On the blow-up solutions for the fractional nonlinear Schr\"odinger equation with combined power-type nonlinearities

This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schr\"odinger equation with combined power-type nonlinearities \[ i\partial_t u-(-\Delta)^su+\lambda_1|u|^{2p_1}u+\lambda_2|u|^{2p_2}u=0, \] where $0<p_1<p_2<\frac{2s}{N-2s}$. Firstly, we obtain some sufficient conditions about existence of blow-up solutions, and then derive some sharp thresholds of blow-up and global existence by constructing some new estimates. Moreover, we find the sharp threshold mass of blow-up and global existence in the case $0<p_1<\frac{2s}{N}$ and $p_2=\frac{2s}{N}$. Finally, we investigate the dynamical properties of blow-up solutions, including $L^2$-concentration, blow-up rate and limiting profile.


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 [22,23] 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, f (u) = |u| 2p u. The fractional differential operator (−∆) s is defined by Recently, equation (1.1) has attracted more and more attentions in both the physics and mathematics fields, see [3,4,5,6,7,12,13,17,19,36,38,40]. For the Hartree-type nonlinearity (|x| −γ * |u| 2 )u, Cho et al. in [3] proved existence and uniqueness of local and global solutions of (1.1). They also showed the existence of blow-up solutions in [6]. The dynamical properties of blow-up solutions have been investigated in [5,38]. Zhang and Zhu in [36] studied the stability and instability of standing waves. For the local nonlinearity |u| 2p u, the well-posedness and illposedness in the Sobolev space H s have been investigated in [7,19]. In [1], Boulenger et al. have obtained a general criterion for blow-up of radial solution of (1.1) with p ≥ 2s N in R N with N ≥ 2. Although a general existence theorem for blow-up solutions of this problem has remained an open problem, it has been strongly supported by numerical evidence [20]. The orbitally stability of standing waves for other kinds of fractional Schrödinger equations has been studied in [12,13,4,40].
In this paper, we consider the following fractional nonlinear Schrödinger equation with combined power-type nonlinearities    i∂ t u − (−∆) s u + λ 1 |u| 2p 1 u + λ 2 |u| 2p 2 u = 0, u(0, x) = u 0 (x), (1.2) where u = u(t, x) : [0, T * ) × R N → C is a complex valued function, 0 < s < 1, λ 1 , λ 2 ∈ R, 0 < p 1 < p 2 < 2s N −2s . This equation has Hamiltonian Because of important applications in physics, nonlinear Schrödinger equations received a great deal of attention from mathematicians in the past decades, see [2,30,31] for a review. Ginibre and Velo [15] established the local well-posedness of (1.4) in H 1 ( see [2] for a review). When λ 1 < 0 and 2 N ≤ p 1 ≤ 2 N −2 , Glassey [16] proved the existence of blow-up solutions for the negative energy and |x|u 0 ∈ L 2 . Ogawa and Tsutsumi [29] proved the existence of blow-up solutions in radial case without the restriction |x|u 0 ∈ L 2 . A natural question appears for p 1 ≥ 2 N : can one find some sharp criteria for blow-up and global existence of (1.4)? Weinstein [33] gave a crucial criterion in terms of L 2 -mass initial data. Also, some sharp criteria in terms of the energy of the initial data were obtained (see [24,35]). Cazenave also mentioned this topic in their monographs [2]. From the view point of physics, this problem is also pursued strongly (see [21] and the references therein). In addition, for the L 2 -critical nonlinearity, i.e., [34] studied the structure and formation of singularity of blow-up solutions with critical mass by the concentration compact principle: the blow-up solution is close to the ground state in H 1 up to scaling and phase parameters, and also translation in the non-radial case. Applying the variational methods, Merle and Raphaël [26] improved Weinstein's results and obtained the sharp decomposition of blow-up solutions with small super-critical mass. By this sharp decomposition and spectral properties, Merle and Raphaël [25,26,27,28] obtained a large body of breakthrough works, such as sharp blow-up rates, profiles, etc. Hmidi and Keraani [18] established the profile decomposition of bounded sequences in H 1 and gave a new and simple proof of some dynamical properties of blow-up solutions in H 1 . These results have been generalized to other kinds of Schrödinger equations, see [10,11,14,24,37,38,39].
