Sharp Criteria of Scattering for the Fractional NLS
aa r X i v : . [ m a t h . A P ] J un SHARP CRITERIA OF SCATTERING FOR THE FRACTIONAL NLS
QING GUO AND SHIHUI ZHU
Abstract.
In this paper, the sharp threshold of scattering for the fractional nonlinearSchr¨odinger equation in the L -supercritical case is obtained, i.e., if 1+ sN < p < sN − s ,and M [ u ] s − scsc E [ u ] < M [ Q ] s − scsc E [ Q ] , M [ u ] s − scsc k u k H s < M [ Q ] s − scsc k Q k H s then the solution u ( t ) is globally well-posed and scatters. This condition is sharp in thesense that if 1 + sN < p < sN − s and M [ u ] s − scsc E [ u ] < M [ Q ] s − scsc E [ Q ] , M [ u ] s − scsc k u k H s > M [ Q ] s − scsc k Q k H s , then the corresponding solution u ( t ) blows up in finite time, according to Boulenger,Himmelsbach, and Lenzmann’s results in [2]. MSC: 35Q55, 47J30Keywords: Fractional Schr¨odinger equation; Power-type nonlinearity; L -supercritical;Scattering. 1. Introduction
From expanding the Feynman path integral from the Brownian-like to the L´evy-likequantum mechanical paths, Laskin in [28, 29] established the fractional Schr¨odinger equa-tions from the viewpoint of Physics, which have physical applications in the energy spec-trum for a hydrogen-like atom-fractional Bohr atom. The studying of the fractional non-linear Schr¨odinger equations (fractional NLS, for short) attacking more and more Mathe-matical researchers (see[1, 2, 7, 10, 12, 13, 14, 21, 24, 31, 33]). In the present paper, weinvestigate the following Cauchy problem of the L -supercritical fractional NLS. iu t − ( −△ ) s u + | u | p − u = 0 , (1.1) u (0 , x ) = u ∈ H s , (1.2)where 0 < s < −△ ) s is defined by( −△ ) s u = 1(2 π ) N Z e ix · ξ | ξ | s b u ( ξ ) dξ = F − [ | ξ | s F [ u ]( ξ )] , where F and F − are the Fourier transform and the Fourier inverse transform in R N ,respectively. u = u ( t, x ): R × R N → C is the wave function. The power exponent1 + sN < p < sN − s (when N ≤ s , 1 + sN < p < ∞ ).When 1 + sN < p < sN − s for N > s , and 1 + sN < p < ∞ for N ≤ s , Eq. (1.1)is the L -supercritical fractional NLS due to the scaling invariance. Indeed, if u ( t, x ) is asolution of Eq.(1.1), then u λ ( t, x ) = λ sp − u ( λ s t, λx ) is also a solution of Eq.(1.1). Then,we see the following invariant norms.(1) k u λ k L pc = k u k L pc , where p c = N ( p − s . We remark that p c > p − > sN , andthen Eq. (1.1) is called the L -supercritical NLS.(2) ˙ H s c -norm is invariant for Eq. (1.1), i.e., k u λ k ˙ H sc = k u k ˙ H sc , where s c = N − sp − .Recently, the Cauchy problem (1.1)-(1.2) has been widely studied in the recent yearsbut is not completely settled yet, see, e.g. [8] and [22]. Let N ≥ ≤ s < sN < p < sN − s . If u ∈ H s , then the Cauchy problem (1.1)-(1.2) has a uniquesolution u ( t, x ) on I = [0 , T ) satisfying u ( t, x ) ∈ C ( I ; H s ) T C ( I ; H − s ). Moreover, either T = + ∞ (global existence) or both 0 < T < + ∞ and lim t → T k u ( t, x ) k H s = + ∞ (blow-up).Furthermore, ∀ t ∈ I , u ( t, x ) has two important conservation laws.(i) Conservation of energy: E [ u ( t )] = 12 Z R N u ( −△ ) s udx − p + 1 Z R N | u ( x, t ) | p +1 dx = E [ u ] . (1.3)(ii) Conservation of mass: M [ u ( t )] = Z R N | u ( t, x ) | dx = M [ u ] . (1.4)Guided by a analogy to classical NLS, the sufficient criteria for blowup of the solution canbe found in [2] in terms of quantities of the ground states Q ∈ H s ( R N ), by solving( −△ ) s Q + Q − | Q | p − Q = 0 , Q ∈ H s ( R N ) (1.5)and the Gagliardo-Nirenberg inequality (see Theorem 3.2 in [36]) Z | v ( x ) | p +1 dx ≤ C GN k v k p +1 − N ( p − s k v k N ( p − s ˙ H s (1.6)with C GN = 2 s ( p + 1) N ( p −
1) 1 k Q k p +1 − N ( p − s k Q k N ( p − s − H s . (1.7)The blow-up and long-time dynamics of the fractional NLS turn out to be very interestingproblems. To the best of the authors’ knowledge, the cases that have been successfullyaddressed by now are: i) for the fractional NLS with nonlocal Hartree-type nonlinearites CATTERING FOR FNLS 3 and radial data, see, e.g. [7, 30]. Recently, Guo and Zhu [18] obtained a sharp threshold ofthe scattering versus blow-up for the focusing L -supercritical case. ii) for the power-typenonlinearities, Boulenger, Himmelsbach, Lenzmann [2] derived a general blowup result for(1.1) in both L -supercritical and L -critical cases respectively, subject to certain threshold.Recently, the authors in [20] performed Kenig-Merle type argument [26] to show the globalwell-posedness of radial solutions and scattering below sharp threshold of ground statesolutions. In [33], the authors adapt the strategy in [9] to prove a similar scattering resultfor the 3D radial focusing cubic fractional NLS, under the restriction that s ∈ ( , s ∈ ( N N − , L -supercritical NLS Eq. (1.1) in terms of the arguments in [15, 23, 26], as follows. Theorem 1.1.
