Global Dynamics of solutions with group invariance for the nonlinear Schrödinger equation
aa r X i v : . [ m a t h . A P ] J un GLOBAL DYNAMICS OF SOLUTIONS WITH GROUPINVARIANCE FOR THE NONLINEAR SCHR ¨ODINGEREQUATION
TAKAHISA INUI
Abstract.
We consider the focusing mass-supercritical and energy-subcriticalnonlinear Schr¨odinger equation (NLS). We are interested in the global behaviorof the solutions to (NLS) with group invariance. By the group invariance, we candetermine the global behavior of the solutions above the ground state standingwaves.
Contents
1. Introduction 11.1. Background 11.2. Main Result 31.3. Idea of proof 52. Variational Argument and Blow-Up Result 53. Preliminaries for the Proof of the Scattering Result 74. Proof of the Scattering Result for the finite group invariant solutions 94.1. Linear Profile Decomposition with finite group invariance 94.2. Construction of a critical element and Rigidity 215. Proof of the Scattering Result for the infinite group invariant solutions 27Appendix A. Lemmas 27Appendix B. Applications 33Acknowledgement 34References 341.
Introduction
Background.
We consider the focusing mass-supercritical and energy-subcriticalnonlinear Schr¨odinger equation:(NLS) (cid:26) i∂ t u + ∆ u + | u | p − u = 0 , ( t, x ) ∈ R × R d ,u (0 , x ) = u ( x ) , x ∈ R d , where d ∈ N and 1 + 4 /d < p < / ( d − / ( d − ∞ if d = 1 ,
2. It is known (see [19] and the standard texts [7, 47, 35]) that this
Key words and phrases. global dynamics, orthogonal matrix, nonlinear Schr¨odinger equation,group invariance. equation (NLS) is locally well-posed in H ( R d ) and the energy, the mass, and themomentum, which are defined as follows, are conserved. E ( u ) := 12 k∇ u k L − p + 1 k u k p +1 L p +1 , (Energy) M ( u ) := k u k L , (Mass) P ( u ) := Im Z R d u ( x ) ∇ u ( x ) dx. (Momentum)Since a pioneer work by Kenig and Merle [25], many researchers have studiedthe global dynamics for (NLS). For the 3d cubic Schr¨odinger equation, Holmer andRoudenko [21] proved that the following two statements hold if the initial data u ∈ H is radially symmetric and satisfies the mass-energy condition M ( u ) E ( u )
Let ω > , u ∈ H ( R d ) satisfy S ω ( u ) < l ω , and u be the solutionof (NLS) with the initial data u . Then, the following statements hold. (1) If K ( u ) ≥ , then the solution u scatters. (2) If K ( u ) < , then the solution u blows up in finite time or grows up atinfinite time. More precisely, one of the following four cases occurs. (a) u blows up in finite time in both directions. LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 3 (b) u blows up in positive finite time and u is global in the negative timedirection and lim sup t →−∞ k∇ u ( t ) k L = ∞ . (c) u blows up in negative finite time and u is global in the positive timedirection and lim sup t →∞ k∇ u ( t ) k L = ∞ . (d) u is global in both time directions and lim sup t →±∞ k∇ u ( t ) k L = ∞ . In the present paper, we weaken the condition S ω < l ω in Theorem 1.1 by groupinvariance.For other studies of global dynamics of dispersive equations, see [16, 41, 17] (otherglobal dynamics of (NLS)), [25, 15, 33, 30, 9, 40] (energy-critical NLS), [31, 29, 10](mass-critical NLS), [38, 39, 27] (mass-subcritical NLS), [26, 14, 13] (wave equations),[23, 28, 42, 24] (nonlinear Klein-Gordon equations), and references therein.1.2. Main Result.
We are interested in the global behavior of the solutions withgroup invariance for (NLS). Let O ( d ) denote the set of d × d orthogonal matrices, i.e. O ( d ) := {R ∈ M d ( R ) : R T R = I d } , where the transpose of a matrix A is written by A T and I d denotes the d × d identitymatrix. R / π Z × O ( d ) is a group with the binary operation (+ , · ). Let a subgroup G of R / π Z × O ( d ) satisfy the following assumption throughout this paper.(A). For ( θ , G ) , ( θ , G ) ∈ G , if G = G , then we have θ = θ .Due to the assumption (A), we can use the notation G without confusion to denotenot only a matrix but also an element of G . For a subgroup G of R / π Z × O ( d ), wesay that a function ϕ is G -invariant (or with G -invariance) if ϕ = G ϕ for all G ∈ G ,where G ϕ ( x ) := e − iθ ( ϕ ◦ G − )( x ) = e − iθ ϕ ( G − x ) for G = ( θ, G ) ∈ R / π Z × O ( d ). Wedefine the Sobolev space with G -invariance by H G := { ϕ ∈ H ( R d ) : ϕ = G ϕ, ∀G ∈ G } . If the initial data u belongs to H G , then the corresponding solution to (NLS) is also G -invariant since the Laplacian ∆ is invariant for group actions by R / π Z × O ( d )and (NLS) is gauge invariant. We remark that if (A) does not hold, then the resultsin the present paper is trivial since we have H G = { } .For a subgroup G of R / π Z × O ( d ), we consider the restricted minimizing problem l Gω := inf { S ω ( ϕ ) : ϕ ∈ H G \ { } , K ( ϕ ) = 0 } , and we define subsets K ± G,ω in H ( R d ) by K + G,ω := { ϕ ∈ H G : S ω ( ϕ ) < l Gω , K ( ϕ ) ≥ } , K − G,ω := { ϕ ∈ H G : S ω ( ϕ ) < l Gω , K ( ϕ ) < } . We define a critical action for the data with G -invariance by S Gω := sup { S ∈ R : ∀ ϕ ∈ K + G,ω , S ω ( ϕ ) < S ⇒ the solution to (NLS) with the initial data ϕ belongs to L α ( R : L r ( R d )) } . See (3.1) below for the definition of α and r . We remark that u ∈ L α ( R : L r ( R d ))implies that the solution u scatters (see Proposition 3.2). Here, we say that the T. INUI solution u to (NLS) scatters if and only if there exist ϕ ± ∈ H ( R d ) such that (cid:13)(cid:13) u ( t ) − e it ∆ ϕ ± (cid:13)(cid:13) H → t → ±∞ , where e it ∆ denotes the free propagator of the Schr¨odinger equation. We say that asubgroup G ′ of G satisfies ( ∗ ) if there exists a sequence { x n } ⊂ R d such that (cid:26) { x n − G ′ x n } is bounded for all G ′ ∈ G ′ , | x n − G x n | → ∞ as n → ∞ , for all G ∈ G \ G ′ . For a finite group G , we define m Gω := min G ′ ( G satisfying ( ∗ ) G G ′ S G ′ ω , where X denotes the number of the elements in a set X .Our aim in the present paper is to prove the following theorem. Theorem 1.2.
Let ω > , G be a subgroup of R / π Z × O ( d ) , u ∈ H G satisfy S ω ( u ) < l Gω , and u be the solution of (NLS) with the initial data u . Then, thefollowing statements hold. (1) In addition, we assume either that (i) G is a finite group and u satisfies S ω ( u ) < m Gω or that (ii) G is an infinite group such that the embedding H G ֒ → L p +1 ( R d ) is compact. If K ( u ) ≥ , then the solution u scatters. (2) If K ( u ) < , then the solution u blows up in finite time or grows up atinfinite time. More precisely, one of (a)–(d) in Theorem 1.1 occurs If G = { (0 , I d ) } , then Theorem 1.2 is nothing but Theorem 1.1 ([18, 1]). The-orem 1.2 means that we can classify the solution whose mass-energy is larger thanthat of the ground state standing waves into scattering and blow-up (grow-up) bygroup invariance. See Appendix B for the applications of Theorem 1.2. Remark 1.1. (1)
For Theorem 1.2 (1): (i) If G is finite, then the embedding H G ֒ → L p +1 ( R d ) is not compact. (ii) If G satisfies that for all x ∈ R d \ { } , Gx := {G x : G ∈ G } has infinitelymany elements, then the embedding H G ֒ → L p +1 ( R d ) is compact (see [3] ). (iii) In the case that G is an infinite group such that the embedding H G ֒ → L p +1 ( R d ) is not compact, the scattering result for the solutions with G -invariance is an open problem. (2) For Theorem 1.2 (2): (i)
If the solution blows up in finite time, then k∇ u ( t ) k L diverges at themaximal existence time by the local well-posedness. (ii) If G = { (0 , G ) : G ∈ O ( d ) } , that is, the solution is radially symmetric,then the solution blows up in finite time in both time directions. See [43] and [21, Theorem 1.1 (2)] . (iii) If the initial data u satisfies | x | u ∈ L ( R d ) , then the correspondingsolution u blows up in finite time in both time directions. This statementfollows from Glassey’s argument (see [20] ) and the virial identity (1.1) d dt k xu ( t ) k L = 4 dK ( u ( t )) . LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 5
See [36] for other sufficient conditions to blow up. (iv)
For the cubic nonlinear Schr¨odinger equation in two dimensions, Marteland Rapha¨el [37] obtained a grow-up solution. However, we don’t knowwhether a grow-up solution exists or not for (NLS).
Idea of proof.
The proof of the scattering part, Theorem 1.2 (1), is basedon the contradiction argument by Kenig and Merle [25]. That is, we find a criticalelement, whose orbit is precompact in H G , by assuming that Theorem 1.2 (1) failsand using a concentration compactness argument, and we eliminate it by a rigidityargument. In the argument, we use the non-admissible Strichartz estimate, whichwas also used in Fang, Xie, and Cazenave [18]. However, unlike [18] and [1], we use alinear profile decomposition lemma for functions with group invariance to extend themass-energy condition. See Proposition 4.1 in the case that G is finite. That is whywe need to modify the construction of a critical element and the rigidity argument.If G satisfies that the embedding H G ֒ → L p +1 ( R d ) is compact, we can eliminate thetranslation parameter in the linear profile decomposition by the compactness of theembedding. In this case, the same argument as in the radial case does work.The blow-up part, Theorem 1.2 (2), follows directly from the result of Du, Wu,and Zhang [11].The rest of the present paper is organized as follows. In Section 2, we reorganizevariational argument for the data with G -invariance and show the blow-up result,Theorem 1.2 (2), by the method of [11]. Section 3 is devoted to preliminaries forthe proof of the scattering result, Theorem 1.2 (1). In Section 4, we consider thecase that G is finite. In Section 4.1, we show the linear profile decomposition lemmafor functions with the finite group invariance, which is a key ingredient. In Section4.2, we prove Theorem 1.2 (1) (i) by constructing a critical element and the rigidityargument. In Section 5, we prove Theorem 1.2 (1) (ii). Its proof is similar to in theradial case. We collect useful lemmas in Appendix A. In Appendix B, we introducesome applications of Theorem 1.2.2. Variational Argument and Blow-Up Result
Lemma 2.1. If K ( ϕ ) ≥ , then we have (2.1) S ω ( ϕ ) ≤ k∇ ϕ k L + ω k ϕ k L ≤ d ( p − d ( p − − S ω ( ϕ ) . Proof.