In [32], Tao et al. undertook a comprehensive study for the following nonlinear Schrödinger equation with combined power-type nonlinearities More precisely, they addressed questions related to local and global well-posedness, finite time blow-up, and asymptotic behaviour. Recently, in [9], we prove the existence of blow-up solutions and find the sharp threshold mass of blow-up and global existence for (1.5) with p 1 = 2 N and 0 < p 2 < 2 N , which is a complement to the result in [32]. As far as we know, the existence of blow-up solutions of (1.2) has not been proved yet. In particular, the dynamical properties of blow-up solutions have not been proved even when λ 1 = 0.
In this paper, we will focus on the blow-up solutions of (1.2). More precisely, we are interested in sufficient conditions about the existence of blow-up solutions, sharp thresholds of blow-up and global existence, the dynamical properties of blow-up solutions, including L 2 -concentration, blow-up rates, and limiting profile.
To solve these problems, we mainly use the ideas from Boulenger et al. [1] and Keraani [18].
The existence of blow-up solutions for the fractional nonlinear Schrödinger equation (1.1) with the local nonlinearity |u| 2p u has been investigated in [1]. The dynamical properties of blow-up solutions for the L 2 -critical nonlinear Schrödinger equation (1.4) have been discussed in [18]. In these papers, the study of blow-up solutions relies heavily on the scaling invariance of (1.1) and (1.4). Hence, the study of blow-up solutions for (1.2), which has no the scaling invariance, is of particular interest.
Firstly, we will investigate sufficient conditions about the existence of blow-up solutions for . But there is no scaling invariance for equation (1.2). Therefore, we must construct some new estimates to obtain some sharp thresholds of blow-up and global existence.
When 0 < p 1 < 2s N and p 2 = 2s N , by using the scaling argument and the variational characteristic provided by the sharp Gagliardo-Nirenberg inequality (2.1), we find the sharp threshold mass Q L 2 of blow-up and global existence for (1.2) in the following sense, where Q is the ground state solution of (2.2) with p = 2s N . (i) If u 0 L 2 < Q L 2 , then the solution of (1.2) exists globally in H s .
(ii) If u 0 L 2 ≥ Q L 2 , we can construct a class of initial data, and the corresponding solution u(t) of (1.2) must blow up.
Finally, in order to overcome the loss of scaling invariance, we use the ground state solution Q of (2.2) to describe the dynamical behaviour of the blow-up solutions to (1.2) with 0 < p 1 < 2s N and p 2 = 2s N , including L 2 -concentration, blow-up rates, and limiting profile. Our method can be easily applied to study the dynamical behaviour of the blow-up solutions to (1.2) with λ 1 = 0 and p 2 = 2s N . Our results are new even for (1.2) with λ 1 = 0 and p 2 = 2s N . This paper is organized as follows: in Section 2, we present some preliminaries. In section 3, we will establish some sufficient conditions of the existence of blow-up solutions for (1.2), and then obtain some sharp thresholds of blow-up and global existence. Moreover, we find the sharp threshold mass of blow-up and global existence for (1.2). In section 4, we will consider some dynamical properties of blow-up solutions of (1.2) with p 2 = 2s N and 0 < p 1 < 2s N , including L 2 -concentration, blow-up rate, and limiting profile.
Notation. Throughout this paper, we use the following notation. C > 0 will stand for a constant that may be different from line to line when it does not cause any confusion. We often abbreviate L q (R N ), · L q (R N ) and H s (R N ) by L q , · L q and H s , respectively.

Preliminaries
Firstly, by a similar argument as that in [7,19], we can establish the local theory for the Cauchy problem (1.2), see also [40].
Moreover, for all 0 ≤ t < T * , the solution u(t) satisfies the following conservation of mass and energy where E(u(t)) defined by (1.3).