Let N ≥ and sN < p < sN − s Suppose that u ∈ H s is radialand M [ u ] s − scsc E [ u ] < M [ Q ] s − scsc E [ Q ] , where Q is the ground-state solution of (1.5) . If N N − ≤ s < and M [ u ] s − scsc k u k H s < M [ Q ] s − scsc k Q k H s , then the corresponding solution u ( t ) of (1.1)-(1.2) exists globally in H s . Moreover, u ( t ) scatters in H s . Specifically, there exists φ ± ∈ H s such that lim t →±∞ k u ( t ) − e − it ( − ∆) s φ ± k H s = 0 . We should point out that the sharp criteria of scattering for the nonlinear Schr¨odingerequation is a quite important and interesting problems, and many researchers have devotedon this topics (see e.g. [4, 9, 15, 18, 23, 26, 33]). The scattering involves in the Strichartzestimates and the choice of admissible pairs, which is quite different and difficult withrespect to different nonlinearities. Although in [18], we have proved the scattering for thefractional Hartree equation in the L supercritical case, that for the fractional NLS (1.1)with power-type nonlinearity is a nontrivial extension(e.g. Proposition 2.6, Theorem 5.1).At the end of this section, we introduce some notations. L q := L q ( R N ), k·k q := k·k L q ( R N ) ,the time-space mixed norm k u k L q X := (cid:18)Z R k u ( t, · ) k qX dt (cid:19) q ,H s := H s ( R N ), ˙ H s := ˙ H s ( R N ), and R · dx := R R N · dx . F v = b v denotes the Fouriertransform of v , which for v ∈ L ( R N ) is given by F v = b v ( ξ ) := R e − ix · ξ v ( x ) dx for all ξ ∈ R N , and F − v is the inverse Fourier transform of v ( ξ ). ℜ z and ℑ z are the real andimaginary parts of the complex number z , respectively. z denotes the complex conjugateof the complex number z . The various positive constants will be denoted by C or c . QING GUO AND SHIHUI ZHU Local theory and Strichartz estimate
In fact, the Cauchy problem (1.1)-(1.2) has the following integral equation: u ( t ) = U ( t ) u + i Z t U ( t − t ) | u | p − u ( t ) dt where U ( t ) φ ( x ) = e − i ( −△ ) s t φ ( x ) = 1(2 π ) N Z e i ( x · ξ −| ξ | s ) b φ ( ξ ) dξ. First, we recall the local theory for Eq. (1.1) by the radial Strichartz estimate ([19]).
Definition 2.1.
For the given θ ∈ [0 , s ), we state that the pair ( q, r ) is θ -level admissible,denoted by ( q, r ) ∈ Λ θ , if q, r ≥ , sq + Nr = N − θ (2.1)and4 N + 22 N − ≤ q ≤ ∞ , q ≤ N −
12 ( 12 − r ) , or ≤ q < N + 22 N − , q < N −
12 ( 12 − r ) . (2.2)Correspondingly, we denote the dual θ -level admissible pair by ( q ′ , r ′ ) ∈ Λ ′ θ if ( q, r ) ∈ Λ − θ with ( q ′ , r ′ ) is the H¨older dual to ( q, r ) . Proposition 2.2. (see [19] ) Assume that N ≥ and that u , f are radial; then for q j , r j ≥ , j = 1 , , k U ( t ) φ k L q L r ≤ C k D θ φ k , (2.3) where D θ = ( −△ ) θ , k Z t U ( t − t ) f ( t ) dt k L q L r ≤ C k f k L q ′ L r ′ , (2.4) in which θ ∈ R , the pairs ( q j , r j ) satisfy the range conditions (2.2) and the gap condition sq + Nr = N − θ, sq + Nr = N θ. Definition 2.3.