The left inequality holds obviusly. We prove the right inequality. We have0 ≤ K ( ϕ ) = (cid:18) d − p − (cid:19) k∇ ϕ k L + ( p − E ( ϕ ) . Adding ω ( p − M ( ϕ ) /
2, we obtain(2.2) (cid:18) p − − d (cid:19) k∇ ϕ k L + ω p − M ( ϕ ) ≤ ( p − S ω ( ϕ ) . Therefore,(2.3) (cid:18) p − − d (cid:19) (cid:26) k∇ ϕ k L + ω M ( ϕ ) (cid:27) ≤ ( p − S ω ( ϕ ) . T. INUI
This completes the proof. (cid:3)
Lemma 2.2. If u ∈ K + G,ω , then the corresponding solution u ( t ) belongs to K + G,ω forall existence time t . Moreover, if u ∈ K − G,ω , then the corresponding solution u ( t ) belongs to K − G,ω for all existence time t .Proof. Let u ∈ K + G,ω . Since the energy and the mass are conserved and the solutionbelongs to H G , we have u ( t ) ∈ K + G,ω ∪ K − G,ω for all existence time t . We assumethat there exists t > u ( t ) ∈ K − G,ω . By the continuity of the solutionin H ( R d ), there exists t ∈ (0 , t ) such that K ( u ( t )) = 0 and K ( u ( t )) < t ∈ ( t , t ]. By the definition of l Gω , if u ( t ) = 0, then we see that l Gω > E ( u ) + ω M ( u ) = E ( u ( t )) + ω M ( u ( t )) ≥ l Gω . This is a contradiction. Thus, u ( t ) = 0. By the uniqueness of the solution, u = 0for all time. However, this contradicts u ( t ) ∈ K − G,ω . Thus, we see that u ( t ) ∈ K + G,ω for all t . The second statement follows from the same argument. (cid:3) Remark 2.1.
By Lemmas 2.1 and 2.2, we obtain the global existence of the solutionin K + G,ω . Lemma 2.3.
Let ϕ ∈ H G satisfy S ω ( ϕ ) < l Gω . Then, one of the following holds. (2.4) K ( ϕ ) ≥ min { l Gω − S ω ( ϕ )) /d, δ k∇ ϕ k L } , or K ( ϕ ) ≤ − l Gω − S ω ( ϕ )) /d, for some δ > .Proof. Since the statement holds if ϕ = 0, we may assume that ϕ = 0. Let s ( λ ) := S ω ( ϕ λ ), where ϕ λ ( x ) = e λ ϕ ( e d λ ). Then, s (0) = S ω ( ϕ ) and s ′ (0) = K ( ϕ ). By directcalculations, s ( λ ) = 12 e d λ k∇ ϕ k L + ω k ϕ k L − p + 1 e ( p − λ k ϕ k p +1 L p +1 , (2.5) s ′ ( λ ) = 2 d e d λ k∇ ϕ k L − p − p + 1 e ( p − λ k ϕ k p +1 L p +1 , (2.6) s ′′ ( λ ) = 8 d e d λ k∇ ϕ k L − ( p − p + 1 e ( p − λ k ϕ k p +1 L p +1 . (2.7)Thus, we have s ′′ ≤ s ′ /d . First, we consider the case of K <
0. Let λ be definedby(2.8) λ := (cid:18) p − − d (cid:19) − log d k∇ ϕ k L p − p +1 k ϕ k p +1 L p +1 ! . Then, s ′ ( λ ) = 0. Moreover, λ < K <
0. Integrating s ′′ ≤ s ′ /d on [ λ , s ′ (0) − s ′ ( λ ) ≤ d ( s (0) − s ( λ )) . LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 7
This completes the proof in the case of
K <
0. Next, we consider the case of
K > λ := (cid:18) p − − d (cid:19) − log d k∇ ϕ k L p − p +1 (cid:0) p − d (cid:1) k ϕ k p +1 L p +1 ! . Then, s ′′ ( λ ) + 4 s ′ ( λ ) /d = 0. If λ ≥
0, then, by the definition of λ , we obtain(2.11) K ( ϕ ) ≥ (cid:18) p − d (cid:19) − d (cid:18) p − − d (cid:19) k∇ ϕ k L . Letting δ := 2( p − − /d ) / { d ( p − /d ) } , we obtain K ( ϕ ) ≤ δ k∇ ϕ k L . If λ < s ′′ ( λ ) < − s ′ ( λ ) /d for λ ∈ [0 , λ ], where we note that λ > K ≥ s ′′ ( λ )+4 s ′ ( λ ) /d < λ > λ . Integrating the inequality s ′′ ( λ ) < − s ′ ( λ ) /d on [0 , λ ], this completes the proof. (cid:3) Proof of Theorem 1.2 (2).
By Lemmas 2.2 and 2.3, if u ∈ K − G,ω , then the solution u satisfies K ( u ( t )) < − l Gω − S ω ( u )) /d < t . Therefore,Theorem 1.2 (2) follows directly from Theorem 2.1 in [11]. (cid:3) Preliminaries for the Proof of the Scattering Result
In this subsection, we introduce some basic facts used to prove the scatteringresult. Their proofs also can be found in [18]. Let α := p − p +1)4 − ( d − p − , β := p − p +1) d ( p − +( d − p − − ,q := p +1) d ( p − , r := p + 1 , s := d − p − . (3.1)Moreover, let β ′ and r ′ denote the H¨older exponents of the exponent β and r , re-spectively. Lemma 3.1 (Strichartz estimates) . The following estimates are vaild. (cid:13)(cid:13) e it ∆ ϕ (cid:13)(cid:13) L q ( R : L r ) . k ϕ k L , (3.2) (cid:13)(cid:13) e it ∆ ϕ (cid:13)(cid:13) L α ( R : L r ) . k ϕ k ˙ H s , (3.3) (cid:13)(cid:13)(cid:13)(cid:13)Z t e i ( t − t ′ )∆ f ( t ′ ) dt ′ (cid:13)(cid:13)(cid:13)(cid:13) L α ( I : L r ) . k f k L β ′ ( I : L r ′ ) , (3.4) where I is a time interval and the implicit constant is independent of I .Proof. The first estimate is a standard Strichartz estimate. The second one is ob-tained by the Sobolev inequality and the Strichartz estimate. The third one is anon-admissible Strichartz estimate (see [8, Lemma 2.1] or [7, Proposition 2.4.1]). (cid:3)
Proposition 3.2.
Let u ∈ H ( R d ) and u be the solution to (NLS) with the initialdata u . If the solution u is positively global and u ∈ L α ((0 , ∞ ) : L r ( R d )) , then thesolution scatters in the positive time direction. Moreover, the same statement holdsin the negative case. T. INUI
Proof.
Proposition 2.3 in [8] implies u belongs to L η ((0 , ∞ ) : L ρ ( R d )) for any admis-sible pair ( η, ρ ), and then a standard argument gives us the fact that u scatters inthe positive time directions (see the argument in Theorem 7.8.1 in [7]). (cid:3) Proposition 3.3.
There exists ε sd > such that if u ∈ H ( R d ) and (cid:13)(cid:13) e it ∆ u (cid:13)(cid:13) L α ((0 , ∞ ): L r ) ≤ ε sd , then the solution u of (NLS) with the initial data u is positively global and wehave (3.5) k u k L α ((0 , ∞ ): L r ) . ε sd . In particular, if k u k H ≤ ε sd , then the solution u is global and we have (3.6) k u k L α ( R : L r ) . k u k H . Proof.
The first statement follows form Proposition 2.4 in [8]. By (3.3), we obtain (cid:13)(cid:13) e it ∆ u (cid:13)(cid:13) L α ( R : L r ) . k u k ˙ H s ≤ ε sd . Applying Proposition 2.4 in [8], we obtain thesecond statement. (cid:3) Lemma 3.4. If ψ ∈ H G satisfies k∇ ψ k L / ωM ( ψ ) / < l Gω , then there exists aglobal solution U + to (NLS) such that U + (0) ∈ K + G,ω and (cid:13)(cid:13) U + ( t ) − e it ∆ ψ (cid:13)(cid:13) H → as t → ∞ . Moreover, the same statement holds in the negative case.Proof. We may assume that ψ = 0 since the statement is true if ψ = 0. It is knownin [46, Theorem 17] (see also [45, Theorem 8]) that there exist T ∈ R and a uniquesolution u ∈ C (( T, ∞ ) : H ( R d )) of (NLS) such that(3.7) (cid:13)(cid:13) U + ( t ) − e it ∆ ψ (cid:13)(cid:13) H → t → ∞ . The uniqueness and the assumption that ψ is G -invariant imply that the solution U + is also G -invariant. By the triangle inequality, the Sobolev embedding, (3.7), and (cid:13)(cid:13) e it ∆ ψ (cid:13)(cid:13) L p +1 → t → ∞ (see [7, Corollary 2.3.7]), we have k U + ( t ) k L p +1 ≤ (cid:13)(cid:13) U + ( t ) − e it ∆ ψ (cid:13)(cid:13) L p +1 + (cid:13)(cid:13) e it ∆ ψ (cid:13)(cid:13) L p +1 . (cid:13)(cid:13) U + ( t ) − e it ∆ ψ (cid:13)(cid:13) H + (cid:13)(cid:13) e it ∆ ψ (cid:13)(cid:13) L p +1 → , as t → ∞ . Therefore, by the conservation laws and the assumption, we obtain S ω ( U + ) = lim t →∞ S ω ( U + ( t )) = 12 k∇ ψ k L + ω M ( ψ ) < l Gω and lim t →∞ K ( U + ( t )) = 2 d k∇ ψ k L > . Thus, U + ( t ) belongs to K + G,ω for large t > T . This statement, Lemmas 2.1, and 2.2,imply that U + is global in both time directions and U + (0) ∈ K + G,ω . (cid:3) Lemma 3.5 (Perturbation Lemma) . Given A ≥ , there exist ε ( A ) > and C ( A ) > with the following property. If u ∈ C ([0 , ∞ ) : H ( R d )) is a solution of (NLS), if ˜ u ∈ C ([0 , ∞ ) : H ( R d )) and e ∈ L loc ([0 , ∞ ) : H − ( R d )) satisfy i∂ t ˜ u + ∆˜ u + | ˜ u | p − ˜ u = LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 9 e , for a.e. t > , and if k ˜ u k L α ([0 , ∞ ): L r ) ≤ A, (3.8) k e k L β ′ ([0 , ∞ ): L r ′ ) ≤ ε ( A ) , (3.9) (cid:13)(cid:13) e it ∆ ( u (0) − ˜ u (0)) (cid:13)(cid:13) L α ([0 , ∞ ): L r ) ≤ ε ≤ ε ( A ) , (3.10) then u ∈ L α ((0 , ∞ ) : L r ( R d )) and k u − ˜ u k L α ([0 , ∞ ): L r ) ≤ Cε . See [18, Proposition 4.7] for the proof.4.
