to (1.2) blows up in finite time 0 < T < +∞. Let a be a real valued non-negative function defined on [0, T ) satisfying where Q is the unique (up to symmetries) positive radial solution to (2.11) with α = 4s d . Remark 8.
• This result shows that the L 2 -norm of blow-up solutions must concentrate by an amount which is bounded from below by Q L 2 at the blow-up time. Moreover, as mentioned in [23], the rate of L 2 -concentration of blow-up solutions is u(t) s . Proof of Theorem 9.1. Without loss of generality, we may assume µ 1 ∈ {±1} and µ 2 = −1. Let (t n ) n≥1 be a sequence such that t n ↑ T . Set λ n := Q Ḣs u(t n ) Ḣs  By the blow-up alternative, we have u(t n ) Ḣs → ∞ as n → ∞, so λ n → 0 as n → ∞. Moreover, On the other hand,

VAN DUONG DINH AND BINHUA FENG
Applying the sharp Gagliardo-Nirenberg inequality and using the conservation of mass, we obtain Since dα1 2s < 2 and u(t n ) Ḣs → ∞ as n → ∞, we learn that |H(v n )| → 0 as n → ∞. From this and the fact v n Ḣs = Q Ḣs , we have as n → ∞. The sequence (v n ) n≥1 satisfies the assumptions of Lemma 2.6 with Thus, there exists a sequence (x n ) n≥1 in R d such that up to a subsequence, We thus have for every R > 0, By a change of variables, By (9.1), we also have as n → ∞. We thus get for every R > 0, which implies that Since (t n ) n≥1 is arbitrary, we infer that Observe that for every t ∈ [0, T ), the function y → |x−y|≤a(t) |u(t, x)| 2 dx is continuous and tends to zero as |y| tends to infinity. Therefore, there exists a function This combined with (9.6) show (9.2). The proof is complete. In order to study the limiting profile of blow-up H s solutions with minimal mass Q L 2 , we need the following characterization of the ground state.
Lemma 9.2 (Characterization of ground state [20]). Let d ≥ 1 and 0 < s < 1. If u ∈ H s is such that u L 2 = Q L 2 and where Q is the unique (up to symmetries) positive radial solution to (2.11) with α = 4s d . Assume that the corresponding solution u to (1.2) blows up in finite time 0 < T < +∞. Then there exist θ(t) ∈ R d , λ(t) > 0 and x(t) ∈ R d such that e iθ(t) λ d 2 (t)u(t, λ(t) · +x(t)) → Q strongly in H s , as t ↑ T .
Proof. We use the notations given in the proof of Theorem 9.1. We see that v(· + x n ) = λ d 2 n u(t n , λ n · +x n ) V weakly in H s , as n → ∞ with V L 2 ≥ Q L 2 . By the semi-continuity of weak convergence and (9.3), we have This shows that In particular, v n (· + x n ) → V strongly in L 2 as n → ∞. On the other hand, by the Gagliardo-Nirenberg inequality, we have v n (· + x n ) → V strongly in L 4s d +2 , as n → ∞. Indeed, by (9.4), This shows that there exists V ∈ H s such that The characterization of ground state given in Lemma 9.2 implies V (x) = e iθ λ d 2 Q(λx + x 0 ) for some θ ∈ R, λ > 0 and x 0 ∈ R d . We thus conclude that v n (· + x n ) = λ d 2 n u(t n , λ n · +x n ) → V = e iθ λ d 2 Q(λ · +x 0 ) strongly in H s , as n → ∞. Redefining variablesθ n := −θ,λ n := λ n λ −1 andx n := λ n λ −1 x 0 + x n , we obtain e iθnλ d 2 n u(t n ,λ n · +x n ) → Q strongly in H s , as n → ∞. Since (t n ) n≥1 is arbitrary, we infer that there exist θ(t) ∈ R, λ(t) > 0 and x(t) ∈ R d such that e iθ(t) λ d 2 (t)u(t, λ(t) · +x(t)) → Q strongly in H s , as t ↑ T . The proof is complete. 9.2. Blow-up dynamics in the mass-supercritical case. Theorem 9.4 (Blow-up concentration). Let d ≥ 2, d 2d−1 ≤ s < 1 and µ 1 ∈ R, µ 2 < 0, 4s d < α 2 < 4s d−2s , dα2−4s 2s ≤ α 1 < α 2 and u 0 ∈Ḣ sc ∩Ḣ s be radial. Assume that the corresponding solution u to (1.2) blows up in finite time 0 < T < +∞ and satisfies (4.13). Let a be a real valued non-negative function defined on [0, T ) satisfying a(t) u(t) |u(t, z)| αc dz ≥ L αc gs . (9.10) Remark 9. In [22], the second author proved a similar result for the nonlinear Schrödinger equation with combined power-type nonlinearities. Here we extend his result in the context of the fractional nonlinear Schrödinger equation. Note that since the uniqueness (up to symmetries) of solutions to (2.14) and (2.17) are not yet known, we need to introduce the notions of Sobolev and Lebesgue ground states (see Definition 2.5).

Assume that
Proof. We only treat the first term, the second one is similar. It is enough to show that for any (t n ) n≥1 satisfying t n ↑ T , there exist a subsequence still denoted by (t n ) n≥1 ,g ∈ G, sequences θ n ∈ R, λ n > 0 and y n ∈ R d such that e itθn λ 2s α 2 n u(t n , λ n · +y n ) →g strongly inḢ sc ∩Ḣ s , (9.17) as n → ∞. Using the notation given in the proof of Proposition 9.4, we have v n (· + y n ) = λ 2s α 2 n u(t n , λ n · +y n ) P weakly inḢ sc ∩Ḣ s , as n → ∞ with P Ḣsc ≥ S gs . By the semi-continuity of weak convergence, (9.11) and (9.15), S gs ≤ P Ḣsc ≤ lim inf n→∞ v n Ḣsc ≤ S gs .