We define the following Srichartz norm k u k S (Λ sc ) = sup ( q,r ) ∈ Λ sc k u k L q L r and the dual Strichartz norm k u k S ′ (Λ − sc ) = inf ( q ′ ,r ′ ) ∈ Λ ′ sc k u k L q ′ L r ′ = inf ( q,r ) ∈ Λ − sc k u k L q ′ L r ′ , where ( q ′ , r ′ ) is the H¨older dual to ( q, r ) . CATTERING FOR FNLS 5
Remark . Notice that if s ∈ [ N N − , ⊂ ( 12 , , the gap condition (2.1) with θ = 0 right implies the range condition (2.2), which furthermeans that Λ is nonempty. That is we have a full set of 0-level admissible Strichartzestimates without loss of derivatives in radial case. By taking q c = r c = ( p − N + 2 s )2 s , (2.5)we see that ( q c , r c ) ∈ Λ s c = ∅ is an s c -level admissible pair.When φ, f are radial, from Proposition 2.2, we have the following Strichartz estimates. k U ( t ) φ k S (Λ ) ≤ C k φ k and (cid:13)(cid:13)(cid:13)(cid:13)Z t U ( t − t ) f ( · , t ) dt (cid:13)(cid:13)(cid:13)(cid:13) S (Λ ) ≤ C k f k S ′ (Λ ) . Then, we further obtain k U ( t ) φ k S (Λ sc ) ≤ c k φ k ˙ H sc , (cid:13)(cid:13)(cid:13)(cid:13)Z t U ( t − t ) f ( · , t ) dt (cid:13)(cid:13)(cid:13)(cid:13) S (Λ sc ) ≤ C k D s c f k S ′ (Λ ) and (cid:13)(cid:13)(cid:13)(cid:13)Z t U ( t − t ) f ( · , t ) dt (cid:13)(cid:13)(cid:13)(cid:13) S (Λ sc ) ≤ C k f k S ′ (Λ − sc ) , where we use the Sobolev embedding.Next, we denote S (Λ θ ; I ) to indicate its restriction to a time subinterval I ⊂ ( −∞ , + ∞ ) . Proposition 2.5. (Small data) Let N ≥ and sN < p < sN − s . If k u k ˙ H sc ≤ A is radial, then, there exists δ sd = δ sd ( A ) > such that when k U ( t ) u k S (Λ sc ) ≤ δ sd , thecorresponding solution u = u ( t ) solving (1.1) is global, and k u k S (Λ sc ) ≤ k U ( t ) u k S (Λ sc ) , k D s c u k S (Λ ) ≤ c k u k ˙ H sc . (Note that by the Strichartz estimates, the hypotheses are satisfied if k u k ˙ H sc ≤ cδ sd . )Proof. Denote Φ u ( v ) = U ( t ) u + i Z t U ( t − t ) | v | p − v ( t ) dt . It follows from the Strichartz estimates that k D s c Φ u ( v ) k S (Λ ) ≤ c k u k ˙ H sc + c k D s c [ | v | p − v ] k L q ′ L r ′ QING GUO AND SHIHUI ZHU and k Φ u ( v ) k S (Λ sc ) ≤ k U ( t ) u k S (Λ sc ) + k| v | p − v k L q p L r p ≤ k U ( t ) u k S (Λ sc ) + k v k pL q L r , where ( q ′ , r ′ ) ∈ Λ ′ , ( q , r ) ∈ Λ s c and ( q p , r p ) ∈ Λ ′ s c . Then, by applying the fractionalLeibnitz [7, 25] , we deduce that k D s c [ | v | p − v ] k L q ′ L r ′ ≤ c k| u | p − k L q q ′ q − q ′ L r r ′ r − r ′ k D s c v k L q L r ≤ c k v k p − L q L r k D s c v k L q L r , where the pairs ( q, r ) , ( q , r ) ∈ Λ , ( q , r ) ∈ Λ s c . Now, we take δ sd ≤ (cid:18) min (cid:18) p c , p (cid:19)(cid:19) p − , and define B := (cid:8) v |k v k S (Λ sc ) ≤ k U ( t ) u k S (Λ sc ) , k D s c v k S (Λ ) ≤ c k u k ˙ H sc (cid:9) . Then, we can prove that Φ u is a contraction mapping from B to B , which completes theproof. (cid:3) Proposition 2.6. (Scattering criterion) Let N ≥ and sN < p < sN − s . If u ∈ H s is radial and u ( t ) is global with both bounded s c -level Strichartz norm k u k S (Λ sc ) < ∞ anduniformly bounded H s norm sup t ∈ [0 , ∞ ) k u k H s ≤ B, then u ( t ) scatters in H s as t → + ∞ . Moreprecisely, there exists φ + ∈ H s such that lim t → + ∞ k u ( t ) − U ( t ) φ + k H s = 0 . Proof.
It follows from the integral equation u ( t ) = U ( t ) u + i Z t U ( t − t ) | u | p − u ( t ) dt (2.6)that u ( t ) − U ( t ) φ + = − i Z ∞ t U ( t − t ) | u | p − u ( t ) dt , (2.7)where φ + = u + i Z ∞ U ( − t ) | u | p − u ( t ) dt . CATTERING FOR FNLS 7
Applying Proposition 2.2, we deduce that for 0 ≤ α ≤ s , there exist some ( q, r ) ∈ Λ ,( q , r ) ∈ Λ ′ such that (cid:13)(cid:13)(cid:13)(cid:13) D α (cid:18)Z I U ( t − s ) (cid:0) | u | p − u ( s, x ) (cid:1) ds (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L qI L r ≤ C k D α (cid:0) | u | p − u (cid:1) k L q I L r (2.8) ≤ C k D α u k L qI L r k u k p − L qcI L rc , where I ⊂ [0 , + ∞ ), 1 q = p − q c + 1 q , r = 1 r + p − r c . Due to k u k L qc [0 , ∞ ) L rc < ∞ , we divide [0 , + ∞ ) into N subintervals: I j = [ t j , t j +1 ] , ≤ j ≤ N ,such that k u k L qcIj L rc < δ (for small δ ) on each subinterval I j . Thus, from (2.6) and (2.8),we see that for 0 ≤ α ≤ s, ∀ ≤ j ≤ N , k D α u k L qIj L r ≤ k U ( t ) u ( t j ) k L qIj L r + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) D α Z I j U ( t − s ) (cid:0) | u | p − u ( s, x ) (cid:1) ds !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L qIj L r ≤ k U ( t ) u ( t j ) k L qIj L r + C k D α u k L qIj L r k u k p − L qcIj L rc ≤ CB + Cδ p − k D α u k L qIj L r . Let δ be small and satisfy Cδ p − < . Then k D α u k L qIj L r < ∞ , ≤ j ≤ N , and k D α u k L q L r < ∞ . Moreover, from (2.7), we see that for 0 ≤ α ≤ s , k D α ( u ( t ) − U ( t ) φ + ) k ≤ k u k p − L qc [ t, ∞ ) L rc k D α u k L q [ t, ∞ ) L r . (2.9)Therefore, we can obtain the claim by taking α = 0 and α = s in (2.9) and letting t → + ∞ . (cid:3) Proposition 2.7.