Proof of the Scattering Result for the finite group invariantsolutions
In this section, we prove Theorem 1.2 (1) (i). Let a subgroup G of R / π Z × R d be finite throughout this section.4.1. Linear Profile Decomposition with finite group invariance.
We prove alinear profile decomposition for functions with G -invariance. Let τ y ϕ ( x ) = ϕ ( x − y )throughout this paper. Proposition 4.1 (Linear Profile Decomposition with finite group invariance) . Let { ϕ n } ⊂ H G be a bounded sequence. Then, after replacing a subsequence, for j ∈ N there exist a subgroup G j of R / π Z × O ( d ) , ψ j ∈ H G j , { W jn } ⊂ H G j , { t jn } ⊂ R , and { x jn } ⊂ R d such that (4.1) ϕ n = J X j =1 e it jn ∆ X G∈ G G ( τ x jn ψ j ) G + X G∈ G G W Jn G for every J ∈ N , and the following statements hold. (1) For any fixed j , { t jn } satisfies either t jn = 0 or t jn → ±∞ as n → ∞ , (2) For any fixed j , { x jn } satisfies x jn = G x jn for all G ∈ G j and | x jn − G x jn | → ∞ for all G ∈ G \ G j . (3) We have the orthogonality of the parameters: for j = h , lim n →∞ | t jn − t hn | = ∞ or lim n →∞ |G x jn − G ′ x hn | = ∞ for all G , G ′ ∈ G. (4) We have smallness of the remainder: lim sup n →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e it ∆ X G∈ G G W Jn G (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L α ( R : L r ) → as J → ∞ . (5) We have the orthogonality in norms: for all λ ∈ [0 , , k ϕ n k H λ = J X j =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X G∈ G G ( τ x jn ψ j ) G (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H λ + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X G∈ G G W Jn G (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H λ + o n (1) , (4.2) k ϕ n k p +1 L p +1 = J X j =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e it jn ∆ X G∈ G G ( τ x jn ψ j ) G (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p +1 L p +1 + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X G∈ G G W Jn G (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p +1 L p +1 + o n (1)(4.3) and, in particular, S ω ( ϕ n ) = J X j =1 S ω e it jn ∆ X G∈ G G ( τ x jn ψ j ) G ! + S ω X G∈ G G W Jn G ! + o n (1) , (4.4) K ( ϕ n ) = J X j =1 K e it jn ∆ X G∈ G G ( τ x jn ψ j ) G ! + K X G∈ G G W Jn G ! + o n (1) . (4.5) Lemma 4.2.
Let a > and { ϕ n } ⊂ H G satisfy lim sup n →∞ k ϕ n k H ≤ a < ∞ . If (cid:13)(cid:13) e it ∆ ϕ n (cid:13)(cid:13) L ∞ ( R : L p +1 ) → A as n → ∞ , then there exist a subsequence, which isstill denoted by { ϕ n } n ∈ N , a subgroup G ′ of G , ψ ∈ H G ′ , sequences { t n } n ∈ N ⊂ R , { x n } n ∈ N ⊂ R d , and { W n } n ∈ N ⊂ H G ′ such that (4.6) ϕ n = e it n ∆ X G∈ G G ( τ x n ψ ) G + X G∈ G G W n G , and the following hold. (1) e − it n ∆ τ −G x n ϕ n ⇀ G ψ/ ( G/ G ′ ) in H ( R d ) and e − it n ∆ τ −G x n ˜ W n ⇀ in H ( R d ) for all G ∈ G , where ˜ W n := P G∈ G G W n / G . (2) The sequence { t n } satisfies either t n = 0 or t n → ±∞ as n → ∞ . (3) The sequence { x n } satisfies G ′ x n = x n for all G ′ ∈ G ′ and | x n − G x n | → ∞ for all G ∈ G \ G ′ . (4) We have the orthogonality in norms: k ϕ n k H λ − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X G∈ G G ( τ x n ψ ) G (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H λ − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X G∈ G G W n G (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H λ → as n → ∞ , for all ≤ λ ≤ . k ϕ n k p +1 L p +1 − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e it n ∆ X G∈ G G ( τ x n ψ ) G (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p +1 L p +1 − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X G∈ G G W n G (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p +1 L p +1 → as n → ∞ . (5) We have k ψ k H ≥ νA d − − Λ) a − d − − Λ) , where Λ := d ( p − / { p + 1) } ∈ (0 , min { , d/ } ) and the constant ν > isindependent of a , A , and { ϕ n } n ∈ N . (6) If A = 0 , then for every sequences { t n } n ∈ N ⊂ R , { x n } n ∈ N ⊂ R d , and { W n } n ∈ N ⊂ H ( R d ) satisfying (4.6) and (1), we must have ψ = 0 .Proof. Let b χ ∈ C ∞ ( R d ) satisfy 0 ≤ b χ ≤ b χ ( ξ ) = (cid:26) , if | ξ | ≤ , , if | ξ | ≥ . LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 11
Given ρ >
0, we set c χ ρ ( ξ ) := b χ ( ξ/ρ ). For any u ∈ H ( R d ) and λ ∈ [0 , k u − χ ρ ∗ u k H λ = (cid:13)(cid:13) | ξ | λ ( b u − c χ ρ b u ) (cid:13)(cid:13) L = Z R d | ξ | λ (1 − c χ ρ ) | b u | dξ ≤ Z | ξ |≥ ρ | ξ | λ | b u | dξ = Z | ξ |≥ ρ | ξ | − − λ ) | ξ | | b u | dξ ≤ ρ − − λ ) Z | ξ |≥ ρ | ξ | | b u | dξ ≤ ρ − − λ ) Z R d | ξ | | b u | dξ = ρ − − λ ) k∇ u k L . Therefore, we have(4.7) k u − χ ρ ∗ u k ˙ H λ ≤ ρ − (1 − λ ) k∇ u k L . By Plancherel’s theorem, we have χ ρ ∗ u ( x ) = Z R d χ ρ ( y ) u ( x − y ) dy = ( − d Z R d c χ ρ ( ξ ) e − iy · ξ b u ( ξ ) dξ. Since Λ < d/
2, by the H¨older inequality, we see that, for any u ∈ H ( R d ), | χ ρ ∗ u ( x ) | ≤ Z R d | c χ ρ ( ξ ) || b u ( ξ ) | dξ ≤ Z | ξ |≤ ρ | b u ( ξ ) | dξ = Z | ξ |≤ ρ | ξ | − Λ | ξ | Λ | b u ( ξ ) | dξ (4.8) ≤ (cid:18)Z | ξ |≤ ρ | ξ | − dξ (cid:19) / k u k ˙ H Λ ≤ κρ d − k u k ˙ H Λ , where κ is a constant independent of ρ and u .First, we consider the case of A >
0. It follows from the Sobolev embedding k u k L p +1 ≤ C k u k ˙ H Λ , the isometry of e it ∆ on ˙ H Λ ( R d ), (4.7), 0 ≤ Λ ≤
1, and theassumption of lim sup n →∞ k ϕ n k H ≤ a < ∞ that (cid:13)(cid:13) e it ∆ ϕ n − e it ∆ ( χ ρ ∗ ϕ n ) (cid:13)(cid:13) L p +1 ≤ C (cid:13)(cid:13) e it ∆ ϕ n − e it ∆ ( χ ρ ∗ ϕ n ) (cid:13)(cid:13) ˙ H Λ = C k ϕ n − ( χ ρ ∗ ϕ n ) k ˙ H Λ ≤ Cρ − (1 − Λ) k∇ ϕ n k L ≤ Cρ − (1 − Λ) a, for large n ∈ N . Choosing ρ = (4 Ca/A ) − Λ , we obtain (cid:13)(cid:13) e it ∆ ϕ n − e it ∆ ( χ ρ ∗ ϕ n ) (cid:13)(cid:13) L p +1 ≤ Cρ − (1 − Λ) a ≤ A . Thus, we have(4.9) (cid:13)(cid:13) e it ∆ ϕ n − e it ∆ ( χ ρ ∗ ϕ n ) (cid:13)(cid:13) L ∞ ( R : L p +1 ) ≤ A . By the triangle inequality, (cid:13)(cid:13) e it ∆ ϕ n (cid:13)(cid:13) L ∞ ( R : L p +1 ) → A as n → ∞ , and (4.9), we get (cid:13)(cid:13) e it ∆ ( χ ρ ∗ ϕ n ) (cid:13)(cid:13) L ∞ ( R : L p +1 ) (4.10) ≥ (cid:13)(cid:13) e it ∆ ϕ n (cid:13)(cid:13) L ∞ ( R : L p +1 ) − (cid:13)(cid:13) e it ∆ ϕ n − e it ∆ ( χ ρ ∗ ϕ n ) (cid:13)(cid:13) L ∞ ( R : L p +1 ) ≥ A − A A , for large n . On the other hand, for large n , we have (cid:13)(cid:13) e it ∆ ( χ ρ ∗ ϕ n ) (cid:13)(cid:13) L ∞ ( R : L p +1 ) ≤ (cid:13)(cid:13) e it ∆ ( χ ρ ∗ ϕ n ) (cid:13)(cid:13) d − d L ∞ ( R : L ) (cid:13)(cid:13) e it ∆ ( χ ρ ∗ ϕ n ) (cid:13)(cid:13) d L ∞ ( R : L ∞ ) (4.11) ≤ k ϕ n k d − d L ∞ ( R : L ) (cid:13)(cid:13) e it ∆ ( χ ρ ∗ ϕ n ) (cid:13)(cid:13) d L ∞ ( R : L ∞ ) ≤ (2 a ) d − d (cid:13)(cid:13) e it ∆ ( χ ρ ∗ ϕ n ) (cid:13)(cid:13) d L ∞ ( R : L ∞ ) , where we have used the H¨older inequality, the isometry of e it ∆ on L ( R d ), k χ ρ ∗ u k L ≤k u k L for all u ∈ H ( R d ), and the assumption of lim sup n →∞ k ϕ n k H ≤ a < ∞ . Com-bining (4.10) and (4.11), we get, for large n , (cid:13)(cid:13) e it ∆ ( χ ρ ∗ ϕ n ) (cid:13)(cid:13) d L ∞ ( R : L ∞ ) ≥ (2 a ) − d − d A , or equivalently, (cid:13)(cid:13) e it ∆ ( χ ρ ∗ ϕ n ) (cid:13)(cid:13) L ∞ ( R : L ∞ ) ≥ (2 a ) − d − (cid:18) A (cid:19) d . Therefore, there exist { ˜ t n } n ∈ N ⊂ R and { ˜ x n } n ∈ N ⊂ R d such that(4.12) (cid:12)(cid:12)(cid:12) e i ˜ t n ∆ ( χ ρ ∗ ϕ n )(˜ x n ) (cid:12)(cid:12)(cid:12) ≥ (4 a ) − d − (cid:18) A (cid:19) d , for large n . We consider the following two cases. Case1: { ˜ t n } n ∈ N ⊂ R is unbounded. Case2: { ˜ t n } n ∈ N ⊂ R is bounded. Case1:
Since { ˜ t n } n ∈ N ⊂ R is unbounded, we may assume ˜ t n → ±∞ as n → ∞ taking a subsequence. Let t n := ˜ t n . Taking a subsequence and using Lemma A.1,we obtain a subgroup of G ′ of G such that a subsequence, which is still denoted by { ˜ x n } , satisfies that (cid:26) ˜ x n − G ′ ˜ x n → ¯ x G ′ as n → ∞ , ∀G ′ ∈ G ′ , | ˜ x n − G ˜ x n | → ∞ as n → ∞ , ∀G ∈ G \ G ′ , for some ¯ x G ′ ∈ R d . Using Lemma A.2, we obtain a sequence { x n } such that (cid:26) x n − G ′ x n = 0 , ∀G ′ ∈ G ′ , | x n − G x n | → ∞ as n → ∞ , ∀G ∈ G \ G ′ , and there exists x ∞ ∈ R d such that x n − ˜ x n → x ∞ as n → ∞ . Since k ϕ n k H is bounded, there exists ψ ∈ H ( R d ) such that, after taking a sub-sequence, e − it n ∆ τ − x n ϕ n ⇀ ψ/ ( G/ G ′ ) in H ( R d ) as n → ∞ . Here, we notethat ψ is G ′ -invariant since ϕ n is G -invariant and x n = G ′ x n for all G ′ ∈ G ′ .We prove (5). Now, we have w n := e − i ˜ t n ∆ τ − ˜ x n ϕ n ⇀ τ x ∞ ψ/ ( G/ G ′ ) in H ( R d ) LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 13 as n → ∞ . Since e it ∆ commutes with the convolution with χ ρ , we find that e i ˜ t n ∆ ( χ ρ ∗ ϕ n )(˜ x n ) = χ ρ ∗ w n (0). By (4.8) and (4.12), we have(4 a ) − d − (cid:18) A (cid:19) d ≤ (cid:12)(cid:12)(cid:12)(cid:12) χ ρ ∗ ψ ( − x ∞ ) G/ G ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ κρ d − k ψ k ˙ H Λ G/ G ′ ≤ κρ d − k ψ k H G/ G ′ . Since we take ρ = (4 Ca/A ) − Λ , we obtain the statement (5). We set W n := ϕ n − e it n ∆ τ x n ψ . Since ϕ n is G -invariant, we see that ϕ n = X G∈ G G ϕ n G = X G∈ G G ( e it n ∆ τ x n ψ + W n ) G = X G∈ G e it n ∆ G ( τ x n ψ ) G + X G∈ G G W n G .