Next, we recall a sharp Gagliardo-Nirenberg type inequality established in [1,40]. Lemma 2.2. Let N ≥ 2, 0 < s < 1 and 0 < p < 2s N −2s . Then, for all u ∈ H s , where the optimal constant C opt given by and Q is a ground state solution of In particular, in the L 2 -critical case p = 2s Moreover, the solution Q satisfies the following relations Next, we shall recall the profile decomposition of bounded sequences in H s , which is important to study the dynamical properties of blow-up solutions, see [40].
(ii) for every l ≥ 1 and every Remark. In this proposition, the number of non-zero terms in the right side of (2.5) may be one, finite and infinite, which may correspond to three possibilities (compactness, dichotomy and vanishing) in the concentration compactness principle proposed by Lions. Hence, the profile decomposition may look as another equivalent description of the concentration compactness principle. However, there are two major advantages of the profile decomposition of bounded sequences in H s : one is that the decomposing expression of the bounded sequence {v n } ∞ n=1 is given and we can inject it into our aim functionals, and the other is that the decomposition is orthogonal by (i) and norms of {v n } ∞ n=1 have similar decompositions, for example (2.6). Those properties are useful in the calculus of variational methods.
In this paper, we will use the method in [1] to prove the existence of blow-up solutions to (1.2). In the following, we recall some important results in [1].
with some constant C > 0 that depends only on ∇ϕ W 1,∞ and N .
Let us assume that ϕ : R N → R is a real-valued function with ∇ϕ ∈ W 3,∞ (R). We define the localized virial of u = u(t, x) to be the quantity given by By applying Lemma 2.4, we obtain the bound Hence the quantity M ϕ [u(t)] is well-defined, since u(t) ∈ H s (R N ) with some s ≥ 1 2 by assumption.
To study the time evolution of M ϕ [u(t)], we shall need the following auxiliary function where the constant c s := sin πs π turns out to be a convenient normalization factor. By the smoothing properties of (−∆ + m) −1 , By a similar argument as that in [1], we have the following time evolution of M ϕ [u(t)].
Lemma 2.6. For any t ∈ [0, T * ), we have the identity Let ϕ : R N → R be as above. In addition, we assume that ϕ = ϕ(r) is radial and satisfies const. for r ≥ 10, We readily verify the inequalities By a similar argument as Lemma 2.2 in [1], we obtain the following time evolution of the localized virial M ϕ R [u(t)] with ϕ R as above.
Lemma 2.7. (Localized radial virial estimate) Let N ≥ 2, s ∈ ( 1 2 , 1) and assume in addition that u(t) is a radial solution of (1.2). We then have In order to deal with the L 2 -critical case, we shall need the following refined version of Lemma 2.7 involving the nonnegative radial functions and assume in addition that u(t) is a radial solution of (1.2) for any t ∈ [0, T * ) and p 2 = 2s N . We then have

The existence of blow-up solutions
In this section, 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. Moreover, we find the sharp threshold mass of blow-up and global existence for (1.2). Firstly, we will prove the existence of blow-up solutions of (1.2).

2). Then the solution u(t) blows up in finite time
in the sense that T * < ∞ must hold in each of the following three cases: Proof. In what follows, we will show that the first derivative of M ϕ [u(t)] is negative for positive times t. More precisely, in each of the three cases described in Theorem 3.1, we will show that for a small positive constant c. This implies that the solution u(t) blows up in finite time.
Indeed, suppose that u(t) exists for all times t ≥ 0, i.e., we can take T * = ∞.
Firstly, we claim the lower bound Indeed, if this conclusion does not hold, then there exists some sequence of time t k ∈ [0, ∞) However, by L 2 -mass conservation and the sharp Gagliardo- Thus, we deduce that (3.2) holds.
Next, it follows from (3.1) and On the other hand, we use Lemma 2.4 and L 2 -mass conservation to find that where we used the interpolation estimate |∇| This, together with (3.3), implies that This yields M ϕ [u(t)] ≤ −C(ϕ R )|t − t * | 1−2s for s > 1 2 with some t * < +∞. Therefore, we have M ϕ [u(t)] → −∞ as t → t * . hence the solution u(t) cannot exist for all time t ≥ 0 and consequently we must have that T * < +∞ holds.