For any given A , there exist ǫ = ǫ ( A, N, p ) and c = c ( A ) such that forany ǫ ≤ ǫ , any interval I = ( T , T ) ⊂ R and any ˜ u = ˜ u ( x, t ) ∈ H s satisfying i ˜ u t − ( − ∆) s ˜ u − | ˜ u | p − ˜ u = e. (2.10) If k ˜ u k S (Λ sc ) ≤ A, k e k S ′ (Λ − sc ) ≤ ǫ and k e − i ( t − t )( − ∆) s ( u ( t ) − ˜ u ( t ) k S (Λ sc ) ≤ ǫ, then k u k S (Λ sc ) ≤ c = c ( A ) < ∞ . QING GUO AND SHIHUI ZHU
Proof.
Let u = ˜ u + w , where ˜ u is the solution of (2.10) and w is the solution of i∂ t w − ( − ∆) s w − | w + ˜ u | p − ( w + ˜ u ) + | ˜ u | p − ˜ u + e = 0 . (2.11)For any t ∈ I , I = ( T , t ] ∪ [ t , T ). We need only consider on I + = [ t , T ), since the caseon I − = ( T , t ] can be considered similarly. Since k ˜ u k S (Λ sc ) ≤ A , we can partition [ t , T )into N = N ( A ) intervals I j = [ t j , t j +1 ] such that for each j , the quantity k ˜ u k S (Λ sc ; I j ) < δ issuitably small with δ to be chosen later. The integral equation of w with initial time t j is w ( t ) = e − i ( t − t j )( − ∆) s w ( t j ) − i Z tt j e − i ( t − s )( − ∆) s [ | w + ˜ u | p − ( w + ˜ u ) − | ˜ u | p − ˜ u − e ]( s ) ds. (2.12)Using the inhomogeneous Strichartz estimates (2.4) on I j , we obtain that for some ( q , r ) ∈ Λ − s c , k w k S (Λ sc ; I j ) ≤k e − i ( t − t j )( − ∆) s w ( t j ) k S (Λ sc ; I j ) + c k| w + ˜ u | p − ( w + ˜ u ) + | ˜ u | p − ˜ u k L q ′ ( I ; L r ′ ) + k e k S ′ (Λ − sc ) ≤k e − i ( t − t j )(∆) s w ( t j ) k S (Λ sc ; I j ) + c k ˜ u k p − S (Λ sc ; I j ) k w k S (Λ sc ; I j ) + c k w k pS (Λ sc ; I j ) + k e k S ′ (Λ − sc ) ≤k e − i ( t − t j )( − ∆) s w ( t j ) k S (Λ sc ; I j ) + cδ p − k w k S (Λ sc ; I j ) + c k w k pS (Λ sc ; I j ) + cǫ . If δ ≤ (cid:18) c (cid:19) p − , k e − i ( t − t j )( − ∆) s w ( t j ) k S (Λ sc ; I j ) + cǫ ≤ (cid:18) c (cid:19) p − , (2.13)then k w k S (Λ sc ; I j ) ≤ k e − i ( t − t j )( − ∆) s w ( t j ) k S (Λ sc ; I j ) + cǫ . Now take t = t j +1 in (2.12), and apply e − i ( t − t j +1 )( − ∆) s to both sides to obtain e − i ( t − t j +1 )( − ∆) s w ( t j +1 ) = e − i ( t − t j )( − ∆) s w ( t j ) − i Z t j +1 t j e − i ( t − s )( − ∆) s [ | w + ˜ u | p − ( w + ˜ u ) − | ˜ u | p − ˜ u − e ]( s ) ds. Since the Duhamel integral is confined to I j , using the inhomogeneous Strichart’z estimatesand following a similar argument as above, we obtain that k e − i ( t − t j +1 )( − ∆) s w ( t j +1 ) k S (Λ sc ; I + ) ≤ k e − i ( t − t j )( − ∆) s w ( t j ) k S (Λ sc ; I + ) + cδ p − k w k S (Λ sc ; I j ) + c k w k pS (Λ sc ; I j ) + cǫ ≤ k e − i ( t − t j )( − ∆) s w ( t j ) k S (Λ sc ; I j ) + cǫ . CATTERING FOR FNLS 9
Iterating beginning with j = 0, we obtain k e − i ( t − t j )( − ∆) s w ( t j ) k S (Λ sc ; I + ) ≤ j k e − i ( t − t )( − ∆) s w ( t ) k S (Λ sc ; I j ) + (2 j − cǫ ≤ j +2 cǫ . To accommodate the conditions (2.13) for all intervals I j with 0 ≤ j ≤ N −
1, we require2 N +2 cǫ ≤ ( 14 c ) p − . (2.14)Finally, k w k S (Λ sc ; I + ) ≤ N − X j =0 j +2 cǫ + cN ǫ ≤ c ( N ) ǫ , which implies k w k S (Λ sc ; I + ) ≤ c ( A ) ǫ since N = N ( A ), concluding the proof. (cid:3) Variational Characteristic and Invariant Sets
First, we collect some variational properties of Q , as follows. Lemma 3.1. ( [36] ) Let N ≥ , < s < and sN < p < sN − s . Suppose that Q isthe ground-state solution of (1.5). Then, we have the following properties. ( i ) k Q k p +1 p +1 = s ( p +1) N ( p − k Q k H s = s ( p +1)2 s ( p +1) − N ( p − k Q k . ( ii ) E [ Q ] = R Q ( −△ ) s Qdx − p +1 k Q k p +1 p +1 = N ( p − − s N ( p − k Q k H s . ( iii ) E [ Q ] M [ Q ] s − scsc = N ( p − − s s − ( N − s )( p − k Q k ssc . ( iv ) k Q k H s M [ Q ] s − scsc = N ( p − s − ( N − s )( p − k Q k ssc . ( v ) C GN = k Q k p +1 p +1 k Q k p +1 − N ( p − s k Q k N ( p − s ˙ Hs = s ( p +1) N ( p −
1) 1 k Q k p +1 − N ( p − s k Q k N ( p − s − Hs . Remark . In fact, Caffarelli and Silvestre in [3] first proposed a general fractional Lapla-cian. And then many researchers have been studying the time dependent and independentof fractional nonlinear Schr¨odinger equations (see[5, 11, 16, 17, 32, 34, 35]).Let u ∈ H s \ { } , and define K g = {k u k H s M [ u ] s − scsc < k Q k H s M [ Q ] s − scsc , E [ u ] M [ u ] s − scsc < E [ Q ] M [ Q ] s − scsc } . Proposition 3.3.