This is the statement (4.6). Moreover, W n is G ′ -invariant since ϕ n and τ x n ψ are G ′ -invariant. We check the statement (1). The first statement e − it n ∆ τ −G x n ϕ n ⇀ G ψ/ ( G/ G ′ ) in H ( R d ) follows from the definition of ψ and the G -invariance of ϕ n . We prove the second statement e − it n ∆ τ −G x n ˜ W n ⇀ H ( R d ) for all G ∈ G ,where we recall that ˜ W n = P G∈ G G W n / G . Let {G k } G/ G ′ k =1 be the set of left cosetrepresentatives, that is, we have G = G/ G ′ X k =1 G k G ′ . Since W n is G ′ -invariant, we find that˜ W n = X G∈ G G W n G = G/ G ′ X k =1 G k W n G/ G ′ . Let G = G l G ′ for some l ∈ { , , · · · , G/ G ′ } and G ′ ∈ G ′ . Then, by the definitionof W n and the first statement in (1), we obtain e − it n ∆ τ −G x n ˜ W n = e − it n ∆ τ −G l x n ϕ n − G/ G ′ X k =1 τ −G l x n + G k x n G k ψ G/ G ′ ⇀ G l ψ G/ G ′ − G l ψ G/ G ′ = 0 , where we note that | − G l x n + G k x n | = | − x n + G − l G k x n | → ∞ since G − l G k G ′ if k = l . Thus, we get the second statement in (1). Next, we prove (4). We set˜ ψ n := P G∈ G e it n ∆ G ( τ x n ψ ) / G . We have k ϕ n k H λ = (cid:13)(cid:13)(cid:13) ˜ ψ n + ˜ W n (cid:13)(cid:13)(cid:13) H λ = (cid:13)(cid:13)(cid:13) ˜ ψ n (cid:13)(cid:13)(cid:13) H λ + (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) H λ + 2 (cid:16) ˜ ψ n , ˜ W n (cid:17) ˙ H λ = k ψ k H λ + (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) H λ + 2 (cid:16) ˜ ψ n , ϕ n − ˜ ψ n (cid:17) ˙ H λ , where ( · , · ) ˙ H λ denotes the inner product in ˙ H λ . We calculate (cid:16) ˜ ψ n , ϕ n − ˜ ψ n (cid:17) ˙ H λ . Since τ x n ψ is G ′ -invariant, we observe that˜ ψ n = X G∈ G e it n ∆ G ( τ x n ψ ) G = G/ G ′ X k =1 e it n ∆ G k ( τ x n ψ )( G/ G ′ ) . By this observation, we have (cid:16) ˜ ψ n , ϕ n − ˜ ψ n (cid:17) ˙ H λ = G/ G ′ X k =1 e it n ∆ G k ( τ x n ψ ) G/ G ′ , ϕ n − G/ G ′ X l =1 e it n ∆ G l ( τ x n ψ ) G/ G ′ ˙ H λ = 1( G/ G ′ ) G/ G ′ X k =1 G/ G ′ X l =1 (cid:0) e it n ∆ G k ( τ x n ψ ) , ϕ n − e it n ∆ G l ( τ x n ψ ) (cid:1) ˙ H λ = 1( G/ G ′ ) G/ G ′ X k,l =1 (cid:8)(cid:0) e it n ∆ G k ( τ x n ψ ) , ϕ n (cid:1) ˙ H λ − (cid:0) e it n ∆ G k ( τ x n ψ ) , e it n ∆ G l ( τ x n ψ ) (cid:1) ˙ H λ (cid:9) For the first term, we find that, for all k ∈ { , , · · · , G/ G ′ } ,(4.13) (cid:0) e it n ∆ G k ( τ x n ψ ) , ϕ n (cid:1) ˙ H λ = (cid:0) ψ, e − it n ∆ τ − x n G − k ϕ n (cid:1) ˙ H λ → k ψ k H λ ( G/ G ′ )since ϕ n is G -invariant and e − it n ∆ τ − x n ϕ n weakly converges to ψ/ ( G/ G ′ ) as n →∞ in H ( R d ). For the second term, we obtain (cid:0) e it n ∆ G k ( τ x n ψ ) , e it n ∆ G l ( τ x n ψ ) (cid:1) ˙ H λ = (cid:0) ψ, τ − x n G − k G l ( τ x n ψ ) (cid:1) ˙ H λ (4.14) = (cid:16) ψ, τ − x n + G − k G l x n G − k G l ψ ) (cid:17) ˙ H λ → (cid:26) k ψ k H λ , if k = l, , if k = l. Combining (4.13) with (4.14), we get G/ G ′ X k,l =1 (cid:8)(cid:0) e it n ∆ G k ( τ x n ψ ) , ϕ n (cid:1) ˙ H λ − (cid:0) e it n ∆ G k ( τ x n ψ ) , e it n ∆ G l ( τ x n ψ ) (cid:1) ˙ H λ (cid:9) → G/ G ′ X k,l =1 k ψ k H λ ( G/ G ′ ) − G/ G ′ X k =1 k ψ k H λ = 0 . This implies the first statement of (4). We set f n := (cid:12)(cid:12)(cid:12)(cid:12) k ϕ n k p +1 L p +1 − (cid:13)(cid:13)(cid:13) ˜ ψ n (cid:13)(cid:13)(cid:13) p +1 L p +1 − (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) p +1 L p +1 (cid:12)(cid:12)(cid:12)(cid:12) . LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 15
We recall that, for every
P > l ≥
2, there exists a constant C = C P,l such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l X j =1 z j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P − l X j =1 | z j | P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C X j = k | z j || z k | P − , for all z j ∈ C for 1 ≤ j ≤ l . This implies that || z + z | p +1 − | z | p +1 − | z | p +1 | ≤ C | z || z | ( | z | p − + | z | p − ) . Letting g n = | ˜ ψ n | p − + | ˜ W n | p − , we get f n ≤ C Z R d (cid:12)(cid:12)(cid:12) ˜ ψ n ( x ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ˜ W n ( x ) (cid:12)(cid:12)(cid:12) g n ( x ) dx ≤ C Z R d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G/ G ′ X k =1 e it n ∆ G k ( τ x n ψ )( x ) G/ G ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ˜ W n ( x ) (cid:12)(cid:12)(cid:12) g n ( x ) dx ≤ C G/ G ′ X k =1 Z R d (cid:12)(cid:12) e it n ∆ G k ( τ x n ψ )( x ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ˜ W n ( x ) (cid:12)(cid:12)(cid:12) g n ( x ) dx ≤ C G/ G ′ X k =1 Z R d (cid:12)(cid:12) e it n ∆ ψ ( x ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) τ − x n G − k ˜ W n ( x ) (cid:12)(cid:12)(cid:12) τ − x n G − k g n ( x ) dx. Note that, by the triangle inequality and the Sobolev embedding, (cid:13)(cid:13) τ − x n G − k g n (cid:13)(cid:13) L p +1 p − = k g n k L p +1 p − ≤ (cid:13)(cid:13)(cid:13) | ˜ ψ n | p − (cid:13)(cid:13)(cid:13) L p +1 p − + (cid:13)(cid:13)(cid:13) | ˜ W n | p − (cid:13)(cid:13)(cid:13) L p +1 p − = (cid:13)(cid:13)(cid:13) ˜ ψ n (cid:13)(cid:13)(cid:13) p − L p +1 + (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) p − L p +1 . (cid:13)(cid:13)(cid:13) ˜ ψ n (cid:13)(cid:13)(cid:13) p − H + (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) p − H . k ψ k p − H + k W n k p − H < C where we use { W n } is bounded in H since { ϕ n } is bounded. And (cid:13)(cid:13)(cid:13) τ − x n G − k ˜ W n (cid:13)(cid:13)(cid:13) L p +1 = (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) L p +1 < C . Now, (cid:13)(cid:13) e it n ∆ ψ (cid:13)(cid:13) L p +1 → n → ∞ since t n → ±∞ (see [7, Corollary2.3.7 ]). Therefore, by the H¨older inequality, we get f n . (cid:13)(cid:13) e it n ∆ ψ (cid:13)(cid:13) L p +1 (cid:13)(cid:13)(cid:13) τ − x n G − k ˜ W n (cid:13)(cid:13)(cid:13) L p +1 (cid:13)(cid:13) τ − x n G − k g n (cid:13)(cid:13) L p +1 p − . (cid:13)(cid:13) e it n ∆ ψ (cid:13)(cid:13) L p +1 → . This means the second statement of (4).If A = 0, then (cid:13)(cid:13) e − it n ∆ τ − x n ϕ n (cid:13)(cid:13) L p +1 = (cid:13)(cid:13) e − it n ∆ ϕ n (cid:13)(cid:13) L p +1 ≤ (cid:13)(cid:13) e − it ∆ ϕ n (cid:13)(cid:13) L ∞ ( R : L p +1 ) →
0. On the other hand, if e − it n ∆ τ − x n ϕ n ⇀ ψ/ ( G/ G ′ ) as n → ∞ in H ( R d ),then e − it n ∆ τ − x n ϕ n → ψ/ ( G/ G ′ ) as n → ∞ in L p +1 ( B R ) for any R > ψ = 0. Case2:
Since { ˜ t n } n ∈ N ⊂ R is bounded, we may assume ˜ t n → ¯ t ∈ R as n → ∞ takinga subsequence. Let t n := 0 for all n . Minor modifications imply the statements, (1)–(3) and the first statement of (4). See the argument below (5.22) in [18] for thesecond statement of (4) . (cid:3) Proof of Proposition 4.1.