For the remainder of the proof, we will derive (3.1) in each of the three cases described in Theorem 3.1.
In this case, by a similar argument as (3.7), we obtain provided that R ≫ 1 is taken sufficiently large. This implies (3.1) with c = p 2 N − 2s.
If this conclusion does not hold, by the continuity of (−∆) Thus the contradiction has been produced, the solution u(t) of (1.2) exists globally.
On the other hand, if (−∆) s 2 u 0 L 2 > y 0 , by the same argument, it follows that (−∆) Next, we pick η > 0 sufficiently small such that Thus, by the conservation of energy, (2.11) and (2.1), we deduce that where δ = 2(p 2 N − 2s) and we have chosen ε 1 and ε 2 small enough such that p 1 s + ε 1 < 2 and p 2 s + ε 2 < 2. We thus conclude < 0 for all t ≥ t 1 with some sufficiently large time t 1 ≫ 1. Hence, by By following exactly the steps after (3.3) above, we deduce that u(t) cannot exist for all times t ≥ 0 and consequently we must have that T * < ∞ holds.
Case 2): We define a function g(y) on [0, ∞) by Thus, (3.11) can be expressed by E(u(t)) ≥ g( (−∆) s 2 u(t) L 2 ), g(y) is continuous on [0, ∞) and For the equation f (y) = 0, there is a unique positive solution y 1 . Indeed, by assumption 2s N < p 1 < p 2 < 2s N −2s , for y > 0, we have 20) which implies that f (y) is decreasing on [0, ∞). Due to f (0) = 1, there exists a unique y 1 > 0 such that f (y 1 ) = 0. This implies On the other hand, we deduce from the conservation of energy and the assumption E(u 0 ) < By the same argument as Case 1), we can obtain that if (−∆) Next, we pick η > 0 sufficiently small such that Inserting this bound into the differential inequality (2.11), we obtain with δ = p 1 N − 2s and • R (1) → 0 as R → ∞ uniformly in t. We thus conclude Therefore, by the same argument as Case 1), we can obtain the desired result.
When 0 < p 1 < 2s N and p 2 = 2s N , the existence of blow-up solutions of (1.2) has not been proved yet. In the following, by using the scaling argument and the variational characteristic provided by the sharp Gagliardo-Nirenberg inequality (2.1), we prove the existence of blow-up solutions for (1.2) and find the sharp threshold mass of blow-up and global existence for (1.2).
N . Then, we have the following sharp threshold mass of blow-up and global existence.
with some constants C > 0 and t * > 0 that depend only on u 0 , s, N .
Remark. As far as we know, this result has not been proved when λ 1 = 0. However, our method can be easily applied to the case of λ 1 = 0. Therefore, this result is new even for (1.2) with λ 1 = 0.
Proof. (i) We deduce from the energy conservation (1.3) and the sharp Gagliardo-Nirenberg inequality(2.1) that for all t ∈ [0, T * ) From the hypothesis u 0 L 2 < Q L 2 , there exists a constant C > 0 such that E(u 0 ) = E(u(t)) ≥ C (−∆) s/2 u(t) 2 L 2 for all t ∈ [0, T * ). Then, u(t) is bounded in H s for all t ∈ [0, T * ) by the conservation of mass, and u(t) exists globally in H s by the local well-posedness (see Proposition 2.1). This completes the proof of (i).
(ii) By the definition of initial data u 0 (x) = cρ N 2 Q(ρx) and the Pohozaev identity for equation Now, taking ρ such that This implies E(u 0 ) < 0.
On the other hand, by a similar argument in [1], we can choose ϕ R (r) and η > 0 sufficiently small such that N 2s ≥ 0 f or all r > 0 and R > 0.
Thus if we choose η ≪ 1 sufficiently small and then R ≫ 1 sufficiently large, we can apply Next, we suppose that u(t) exists for all time t ≥ 0, i.e., T * = ∞. It follows from (3.24) that with some sufficiently large time t 0 > 0 and some constant c > 0 depending only on s and E(u 0 ) < 0. On the other hand, if we invoke Lemma 2.4, we see that where we also used the conservation of L 2 -mass together with the interpolation estimate |∇| Then, there exist V ∈ H s and {x n } ∞ n=1 ⊂ R N such that, up to a subsequence, where Q is the ground state solution of (2.2) with p = 2s N .