Let N ≥ , < s < and sN < p < sN − s . Let Q be theground-state solution of (1.5). Then K g is invariant manifold of (1.1). Proof.
It follows from the conservation of energy and the sharp Gagaliardo-Nirenberg in-equality (1.6) that M [ u ] s − scsc E [ u ] = 12 k u ( t ) k s − sc ) sc k D s u ( t ) k − p + 1 k u k p +1 p +1 k u k s − sc ) sc ≥
12 ( k u ( t ) k s − scsc k D s u ( t ) k ) − C GN p + 1 ( k u ( t ) k s − scsc k D s u ( t ) k ) γs . Now, we define f ( y ) = y − p +1 C GN y N ( p − s . We find that f ( y ) has the following properties: f ′ ( y ) = y (cid:16) − C GN N ( p − s ( p +1) y N ( p − − s s (cid:17) , and thus, y = 0 and y = k Q k s − scsc k D s Q k are tworoots of f ′ ( y ) = 0, which implies that f has a local minimum at y and a local maximumat y . From Lemma 3.1, we have f max = f ( y ) = M [ Q ] s − scsc E [ Q ], and for all t ∈ If ( k u ( t ) k s − scsc k D s u ( t ) k ) ≤ M [ u ( t )] s − scsc E [ u ( t )] = M [ u ] s − scsc E [ u ] < f ( y ) . (3.1)If u ∈ K g , i.e., k u k s − scsc k D s u k < y , then by (3.1) and the continuity of k D s u ( t ) k in t , we claim that t ∈ R , k u ( t ) k H s M [ u ( t )] s − scsc < k Q k H s M [ Q ] s − scsc . (3.2)Indeed, if (3.2) is not true, there must be t ∈ I such that k u ( t ) k s − scsc k D s u ( t ) k ≥ y .Since u ( t, x ) ∈ C ( I ; H s ) is continuous with respect to t , we can find a 0 < t ≤ t such that k u ( t ) k s − scsc k D s u ( t ) k = y . Thus, by injecting E [ u ( t )] = E [ u ] and k u ( t ) k s − scsc k D s u ( t ) k = y into (3.1), we see that f ( y ) > M [ u ] s − scsc E [ u ] = M [ u ( t )] s − scsc E [ u ( t )] ≥ f ( k u ( t ) k s − scsc k D s u ( t ) k ) = f ( y ) . This is a contradiction. This completes the proof. (cid:3)
Remark . In fact, using the same argument in Proposition 3.3, we can obtain a preciseestimate. Specially, if the initial data is such that k u k H s M [ u ] s − scsc < k Q k H s M [ Q ] s − scsc , then, we can chose a δ > M [ u ] s − scsc E [ u ] < (1 − δ ) M [ Q ] s − scsc E [ Q ]. Moreover,for the solution u = u ( t ) with the above initial data we can find δ = δ ( δ ) such that k u ( t ) k s − scsc k D s u ( t ) k < (1 − δ ) k Q k s − scsc k D s Q k .By the invariance of K g , we see that (3.2) is true. In particular, the H s -norm of thesolution u is bounded, which proves the global existence of the solution in this case. CATTERING FOR FNLS 11
Theorem 3.5.
Let N ≥ , < s < and sN < p < sN − s . Assume that u ∈ H s ,and I = ( T − , T + ) is the maximal existence interval of u ( t ) solving (1.1) . If u ∈ K g ,then I = ( −∞ , + ∞ ) , i.e., u ( t ) exists globally in time. Lemma 3.6.
Let u ∈ K g . Furthermore, take δ > such that M [ u ] s − scsc E [ u ] < (1 − δ ) M [ Q ] s − scsc E [ Q ] . If u is a solution to problem (1.1) with initial data u , then there exists C δ > such that for all t ∈ R , k D s u k − N ( p − s ( p + 1) k u k p +1 p +1 ≥ C δ k D s u k . (3.3) Proof.