We use an induction argument. By the boundedness of { ϕ n } in H ( R d ), let a := lim sup n →∞ k ϕ n k H < ∞ . Moreover, we define A :=lim sup n →∞ (cid:13)(cid:13) e it ∆ ϕ n (cid:13)(cid:13) L ∞ ( R : L p +1 ) . Note that the Sobolev embedding and the bounded-ness of { ϕ n } in H ( R d ) give A < ∞ . Taking a subsequence, we may assume that A = lim n →∞ (cid:13)(cid:13) e it ∆ ϕ n (cid:13)(cid:13) L ∞ ( R : L p +1 ) . By Lemma 4.2, we obtain a subsequence, which isstill denoted by { ϕ n } n ∈ N , a subgroup G of G , ψ ∈ H G , sequences { t n } n ∈ N ⊂ R , { x n } n ∈ N ⊂ R d , and { W n } n ∈ N ⊂ H G such that ϕ n = ˜ ψ n + ˜ W n , where ˜ ψ n := e it n ∆ P G∈ G G ( τ x n ψ ) / G and ˜ W n := P G∈ G G W n / G and the followingstatements hold. e − it n ∆ τ −G x n ϕ n ⇀ G ψ G/ G , and e − it n ∆ τ −G x n ˜ W n ⇀ H ( R d ) for all G ∈
G.t n = 0 or t n → ±∞ as n → ∞ . G x n = x n for all G ∈ G and | x n − G x n | → ∞ for all G ∈ G \ G . k ϕ n k H λ − (cid:13)(cid:13)(cid:13) ˜ ψ n (cid:13)(cid:13)(cid:13) H λ − (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) H λ → n → ∞ , for all 0 ≤ λ ≤ . k ϕ n k p +1 L p +1 − (cid:13)(cid:13)(cid:13) ˜ ψ n (cid:13)(cid:13)(cid:13) p +1 L p +1 − (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) p +1 L p +1 → n → ∞ . (cid:13)(cid:13) ψ (cid:13)(cid:13) H ≥ νA d − − Λ) (cid:18) lim sup n →∞ k ϕ n k H (cid:19) − d − − Λ) , where ν > W n is G -invariant and A := lim sup n →∞ (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) L ∞ ( R : L p +1 ) < ∞ . Taking a subsequence, we may assume that A = lim n →∞ (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) L ∞ ( R : L p +1 ) . Ap-plying Lemma 4.2 as ϕ n = ˜ W n , we obtain a subsequence, which is still denoted by { ˜ W n } , a subgroup G of G , ψ ∈ H G , sequences { t n } n ∈ N ⊂ R , { x n } n ∈ N ⊂ R d , and { W n } n ∈ N ⊂ H G such that ˜ W n = ˜ ψ n + ˜ W n , LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 17 where ˜ ψ n := e it n ∆ P G∈ G G ( τ x n ψ ) / G and ˜ W n := P G∈ G G W n / G and the followingstatements hold. e − it n ∆ τ −G x n ˜ W n ⇀ G ψ G/ G , and e − it n ∆ τ −G x n ˜ W n ⇀ H ( R d ) for all G ∈
G.t n = 0 or t n → ±∞ as n → ∞ . G x n = x n for all G ∈ G and | x n − G x n | → ∞ for all G ∈ G \ G . (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) H λ − (cid:13)(cid:13)(cid:13) ˜ ψ n (cid:13)(cid:13)(cid:13) H λ − (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) H λ → n → ∞ , for all 0 ≤ λ ≤ . (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) p +1 L p +1 − (cid:13)(cid:13)(cid:13) ˜ ψ n (cid:13)(cid:13)(cid:13) p +1 L p +1 − (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) p +1 L p +1 → n → ∞ . (cid:13)(cid:13) ψ (cid:13)(cid:13) H ≥ νA d − − Λ) (cid:18) lim sup n →∞ (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) H (cid:19) − d − − Λ) , where ν > { ˜ W j − n } , a subgroup G j of G , ψ j ∈ H G j , sequences { t jn } n ∈ N ⊂ R , { x jn } n ∈ N ⊂ R d , and { W jn } n ∈ N ⊂ H G j such that˜ W j − n = ˜ ψ jn + ˜ W jn , where ˜ ψ jn := e it jn ∆ P G∈ G G ( τ x jn ψ j ) / G and ˜ W jn := P G∈ G G W jn / G and the followingstatements hold. e − it jn ∆ τ −G x jn ˜ W j − n ⇀ G ψ j G/ G j , and e − it jn ∆ τ −G x jn ˜ W jn ⇀ H ( R d ) for all G ∈
G.t jn = 0 or t jn → ±∞ as n → ∞ . G j x jn = x jn for all G j ∈ G j and | x jn − G x jn | → ∞ for all G ∈ G \ G j . (cid:13)(cid:13)(cid:13) ˜ W j − n (cid:13)(cid:13)(cid:13) H λ − (cid:13)(cid:13)(cid:13) ˜ ψ jn (cid:13)(cid:13)(cid:13) H λ − (cid:13)(cid:13)(cid:13) ˜ W jn (cid:13)(cid:13)(cid:13) H λ → n → ∞ , for all 0 ≤ λ ≤ . (cid:13)(cid:13)(cid:13) ˜ W j − n (cid:13)(cid:13)(cid:13) p +1 L p +1 − (cid:13)(cid:13)(cid:13) ˜ ψ jn (cid:13)(cid:13)(cid:13) p +1 L p +1 − (cid:13)(cid:13)(cid:13) ˜ W jn (cid:13)(cid:13)(cid:13) p +1 L p +1 → n → ∞ . (cid:13)(cid:13) ψ j (cid:13)(cid:13) H ≥ νA d − − Λ) j (cid:18) lim sup n →∞ (cid:13)(cid:13)(cid:13) ˜ W j − n (cid:13)(cid:13)(cid:13) H (cid:19) − d − − Λ) , where ν > j thproperties. Here, we regard ˜ W n as ϕ n . Combining 1st, 2nd, · · · , and J th properties,we obtain the statements (1), (2), and (5). Note that the orthogonalities of thefunctionals S ω and K follows from the orthogonalities in norms. We prove (4). By theorthogonality of H -norm, we have lim sup n →∞ (cid:13)(cid:13)(cid:13) ˜ W jn (cid:13)(cid:13)(cid:13) H ≤ lim sup n →∞ k ϕ n k H = a for all j . Thus we see that(4.15) (cid:13)(cid:13) ψ j (cid:13)(cid:13) H ≥ νA d − − Λ) j a − d − − Λ) . By the orthogonality of H -norm and Lemma A.3, we also have P Jj =1 k ψ j k H ≤ a .Taking the limit J → ∞ , we obtain P ∞ j =1 k ψ j k H ≤ a . Combining this with (4.15), we obtain νa − d − − Λ) ∞ X j =1 A d − − Λ) j ≤ ∞ X j =1 (cid:13)(cid:13) ψ j (cid:13)(cid:13) H ≤ a. This gives us P ∞ j =1 A d − − Λ) j . a d − − Λ) < ∞ so that A J → J → ∞ . By theH¨older inequality and the Strichartz estimate (3.2), we obtainlim sup n →∞ (cid:13)(cid:13)(cid:13) e it ∆ ˜ W Jn (cid:13)(cid:13)(cid:13) L α ( R : L r ) ≤ lim sup n →∞ (cid:18)(cid:13)(cid:13)(cid:13) e it ∆ ˜ W Jn (cid:13)(cid:13)(cid:13) θL q ( R : L r ) (cid:13)(cid:13)(cid:13) e it ∆ ˜ W Jn (cid:13)(cid:13)(cid:13) − θL ∞ ( R : L r ) (cid:19) . lim sup n →∞ (cid:13)(cid:13)(cid:13) ˜ W Jn (cid:13)(cid:13)(cid:13) θH lim sup n →∞ (cid:13)(cid:13)(cid:13) e it ∆ ˜ W Jn (cid:13)(cid:13)(cid:13) − θL ∞ ( R : L r ) . a θ lim sup n →∞ (cid:13)(cid:13)(cid:13) e it ∆ ˜ W Jn (cid:13)(cid:13)(cid:13) − θL ∞ ( R : L r ) → J → ∞ where θ = 2 { − ( d − p − } / { d ( p − } ∈ (0 , ψ J = 0 for some J implies that ψ j = 0 for j ≥ J . Therefore, there exists J ∈ N ∪ {∞} such that ψ j = 0for j ≤ J and ψ j = 0 for j > J . In the case of J = 1, there is nothing to prove. Wesuppose J ≥
2. By 1st properties, we have e − it n ∆ τ −G x n ˜ W n ⇀ H ( R d ) for all G ∈ G and e − it n ∆ τ −G ′ x n ˜ W n ⇀ G ′ ψ G/ G = 0 in H ( R d ) for all G ′ ∈ G. By Lemma A.5 as f n = e − it n ∆ τ −G x n ˜ W n , we get | t n − t n | + |G x n − G ′ x n | → ∞ for all G , G ′ ∈ G . We suppose J ≥ ≤ j = h ≤ µ − µ < J . For j ∈ { , , · · · , µ − } , we have˜ W µ − n − ˜ W j − n = ˜ W µ − n − ˜ ψ µ − n − ˜ W j − n = ˜ W µ − n − ˜ ψ µ − n − ˜ ψ µ − n − ˜ W j − n = · · · = − µ − X ν = j ˜ ψ νn . Therefore, we see that e − it jn ∆ τ −G x jn (cid:16) ˜ W µ − n − ˜ W j − n (cid:17) = − e − it jn ∆ τ −G x jn ˜ ψ jn − µ − X ν = j +1 e − it jn ∆ τ −G x jn ˜ ψ νn . (4.16)Let {G ( j ) k } G/ G j k =1 be the set of left coset representatives and G = G ( j ) l G j for some l ∈ { , , · · · , G/ G j } and G j ∈ G j . For the first term of the right hand side in LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 19 (4.16), we have e − it jn ∆ τ −G x jn ˜ ψ jn = τ −G x jn G/ G j X k =1 G ( j ) k ( τ x jn ψ j ) G/ G j = G ψ j G/ G j + G/ G j X k =1 k = l τ −G ( j ) l x jn + G ( j ) k x jn G ( j ) k ψ j G/ G j ⇀ G ψ j G/ G j . The second term of the right hand side in (4.16) weakly converges to 0 in H ( R d )since we suppose that the statement (3) holds for 1 ≤ j = h ≤ µ −
1. Since e − it jn ∆ τ −G x jn ˜ W j − n ⇀ G ψ j / ( G/ G j ), we have e − it jn ∆ τ −G x jn ˜ W µ − n ⇀ H ( R d ) for all G ∈ G, and e − it µn ∆ τ −G ′ x µn ˜ W µ − n ⇀ G ′ ψ G/ G = 0 in H ( R d ) for all G ′ ∈ G. By Lemma A.5 as f n = e − it jn ∆ τ −G x jn ˜ W µ − n , we get | t jn − t µn | + |G x jn − G ′ x µn | → ∞ forall G , G ′ ∈ G . Therefore, we obtain the statement (3). This completes the proof. (cid:3) Lemma 4.3.