Proof. We deduce from the profile decomposition (Proposition 2.3) that with lim sup n→∞ v l n L q → 0 as l → ∞.
From (4.1), (2.1) and Proposition 2.3, we obtain On the other hand, we observe that Therefore, it follows from (4.2) and (4.3) that Since the series ∞ j=1 V j 2 L 2 is convergent, there exists j 0 ≥ 1 such that From (2.5), a change of variables x = x + x j 0 n gives Using the pairwise orthogonality of {x j n } ∞ j=1 , we have weakly in H s f or every j = j 0 .
Hence, we have u n (· + x j 0 n ) ⇀ V j 0 +ṽ l , weakly in H s . whereṽ l denote the weak limit of v l n (x + x j 0 n ). However, Thus, it follows from uniqueness of weak limit thatṽ l = 0 for all l ≥ J 0 . Therefore, This completes the proof.
By applying the refined compactness Lemma 4.1, we can obtain the following L 2 -concentration and rate of L 2 -concentration of blow-up solutions of (1.2).
where Q is the ground state solution of (2.2) with p = 2s N .
Remark. if u is a blow-up solution of (1.2) and T * its blow-up time, then for every r > 0, there exists a function x(t) ∈ R N such that lim inf Meanwhile, it follows from the choice of a(t) that for any function 0 < a(t) ≤ (4.4) holds, which implies that the rate of L 2 -concentration of blow-up solutions of (1.2) is Proof. Set ρ s (t) = (−∆) s/2 Q L 2 / (−∆) s/2 u(t) L 2 and v(t, x) = ρ N 2 (t)u(t, ρ(t)x).
Let {t n } ∞ n=1 be an any time sequence such that t n → T * , ρ n := ρ(t n ) and v n (x) := v(t n , x).
Then, the sequence {v n } satisfies Observe that Applying the following Gagliardo-Nirenberg inequality Then it follows from Lemma 4.1 that there exist V ∈ H s and {x n } ∞ n=1 ⊂ R N such that, up to a subsequence, v n (· + x n ) = ρ N/2 n u(t n , ρ n (· + x n )) ⇀ V weakly in H s (4.7) with Note that a(t n ) ρ n = a(t n ) (−∆) s/2 u(t n ) Then for every r > 0, there exists n 0 > 0 such that for every n > n 0 , rρ n < a(t n ). Therefore, using (4.7), we obtain Since the sequence {t n } ∞ n=1 is arbitrary, it follows from (4.8) that Observe that for every t ∈ [0, T * ), the function g(y) := |x−y|≤a(t) |u(t, x)| 2 dx is continuous on y ∈ R N and g(y) → 0 as |y| → ∞. So there exists a function x(t) ∈ R N such that for every This and (4.9) yield (4.4).
In the following theorem, we study the limiting profile of blow-up solutions of (1.2).
Next, we will prove that v n (· + x n ) converges to V strongly in H s . For this aim, we estimate as follows: Using the inequality (2.1), we infer that On the other hand, we deduce from (4.5) that (−∆) s/2 V L 2 ≤ lim inf n→∞ (−∆) s/2 v n (· + x n ) L 2 = (−∆) s/2 Q L 2 . Hence, we have Q H s = V H s and v n (· + x n ) → V strongly in H s as n → ∞. (4.14) This and (4.12) imply that Up to now, we have verified that V L 2 = Q L 2 , (−∆) s/2 V L 2 = (−∆) s/2 Q L 2 and H(V ) = 0.
The variational characterization of the ground state implies that there exist x 0 ∈ R N and θ ∈ [0, 2π) such that V (x) = e iθ Q(x + x 0 ), and ρ N/2 n u(t n , ρ n (· + x n )) → e iθ Q(· + x 0 ) strongly in H s as n → ∞.