Let δ = δ ( δ ) > t ∈ R , we have k u ( t ) k s − scsc k D s u ( t ) k < (1 − δ ) k Q k s − scsc k D s Q k . (3.4)Denote H ( t ) = 1 k Q k s − sc ) sc k D s Q k ( k u ( t ) k s − sc ) sc k D s u ( t ) k − N ( p − s ( p + 1) k u k p +1 p +1 k u ( t ) k s − sc ) sc )and G ( y ) = y − y N ( p − s . Applying the sharp Gagliardo-Nirenberg inequality in (1.6), wededuce that H ( t ) ≥ G k u ( t ) k s − scsc k D s u ( t ) k k Q k s − scsc k D s Q k . When 0 ≤ y ≤ − δ , by the properties of the function G ( y ), we deduce that there existsa constant C δ such that g ( y ) ≥ C δ y provided 0 ≤ y ≤ − δ . This completes the proof. (cid:3) Lemma 3.7. (Comparability of gradient and energy) Let u ∈ K g . Then, N ( p − − s N ( p − k D s u ( t ) k ≤ E [ u ( t )] ≤ k D s u ( t ) k . Proof.
The second inequality can be obtained directly by the expression of E [ u ( t )]. Thefirst inequality follows from (1.6), (1.7) and (3.2) that12 k D s u k L − p + 1 k u k p +1 p +1 ≥ k D s u k L − sN ( p − k D s u k k u k s − scsc k D s Q k k Q k s − scsc scs ≥ N ( p − − s N ( p − k D s u k . (cid:3) At the end of this section, we prove the existence result of the wave operator Ω + : φ + v . This is important to establish the scattering theory. Proposition 3.8. (Existence of wave operators) Suppose that φ + ∈ H s and that M [ φ + ] s − scsc k D s φ + k < M [ Q ] s − scsc E [ Q ] . (3.5) Then, there exists v ∈ H s such that v globally solves (1.1) with initial data v satisfying k D s v ( t ) k k v k s − scsc ≤ k D s Q k k Q k s − scsc , M [ v ] = k φ + k , E [ v ] = 12 k D s φ + k , and lim t → + ∞ k v ( t ) − U ( t ) φ + k H s = 0 . Moreover, if k U ( t ) φ + k S (Λ sc ) ≤ δ sd , where δ sd is definedin Proposition 2.5, then k v k S (Λ sc ) ≤ k U ( t ) φ + k S (Λ sc ) , k D s c v k S (Λ ) ≤ c k φ + k ˙ H sc . Proof.
Let v ( t ) = F N LS ( t ) v be the solution v ( t ) of the fractional NLS (1.1) with the ini-tial data v (0) = v . According to the scattering theory of small initial data(see Proposition2.5), we consider the integral equation v ( t ) = U ( t ) φ + − i Z ∞ t U ( t − t ) | v | p − v ( t ) dt (3.6)for t ≥ T with T large. From Proposition 2.5, for sufficiently large T , we deduce that k v k S (Λ sc ;[ T, ∞ )) ≤ δ sd , and k v k S (Λ ;[ T, ∞ )) + k D s v k S (Λ ;[ T, ∞ )) < c k φ + k H s . Thus, by a similar argument when t > T , we obtain k v − U ( t ) φ + k S (Λ ;[ T, ∞ )) + k D s ( v − e it ∆ φ + ) k S (Λ ;[ T, ∞ )) → T → ∞ . Hence, v ( t ) − U ( t ) φ + → H s , and thus, M [ v ] = k φ + k . By the fact U ( t ) φ + → L q for any q ∈ (2 , NN − s ] as t → ∞ , we have k U ( t ) φ + k p +1 →
0. Moreover, combining this withthat k D s U ( t ) φ + k is conserved, we deduce that E [ v ] = lim t →∞ ( 12 k D s U ( t ) φ + k − p + 1 k U ( t ) φ + k p +1 p +1 ) = 12 k D s φ + k . By the assumption (3.5), then we obtain M [ v ] s − scsc E [ v ] < E [ Q ] M [ Q ] s − scsc . According to(3.5) and Remark 3.1 , we deduce thatlim t →∞ k v ( t ) k s − sc ) sc k D s v ( t ) k = lim t →∞ k U ( t ) φ + k s − sc ) sc k D s U ( t ) φ + k = k φ + k s − sc ) sc k D s φ + k ≤ E [ Q ] M [ Q ] s − scsc = N ( p − − sN ( p − k Q k s − sc ) sc k D s Q k . Finally, we can evolve v ( t ) from T back to time 0 as in Theorem 3.5. (cid:3) CATTERING FOR FNLS 13 Critical solution and compactness
In the previous, we have proved the the global existence part of Theorem 1.1(see Theorem3.5). From now on, we begin to prove the scattering part of Theorem 1.1. u ( t ) is globallywell-posed. According to Proposition 2.6, we just need to show that k u k S (Λ sc ) < ∞ . (4.1)Then, the H s scattering of the solution for Eq.(1.1) follows. We say that SC ( u ) holds if (4.1) is true for the solution u with the initialdata u . From Proposition 2.5, we see that there exists δ > E [ u ] M [ u ] s − scsc < δ and k u k s − scsc k D s u k < k Q k s − scsc k D s Q k , then (4.1) holds. Now, for each δ , we define theset S δ to be the collection of all such initial data in H s : S δ = { u ∈ H s : E [ u ] M [ u ] s − scsc < δ and M [ u ] s − scsc k D s u k < M [ Q ] s − scsc k D s Q k } . We also define that (
M E ) c = sup { δ : u ∈ S δ ⇒ SC ( u ) holds } . If (