Let k be a nonnegative integer and ϕ j ∈ H G ( R ) for j ∈ { , , · · · , k } satisfy S ω ( P kj =1 ϕ j ) ≤ l Gω − δ, S ω ( P kj =1 ϕ j ) ≥ P kj =1 S ω ( ϕ j ) − ε,K ( P kj =1 ϕ j ) ≥ − ε, K ( P kj =1 ϕ j ) ≤ P kj =1 K ( ϕ j ) + ε, for δ , ε satisfying ε < δ . Then we have ≤ S ω ( ϕ j ) < l Gω and K ( ϕ j ) ≥ for all j ∈ { , , · · · , k } . Namely, we see that ϕ j ∈ K + G,ω for all j ∈ { , , · · · , k } .Proof. Let J ω := S ω − dK/
4. First, we prove the following equality. l Gω = inf (cid:8) J ω ( ϕ ) : ϕ ∈ H G \ { } , K ( ϕ ) ≤ (cid:9) . Let l ′ denote the right hand side. By the definition of l Gω and J ω , we have l Gω = inf (cid:8) J ω ( ϕ ) : ϕ ∈ H G \ { } , K ( ϕ ) = 0 (cid:9) . Therefore, we have l Gω ≥ l ′ . We prove l Gω ≤ l ′ . Take ϕ ∈ H G such that K ( ϕ ) ≤ λ ≤ K ( ϕ λ ) = 0. Thus, we see that l Gω ≤ S ω ( ϕ λ ) = J ω ( ϕ λ ) ≤ J ω ( ϕ ) . Taking the infimum for ϕ ∈ H G such that K ( ϕ ) ≤
0, we obtain l Gω ≤ l ′ . Thus, wehave l Gω = l ′ . Next, we prove the statement of the present lemma. We assume thatthere exists an j ∈ { , , · · · , k } such that K ( ϕ j ) <
0. By l Gω = l ′ and the positivity of J ω , we obtain l Gω ≤ k X j =1 J ω ( ϕ j )= k X j =1 (cid:18) S ω ( ϕ j ) − d K ( ϕ j ) (cid:19) = k X j =1 S ω ( ϕ j ) − k X j =1 d K ( ϕ j ) ≤ S ω k X j =1 ϕ j ! + ε − d K k X j =1 ϕ j ! − ε ! ≤ l Gω − δ + ε + ε< l Gω . This is a contradiction. So, K ( ϕ j ) ≥ j ∈ { , , · · · , k } . Moreover, for any j ∈ { , , · · · , k } , we have S ω ( ϕ j ) = J ω ( ϕ j ) + d K ( ϕ j ) ≥ , and S ω ( ϕ j ) ≤ k X j =1 S ω ( ϕ j ) ≤ S ω k X j =1 ϕ j ! + ε ≤ l Gω − δ + ε < l Gω . This completes the proof. (cid:3)
We collect lemmas for nonlinear profiles.
Lemma 4.4.
Let { x n } be a sequence, ψ ∈ H , and U be a solution of (NLS) withthe initial data ψ . Then, we have U n ( t ) = e it ∆ τ x n ψ + i Z t e i ( t − s )∆ ( | U n ( s ) | p − U n ( s )) ds, where U n ( t, x ) := U ( t, x − x n ) . Lemma 4.4 follows from the space translation invariance of the equation (NLS).
Lemma 4.5.
Let { t n } satisfy t n → ±∞ , { x n } be a sequence, ψ ∈ H , and U be asolution of (NLS) satisfying (cid:13)(cid:13) U ± ( t ) − e it ∆ ψ (cid:13)(cid:13) H → as t → ±∞ Then, we have U ± ,n ( t ) = e it ∆ e it n ∆ τ x n ψ + i Z t e i ( t − s )∆ ( | U ± ,n ( s ) | p − U ± ,n ( s )) ds + e ± ,n ( t ) , where U ± ,n ( t, x ) := U ± ( t + t n , x − x n ) and k e ± ,n k L α ( R : L r ) → as n → ∞ . LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 21
Proof.
Since U ± ,n is a solution of (NLS) with the initial data τ x n U ± ( t n ) by the timeand space translation invariance, we have e ± ,n ( t ) = U ± ,n ( t ) − e it ∆ e it n ∆ τ x n ψ − i Z t e i ( t − s )∆ ( | U ± ,n ( s ) | p − U ± ,n ( s )) ds = e it ∆ τ x n U ± ( t n ) − e it ∆ e it n ∆ τ x n ψ. By the Strichartz estimate, k e ± ,n k L α ( R : L r ) . (cid:13)(cid:13) U ± ( t n ) − e it n ∆ ψ (cid:13)(cid:13) H → n → ∞ . This completes the proof. (cid:3)
Construction of a critical element and Rigidity.
By the definition of S Gω ,we have S Gω ≤ l Gω . Lemma 2.1 and Lemma 3.3 give S Gω >
0. We prove S Gω =min { m Gω , l Gω } by contradiction argument so that we suppose S Gω < min { m Gω , l Gω } . Proposition 4.6.
Assume S Gω < min { m ω , l Gω } . Then, there exists a global solution u c to (NLS) with G -invariance such that S ω ( u c ) = S Gω and k u c k L α ( R : L r ) = ∞ . We call u c a critical element. Proof.
By the definition of S Gω and the assumption of S Gω < min { m ω , l Gω } , there existsa sequence { ϕ n } ∈ K + G,ω satisfying S Gω < S ω ( ϕ n ) < min { m ω , l Gω } , S ω ( ϕ n ) ց S Gω ,and u n L α ( R : L r ( R d )), where u n is a global solution with the initial data ϕ n .Since { ϕ n } is bounded in H ( R d ), we apply the linear profile decomposition with G -invariance, Proposition 4.1, to the sequence { ϕ n } and then we obtain ϕ n = J X j =1 ˜ ψ jn + ˜ W Jn , where we recall that ˜ ψ jn = P G∈ G e it jn ∆ G ( τ x jn ψ j ) / G and ˜ W Jn = P G∈ G G W Jn / G . Wealso see that S ω ( ϕ n ) = J X j =1 S ω (cid:16) ˜ ψ jn (cid:17) + S ω (cid:16) ˜ W Jn (cid:17) + o (1) ,K ( ϕ n ) = J X j =1 K (cid:16) ˜ ψ jn (cid:17) + K (cid:16) ˜ W Jn (cid:17) + o (1) , where o (1) → n → ∞ . By these decompositions, we have S ω ( ϕ n ) ≤ l Gω − δ,S ω ( ϕ n ) ≥ J X j =1 S ω (cid:16) ˜ ψ jn (cid:17) + S ω (cid:16) ˜ W Jn (cid:17) − ε,K ( ϕ n ) ≥ > − ε,K ( ϕ n ) ≤ J X j =1 K (cid:16) ˜ ψ jn (cid:17) + K (cid:16) ˜ W Jn (cid:17) + ε, for large n where δ = l Gω − S ω ( ϕ ) and ε > ε < δ . Therefore, Lemma 4.3gives us that ˜ ψ jn ∈ K + G,ω for all j ∈ { , , · · · , J } and ˜ W Jn ∈ K + G,ω . Thus, for any J ,we obtain S Gω = lim n →∞ S ω ( ϕ n ) ≥ J X j =1 lim sup n →∞ S ω (cid:16) ˜ ψ jn (cid:17) . We prove S Gω = lim sup n →∞ S ω ( ˜ ψ jn ) for some j by a contradiction argument. Weassume that S Gω = lim sup n →∞ S ω ( ˜ ψ jn ) fails for all j . Namely, we assume thatlim sup n →∞ S ω ( ˜ ψ jn ) < S Gω for all j . By reordering, we can choose 0 ≤ J ≤ J ≤ J such that 1 ≤ j ≤ J : t jn = 0 , ∀ nJ + 1 ≤ j ≤ J : lim n →∞ t jn = −∞ ,J + 1 ≤ j ≤ J : lim n →∞ t jn = + ∞ . Above we are assuming that if a > b then there is no j such that a ≤ j ≤ b .Note that J ∈ { , } by the orthogonality of the parameter { t jn } (see Proposi-tion 4.1 (3)). We only consider the case J = 1 since the case J = 0 is eas-ier. By the assumption of the contradiction argument and t n = 0, we have 0 < lim sup n →∞ S ω ( P G∈ G G ( τ x n ψ ) / G ) < S Gω . By the choice of { x n } and Lemma A.3,lim sup n →∞ S ω X G∈ G G ( τ x n ψ ) G ! = G G S ω (cid:18) ψ G/ G (cid:19) < S Gω < m Gω . Therefore, S ω ( ψ / ( G/ G )) < S G ω . By the definition of S G ω , the solution U to(NLS) with the initial data ψ / ( G/ G ) belongs to L α ( R : L r ( R d )).For j ∈ [ J + 1 , J ], we have m Gω > S Gω > lim sup n →∞ S ω (cid:16) ˜ ψ jn (cid:17) = lim sup n →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X G∈ G G ( τ x jn ψ j ) G (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H + ω (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X G∈ G G ( τ x jn ψ j ) G (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L − p + 1 lim inf n →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e it jn ∆ X G∈ G G ( τ x jn ψ j ) G (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p +1 L p +1 = G G j (cid:13)(cid:13)(cid:13)(cid:13) ψ j G/ G j (cid:13)(cid:13)(cid:13)(cid:13) H + ω (cid:13)(cid:13)(cid:13)(cid:13) ψ j G/ G j (cid:13)(cid:13)(cid:13)(cid:13) L ! , where we use (cid:13)(cid:13) e it n ∆ φ (cid:13)(cid:13) L p +1 → n → ∞ (see [7, Corollary 2.3.7]) and Lemma A.3.This inequality implies that ψ j / ( G/ G j ) satisfies the assumption of Lemma 3.4as G = G j , where we note that S G j ω ≤ l G j ω . Thus, we obtain the global solution U j − to (NLS) such that U j − (0) ∈ K + G,ω and (cid:13)(cid:13)(cid:13)(cid:13) U j − ( t ) − e it ∆ ψ j G/ G j (cid:13)(cid:13)(cid:13)(cid:13) H → t → −∞ . LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 23