M E ) c = M [ Q ] s − scsc E [ Q ], then we are done. Thus, we assume that( M E ) c < M [ Q ] s − scsc E [ Q ] . (4.2)Then, there exist solutions u n of (1.1) with H s initial data u n, (after rescaling, we mayinquire that u n satisfies k u n k = 1 ) such that k D s u n, k < k Q k s − scsc k D s Q k and E [ u n, ] ↓ ( M E ) c as n → ∞ , and SC ( u ) does not hold for any n .In this section, we will prove that there exists a critical H s solution u c to (1.1) with initialdata u c, such that k u c, k s − scsc k D s u c, k < k Q k s − scsc k D s Q k and M [ u c ] s − scsc E [ u c ] = ( M E ) c for which SC ( u c, ) does not hold. Then, we will show that the set { u c ( · , t ) | ≤ t < + ∞} isprecompact in H s . Finally, we will use these properties to obtain the rigidity theorem inSection 5, which will use to conduct a contradiction. This can be used to finish the proofof Theorem 1.1.First, we will introduce a profile decomposition lemma that is highly similar to that in[23], which were firstly proposed by Keraani [27] for for the cubic Schr¨odinger equation, asfollows. Lemma 4.1. (Profile expansion) Let φ n ( x ) be a radial and uniformly bounded sequence in H s . Then, for each M, there exists a subsequence of φ n , also denoted by φ n , and(1) for each ≤ j ≤ M , there exists a (fixed in n) profile ψ j ( x ) in H s ,(2) for each ≤ j ≤ M , there exists a sequence (in n) of time shifts t jn , (3) there exists a sequence (in n) of remainders W Mn ( x ) in H s such that φ n ( x ) = M X j =1 U ( − t jn ) ψ j ( x ) + W Mn ( x ) . The time and space sequences have a pairwise divergence property, i.e., for ≤ j = k ≤ M ,we have lim n → + ∞ | t jn − t kn | = + ∞ . (4.3) The remainder sequence has the following asymptotic smallness property: lim M → + ∞ [ lim n → + ∞ k U ( t ) W Mn k S (Λ sc ) ] = 0 . (4.4) For fixed M and any ≤ α ≤ s , we have the asymptotic Pythagorean expansion: k φ n k H α = M X j =1 k ψ j k H α + k W Mn k H α + o n (1) . (4.5) Proof.
The proof of the above linear profile decomposition for the fractional NLS is quitesimilar with that for the fourth-order nonlinear Schr¨odinger equation in [15]. Here, weomit the main proof. But we should point out that (4.4) could be improved tolim M → + ∞ [ lim n → + ∞ k U ( t ) W Mn k L q L r ] = 0 , ∀ ( q, r ) satisfies (2.1) with θ = s c . (4.6)More precisely, lim M → + ∞ [ lim n → + ∞ k U ( t ) W Mn k L ∞ L NN − sc ] = 0 . (4.7) (cid:3) Using the profile expansion, similar argument as in [15] could just be applied to obtain thefollowing results: Energy expansion, Existence of a critical solution and Precompactnessof the flow of the critical solution. Note that we have also proved similar counterparts ofthese results with respect to the fractional Hartree equation [18], and we omit the proofhere.
Lemma 4.2. (Energy Pythagorean expansion) In the situation of Lemma 4.1, we have E [ φ n ] = M X j =1 E [ U ( − t jn ) ψ j ] + E [ W Mn ] + o n (1) . (4.8) Proposition 4.3. (Existence of a critical solution) There exists a global solution u c in H s with initial data u c, such that k u c, k = 1 ,E [ u c ] = ( M E ) c < M [ Q ] s − scsc E [ Q ] , k D s u c k < M [ Q ] s − scsc k D s Q k , f or all ≤ t < ∞ , CATTERING FOR FNLS 15 and k u c k S (Λ sc ) = + ∞ . Proposition 4.4. (Precompactness of the flow of the critical solution) Let u c be as inProposition 4.3; then, if k u c k S ([0 , + ∞ );Λ sc ) = ∞ , { u c ( · , t ) | t ∈ [0 , + ∞ ) } ⊂ H s is precompact in H s . A corresponding conclusion is reached if k u c k S (( −∞ , sc ) = ∞ . Corollary 4.5.
Let u = u ( t ) be a solution to (1.1) such that K + = { u ( · , t ) | t ∈ [0 , + ∞ ) } is precompact in H sr . Then, for each ǫ > , there exists R > such that Z | x | >R | D s u ( x, t ) | + | u ( x, t ) | + | u ( x, t ) | p +1 dx ≤ ǫ. Proof.
If not, for any
R >
0, there exists ǫ > t n such that Z | x | >R | D s u ( x, t n ) | + | u ( x, t n ) | + | u ( x, t n ) | p +1 dx ≥ ǫ . By the precompactness of K + , there exists φ ∈ H s such that, up to a subsequence of t n ,we have u ( · , t n ) → φ in H s . Thus, for any R >
0, we obtain Z | x | >R | D s φ ( x ) | + | φ ( x ) | + | φ ( x ) | p +1 dx ≥ ǫ , from which we can easily obtain a contradiction because φ ∈ H s and k φ k p +1 p +1 ≤ c k φ k p +1 H s bythe Sobolev inequality. (cid:3) Rigidity theorem
In order to finish the proof of Theorem 1.1, we need the following rigidity theorem.