Moreover, U j − belongs to L α ( R : L r ( R d )) by the definition of S G j ω since we have S ω ( U j − ) = 12 (cid:13)(cid:13)(cid:13)(cid:13) ψ j G/ G j (cid:13)(cid:13)(cid:13)(cid:13) H + ω (cid:13)(cid:13)(cid:13)(cid:13) ψ j G/ G j (cid:13)(cid:13)(cid:13)(cid:13) L < S G j ω . For j ∈ [ J + 1 , J ], by the similar argument, we obtain a global solution U j + suchthat U j + (0) ∈ K + G,ω , U j + ∈ L α ( R : L r ( R d )), and (cid:13)(cid:13)(cid:13)(cid:13) U j + ( t ) − e it ∆ ψ j G/ G j (cid:13)(cid:13)(cid:13)(cid:13) H → t → ∞ . We define U j := U , if j = 1 ,U j − , if j ∈ [2 , J ] ,U j + , if j ∈ [ J + 1 , J ] , and U jn ( t, x ) := U j ( t + t jn , x − x jn ) . Moreover, we define u Jn := J X j =1 G/ G j X k =1 G ( j ) k U jn , where {G ( j ) k } G/ G j k =1 be the set of left coset representatives. Then u Jn satisfies i∂ t u Jn + ∆ u Jn + (cid:12)(cid:12) u Jn (cid:12)(cid:12) p − u Jn = e Jn ,e Jn = (cid:12)(cid:12) u Jn (cid:12)(cid:12) p − u Jn − J X j =1 G/ G j X k =1 (cid:12)(cid:12)(cid:12) G ( j ) k U jn (cid:12)(cid:12)(cid:12) p − G ( j ) k U jn . Moreover, we have u n (0) − u Jn (0) = J X j =1 G/ G j X k =1 G ( j ) k (cid:18) e it jn ∆ τ x jn ψ j G/ G j − τ x jn U j ( t jn ) (cid:19) + ˜ W Jn . To apply the perturbation lemma, Lemma 3.5, we prove the following inequalitieshold for large n . (cid:13)(cid:13) u Jn (cid:13)(cid:13) L α ( R ; L r ) ≤ A, (4.17) (cid:13)(cid:13) e Jn (cid:13)(cid:13) L β ′ ( R : L r ′ ) ≤ ε ( A ) , (4.18) (cid:13)(cid:13) e it ∆ ( u n (0) − u Jn (0)) (cid:13)(cid:13) L α ( R : L r ) ≤ ε ( A ) . (4.19)We prove (4.17). By the definition of U jn , we have u Jn ( t ) = J X j =1 G/ G j X k =1 G ( j ) k U jn ( t )= G/ G X k =1 G (1) k ( τ x n U ( t )) + J X j =2 G/ G j X k =1 G ( j ) k ( τ x jn U j − ( t + t jn )) + J X j = J +1 G/ G j X k =1 G ( j ) k ( τ x jn U j + ( t + t jn )) . By Lemma A.7, we obtainlim sup n →∞ (cid:13)(cid:13) u Jn (cid:13)(cid:13) L α ( R : L r ) ≤ J X j =1 G/ G j X k =1 (cid:13)(cid:13)(cid:13) G ( j ) k U j (cid:13)(cid:13)(cid:13) L α ( R : L r ) = 2 J X j =1 G G j (cid:13)(cid:13) U j (cid:13)(cid:13) L α ( R : L r ) , where we use | t jn − t hn | → ∞ or |G x jn − G ′ x hn | → ∞ for all G , G ′ ∈ G if j = h and alsouse |G ( j ) k x jn − G ( j ) l x jn | → ∞ if k = l . By (5) in Proposition 4.1 and Lemma A.3, wehave k ϕ n k H = J X j =1 G G j (cid:13)(cid:13)(cid:13)(cid:13) ψ j G/ G j (cid:13)(cid:13)(cid:13)(cid:13) H + (cid:13)(cid:13)(cid:13) ˜ W Jn (cid:13)(cid:13)(cid:13) H + o n (1) . Therefore, sup n ∈ N k ϕ n k H < ∞ implies that there exists a finite set J such that k ψ j / ( G/ G j ) k H < ε sd for j J , where ε sd is a constant appearing in Proposi-tion 3.3. Thus, we getlim sup n →∞ (cid:13)(cid:13) u Jn (cid:13)(cid:13) L α ( R : L r ) . J X j =1 (cid:13)(cid:13) U j (cid:13)(cid:13) L α ( R : L r ) = X j ∈ J (cid:13)(cid:13) U j (cid:13)(cid:13) L α ( R : L r ) + X j J (cid:13)(cid:13) U j (cid:13)(cid:13) L α ( R : L r ) . X j ∈ J (cid:13)(cid:13) U j (cid:13)(cid:13) L α ( R : L r ) + X j J (cid:13)(cid:13)(cid:13)(cid:13) ψ j G/ G j (cid:13)(cid:13)(cid:13)(cid:13) H . X j ∈ J (cid:13)(cid:13) U j (cid:13)(cid:13) L α ( R : L r ) + lim sup n →∞ k ϕ n k H ≤ A < ∞ . We prove (4.19). By the triangle inequality, the Strichartz estimate, the definitionof U j , (4) in Proposition 4.1, and Lemmas 4.4 and 4.5, we have (cid:13)(cid:13) e it ∆ ( u n (0) − u Jn (0)) (cid:13)(cid:13) L α ( R : L r ) ≤ J X j =1 G/ G j X k =1 (cid:13)(cid:13)(cid:13)(cid:13) e it ∆ (cid:18) τ x jn U j ( t jn ) − e it jn ∆ τ x jn ψ j G/ G j (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L α ( R : L r ) + (cid:13)(cid:13)(cid:13) e it ∆ ˜ W Jn (cid:13)(cid:13)(cid:13) L α ( R : L r ) ≤ J X j =1 G/ G j X k =1 (cid:13)(cid:13)(cid:13)(cid:13) τ x jn U j ( t jn ) − e it jn ∆ τ x jn ψ j G/ G j (cid:13)(cid:13)(cid:13)(cid:13) H + (cid:13)(cid:13)(cid:13) e it ∆ ˜ W Jn (cid:13)(cid:13)(cid:13) L α ( R : L r ) ≤ ε ≤ ε ( A ) , for large n and J . We prove (4.18). In general, the following inequality holds. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J X j =1 z j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p − J X j =1 z j − J X j =1 (cid:12)(cid:12) z j (cid:12)(cid:12) p − z j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C J X ≤ j = h ≤ J | z j | p − | z h | . This implies that (cid:13)(cid:13) e Jn (cid:13)(cid:13) L β ′ ( R : L r ′ ) ≤ C J X ≤ j = h ≤ J (cid:13)(cid:13) | U jn | p − | U hn | (cid:13)(cid:13) L β ′ ( R : L r ′ ) . LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 25
An approximation argument and | t jn − t hn | → ∞ or |G x jn −G ′ x hn | → ∞ for all G , G ′ ∈ G if j = h and also use |G ( j ) k x jn −G ( j ) l x jn | → ∞ if k = l give us (cid:13)(cid:13) | U jn | p − | U hn | (cid:13)(cid:13) L β ′ ( R : L r ′ ) → n → ∞ . Thus, we obtain (4.18). Applying Lemma 3.5, we conclude that u n scatters. However, this contradicts the definition of { ϕ n } . Therefore, there exists j such that S Gω = lim sup n →∞ S ω ( ˜ ψ jn ). We may assume j = 1. The linear profiledecomposition as J = 1 and ˜ W n ∈ K + G,ω imply (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) L ∞ ( R : H ) → n → ∞ byLemma 2.1. Therefore, we see that ϕ n = ˜ ψ n + ˜ W n , (cid:13)(cid:13)(cid:13) ˜ W n (cid:13)(cid:13)(cid:13) L ∞ ( R : H ) → , S Gω = lim n →∞ S ω ( ˜ ψ n ) . We assume that there exists G ( G such that x n = G x n for all G ∈ G and | x n − G x n | → ∞ for all G ∈ G \ G . Let U be a global solution of (NLS) with theinitial data ψ / ( G/ G ) if t n = 0 or the final data ψ / ( G/ G ) if | t n | → ∞ .Then, by the definition of S G ω , U belongs to L α ( R : L r ( R d )) since we have, byLemma A.3,lim n →∞ S ω ( ˜ ψ n ) = lim n →∞ G G S ω (cid:18) e it n ∆ ψ G/ G (cid:19) = S Gω < m Gω ≤ G G S G ω . By Lemma 3.5 again, this contradicts that u n does not belong to L α ( R : L r ( R d )).Thus, G = G . This means that ψ and W n are G -invariant, x n = G x n for all G ∈ G ,and we see that ϕ n = e it n ∆ τ x n ψ + W n . Let u c be a global solution of (NLS) with the initial data ψ if t n = 0 or the final data ψ if | t n | → ∞ . Then, u c is G -invariant. We prove k u c k L α ( R : L r ) = ∞ . Suppose that k u c k L α ( R : L r ) < ∞ . We observe that ϕ n − τ x n u c ( t n ) = e it n ∆ τ x n ψ − τ x n u c ( t n ) + W n , sothat we have (cid:13)(cid:13) e it ∆ (cid:0) ϕ n − τ x n u c ( t n ) (cid:1)(cid:13)(cid:13) L α ( R ; L r ) → n → ∞ . By Lemma 3.5, we see that u n ∈ L α ( R : L r ( R )) for large n , which is absurd. Thus,we get k u c k L α ( R : L r ) = ∞ . Moreover, we have S ω ( u c ) = lim n →∞ S ω ( e it n ψ ) = S Gω .Thus, we get a critical element u c . (cid:3) We say that the solution u is a forward critical element if u is a critical elementand satisfies k u k L α ([0 , ∞ ): L r ) = ∞ . In the same manner, we define a backward criticalelement. We only prove extinction of the forward critical element since that of thebackward critical element can be obtained by the similar argument based on timereversibility. The extinction contradicts Proposition 4.6. Lemma 4.7.