Theorem 5.1.
Let N ≥ and sN < p < sN − s . Assume that the initial data u ∈ H s is radial and u ∈ K g , i.e., M [ u ] s − scsc E [ u ] < M [ Q ] s − scsc E [ Q ] , (5.1) and M [ u ] s − scsc k u k H s < M [ Q ] s − scsc k Q k H s . (5.2) Let u = u ( t ) be the riadially global solution of (1.1) with initial data u . If it holds that K + = { u ( · , t ) | t ∈ [0 , + ∞ ) } is precompact in H s , then u = 0 . The same conclusion holdsif K − = { u ( · , t ) : t ∈ ( −∞ , } is precompact in H s . Now, we introduce the localized virial estimate for the radial solutions of (1.1) in termsof the idea in [2]. Let u ∈ H s with s ≥ . we define the auxiliary function u m = u m ( t, x )as u m := c s − ∆ + m u ( t ) = c s F − b u ( t, ξ ) | ξ | + m (5.3)where c s = q sinπsπ . It follows from [2] that, for any u ∈ H s , Z ∞ m s Z R N |∇ u m | dxdm = s k ( − ∆) s u k . (5.4)Then, we obtain the following corollary, which is a counterpart of Corollary 4.5. Corollary 5.2.
Let u = u ( t ) be a solution to (1.1) such that K = { u ( · , t ) | t ∈ [0 , + ∞ ) } isprecompact in H sr . Then, for each ǫ > , there exists R > such that Z ∞ m s Z | x | >R |∇ u m | dxdm + Z | x | >R | u ( x, t ) | + | u ( x, t ) | p +1 dx ≤ ǫ. The Proof of Theorem 5.1.
It suffices to address the K + case, since the K − casefollows similarly.Let ϕ ∈ C ∞ c be a radially real-valued function, and defined by ϕ ( x ) = ( | x | for | x | ≤
10 for | x | ≥ . For any
R >
0, take ϕ R ( x ) = R ϕ ( xR ) and define the localized virial of u ∈ H s by J R ( t ) := 2 Im Z R N ¯ u ( t, x ) ∇ ϕ R ( x ) · ∇ u ( t, x ) dx. Similar to the method in [2], we obtain the identity J ′ R ( t )) = Z ∞ m s Z R N (cid:16) ∂ k u m ( ∂ kl ϕ ( xR )) ∂ l u m − (∆ ϕ R ( x )) | u m | (cid:17) dxdm − p − p + 1 Z R N (∆ ϕ )( xR ) | u | p +1 dx. CATTERING FOR FNLS 17
By the properties of ϕ , we deduce that J ′ R ( t ) ≥ Z ∞ m s Z | x |≤ R |∇ u m | dx + 4 Z ∞ m s Z R< | x | < R ∂ r ϕ (cid:16) xR (cid:17) |∇ u m | dxdm (5.5) − Z ∞ m s Z | x | >R ∆ ϕ R ( x ) | u m | dxdm − p − p + 1 Z | x |≤ R | u | p +1 dx − c Z R< | x | < R | u | p +1 dx ≥ (cid:18) Z ∞ m s Z R N |∇ u m | dx − N ( p − P + 1 Z | u | p +1 dx (cid:19) − (cid:18) Z ∞ m s Z | x | >R |∇ u m | dx − N ( p − P + 1 Z | x | >R | u | p +1 dx (cid:19) − c (cid:18)Z ∞ m s Z R< | x | < R |∇ u m | dx + Z R< | x | < R | u | p +1 dx (cid:19) − cR s k u k . Here, we use the following estimate in the last step(see [2]), Z ∞ m s Z | x | >R ∆ ϕ R ( x ) | u m | dxdm ≤ cR − s k u k . Now, let δ ∈ (0 ,
1) satisfy E [ u ] < (1 − δ ) E [ Q ] M [ Q ] s − scsc . From Lemma 3.6 and Lemma3.7, we see that8 Z ∞ m s Z R N |∇ u m | dx − N ( p − P + 1 Z | u | p +1 dx = 8 sγ k D s u k − N ( p − P + 1 Z | u | p +1 dx ≥ C δ k D s u k . Then, we can take R large enough to obtain the following estimate J ′ R ( t ) ≥ C k D s u k . (5.6)Integrate (5.6) over [0 , t ]. |J R ( t ) − J R (0) | ≥ Ct k D s u k However, by [2], we should have |J R ( t ) − J R (0) | ≤ C R ( k u k H + k u k H ) ≤ C R ( k u k H s + k u k H s ) ≤ C R k Q k H s , which is a contradiction for large t unless u = 0. (cid:3) Now, we can finish the proof of Theorem 1.1.
The Proof of Theorem 1.1.
Note that by Proposition 4.4, the critical solution u c constructed in Section 4 satisfiesthe hypotheses in Theorem 5.1. Therefore, to complete the proof of Theorem 1.1, weshould apply Theorem 5.1 to u c and find that u c, = 0, which contradicts the fact that k u c k S (Λ sc ) = + ∞ . This contradiction shows that SC ( u ) holds. Thus, by Proposition 2.6,we have shown that H s scattering holds. (cid:3) Acknowledgments
This work was supported by the National Natural Science Foundation of China (No.11501395, 11301564, and 11371267) and the Excellent Youth Foundation of Sichuan Sci-entific Committee grant No. 2014JQ0039 in China.
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