Let u be a forward critical element. There exists a continuous function x : [0 , ∞ ) → R d such that G x ( t ) = x ( t ) for all G ∈ G and { u ( t, · − x ( t )) : t ∈ [0 , ∞ ) } is precompact in H ( R d ) . The above lemma can be obtained by the same argument as in [12, Proposition3.2] noting u is G -invariant and { x n } , which appears in the profile decomposition,satisfies G x n = x n for all G ∈ G . Lemma 4.8.
Let u be a solution to (NLS) satisfying that there exists a continuousfunction x : [0 , ∞ ) → R d such that { u ( t, · − x ( t )) : t ∈ [0 , ∞ ) } is precompact in H ( R d ) . Then, for any ε > , there exists R = R ( ε ) > such that (4.20) Z | x + x ( t ) | >R |∇ u ( t, x ) | + | u ( t, x ) | + | u ( t, x ) | p +1 dx < ε for any t ∈ [0 , ∞ ) . It can be obtained by using directly the argument of [12, Corollary 3.3].
Lemma 4.9.
Let u be a forward critical element. Then, the momentum must be ,i.e. P ( u ) = 0 .Proof. First, we prove G P ( u ) = P ( u ) for all G ∈ G . By the G -invariance of u , wesee that P ( u ) = P ( G − u ) = Im Z R d e iθ u ( G x ) ∇{ e iθ u ( G x ) } dx = G Im Z R d u ( G x ) ∇ u ( G x ) dx = G Im Z R d u ( x ) ∇ u ( x ) dx = G P ( u ) . Therefore, the Galilean transformation u ξ ( t, x ) := e i ( x · ξ −| ξ | t ) u ( t, x − tξ ) , where ξ = − P ( u ) /M ( u ), conserves the G -invariance of the solution. The rest of theproof is same as in [12, Proposition 4.1] and [1, Proposition 4.1 (iii)]. (cid:3) We use the following lemma to prove the rigidity lemma, Lemma 4.11.
Lemma 4.10.
Let u be a solution to (NLS) on [0 , ∞ ) such that P ( u ) = 0 andthere exists a continuous x : [0 , ∞ ) → R d such that, for any ε > , there exists R = R ( ε ) > such that Z | x + x ( t ) | >R |∇ u ( t, x ) | + | u ( t, x ) | + | u ( t, x ) | p +1 dx < ε for any t ∈ [0 , ∞ ) . Then, we have x ( t ) t → as t → ∞ . This follows from [12, Lemma 5.1], [18, Proof of Theorem 7.1, Step1].
Lemma 4.11 (Rigidity) . If the solution u with G -invariance satisfies the followingproperties, then u = 0 . (1) u ∈ K + G,ω . (2) P ( u ) = 0 . (3) There exists a continuous x : [0 , ∞ ) → R d such that G x ( t ) = x ( t ) for all t ∈ [0 , ∞ ) and G ∈ G and, for any ε > , there exists R = R ( ε ) > suchthat Z | x + x ( t ) | >R |∇ u ( t, x ) | + | u ( t, x ) | + | u ( t, x ) | p +1 dx < ε for any t ∈ [0 , ∞ ) . LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 27
For the proof of Lemma 4.11, see [12, Theorem 6.1] and [18, Theorem7.1].Combining Lemmas 4.7, 4.8, and 4.9, the forward critical element satisfies theassumption (1)–(3) in Lemma 4.11. The result by Lemma 4.11 contradicts S ω ( u ) = S Gω >
0. This contradiction comes from the assumption S Gω < min { m Gω , l Gω } in Propo-sition 4.6. This means S Gω = min { m Gω , l Gω } , which completes the proof of Theorem 1.2(1) (i).5. Proof of the Scattering Result for the infinite group invariantsolutions
In this section, we prove Theorem 1.2 (1) (ii). Let a subgroup G of R / π Z × O ( d )be infinite and satisfy that the embedding H G ֒ → L p +1 ( R d ) is compact throughoutthis section.First, we prove that the sequence { ˜ x n } in (4.12) is bounded by a contradictionargument. We suppose that { ˜ x n } is unbounded. We may assume that | ˜ x n | → ∞ as n → ∞ . Since { ϕ n } is bounded in H , we can find ˜ ψ ∈ H G and ψ ∈ H ( R d ) suchthat e − i ˜ t n ∆ ϕ n ⇀ ˜ ψ in H ( R d ) ,e − i ˜ t n ∆ τ − ˜ x n ϕ n ⇀ ψ in H ( R d ) . Since the embedding H G ֒ → L p +1 ( R d ) is compact, we see that e − i ˜ t n ∆ ϕ n → ˜ ψ in L p +1 ( R d ). On the other hand, for any R >
0, we have e − i ˜ t n ∆ τ − ˜ x n ϕ n → ψ in L p +1 ( B R ),where B R is a ball of radius R centered at the origin. These limits imply that (cid:13)(cid:13)(cid:13) τ − ˜ x n ˜ ψ − ψ (cid:13)(cid:13)(cid:13) L p +1 ( B R ) ≤ (cid:13)(cid:13)(cid:13) τ − ˜ x n ˜ ψ − e − i ˜ t n ∆ τ − ˜ x n ϕ n (cid:13)(cid:13)(cid:13) L p +1 ( B R ) + (cid:13)(cid:13)(cid:13) e − i ˜ t n ∆ τ − ˜ x n ϕ n − ψ (cid:13)(cid:13)(cid:13) L p +1 ( B R ) → . Since | ˜ x n | → ∞ as n → ∞ , we have k τ − ˜ x n ψ k L p +1 ( B R ) → n → ∞ . Therefore, wesee that, for any R > k ψ k L p +1 ( B R ) ≤ (cid:13)(cid:13)(cid:13) τ − ˜ x n ˜ ψ − ψ (cid:13)(cid:13)(cid:13) L p +1 ( B R ) + k τ − ˜ x n ψ k L p +1 ( B R ) → . This means that ψ = 0. However, we have ψ = 0 by (4.12). This is a contradiction.Thus, we can take x n := 0 for all n ∈ N in the linear profile decomposition lemma.The rest of the proof is same as in the radial case. See [21] for details. Appendix A. Lemmas
Lemma A.1.
Let G be a subgroup of R / π Z × O ( d ) and { ˜ x n } be a sequence. Then,there exists a subgroup G ′ of G such that the sequence { ˜ x n − G ′ ˜ x n } is bounded for all G ′ ∈ G ′ and | ˜ x n − G ˜ x n | → ∞ as n → ∞ for all G ∈ G \ G ′ .Proof. It is easy to check that there exists a subset G ′ of G such that the sequence { ˜ x n − G ′ ˜ x n } is bounded for G ′ ∈ G ′ and the sequence { ˜ x n − G ˜ x n } is unbounded for G ∈ G \ G ′ . We prove that G ′ is a group. Let h G ′ i denote the group generatedby G ′ . For any G ∈ h G ′ i , there exist k ∈ N and {G ′ , · · · , G ′ k } ⊂ G ′ such that G = ( G ′ ) s ( G ′ ) s · · · ( G ′ k ) s k , where s j denotes either 1 or − j ∈ { , , · · · , k } . Bythe triangle inequality, we have | ˜ x n − G ˜ x n | = | ˜ x n − ( G ′ ) s ( G ′ ) s · · · ( G ′ k ) s k ˜ x n |≤ | ˜ x n − ( G ′ ) s ˜ x n | + | ( G ′ ) s ˜ x n − ( G ′ ) s ( G ′ ) s ˜ x n | + | ( G ′ ) s ( G ′ ) s ˜ x n − ( G ′ ) s ( G ′ ) s ( G ′ ) s ˜ x n | + · · · + | ( G ′ ) s ( G ′ ) s · · · ( G ′ k − ) s k − ˜ x n − ( G ′ ) s ( G ′ ) s · · · ( G ′ k ) s k ˜ x n |≤ | ˜ x n − ( G ′ ) s ˜ x n | + | ˜ x n − ( G ′ ) s ˜ x n | + | ˜ x n − ( G ′ ) s ˜ x n | + · · · + | ˜ x n − ( G ′ k ) s k ˜ x n | . Therefore, { ˜ x n − G ˜ x n } is bounded. This means that h G ′ i ⊂ G ′ and thus G ′ is agroup. (cid:3) Lemma A.2.
Let k ∈ N and A be a kd × d -matrix. We assume that a sequence { ˜ x n } ⊂ R d satisfies that there exists ¯ x ∈ R kd such that I ˜ x n − A ˜ x n → ¯ x where I is a kd × d -matrix such that I = I d I d ... I d k. Then, there exist { x n } ⊂ R d and x ∞ ∈ R d such that (cid:26) A x n = I x n ,x n − ˜ x n → x ∞ . Proof.
It is well known that there exist kd × kd -matrix P and d × d -matrix Q suchthat(A.1) P ( I − A ) Q = | {z } r a · · · a d − r ... ... a r · · · a rd − r | {z } d − r ) r } kd − r , where r = rank( I − A ). We set ˜ y n := Q − ˜ x n , B := P ( I − A ) Q , and ¯ y := P ¯ x . Then,we have B ˜ y n → ¯ y since |B ˜ y n − ¯ y | = |P ( I − A ) Q ˜ y n − P ¯ x | = |P ( I − A ) QQ − ˜ x n − P ¯ x |≤ |P|| ( I − A )˜ x n − ¯ x | → n → ∞ . In particular, for i ∈ { , , · · · , r } ,(A.2) | ˜ y in + ( a i ˜ y r +1 n + · · · + a id − r ˜ y dn ) − ¯ y i | → n → ∞ , where z j denotes the j -th component of z ∈ R d . We take { y n } ⊂ R d satisfying thefollowing properties. (cid:26) y in := − ( a i ˜ y r +1 n + · · · + a id − r ˜ y dn ) ,y jn := ˜ y jn , LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 29 for i ∈ { , , · · · , r } and j ∈ { r + 1 , r + 2 , · · · , d } . Then, we have B y n = 0 forall n ∈ N by the definition of { y n } . Moreover, we have for i ∈ { , , · · · , r } and j ∈ { r + 1 , r + 2 , · · · , d } ,˜ y in − y in = ˜ y in + ( a i ˜ y r +1 n + · · · + a id − r ˜ y dn ) − ( a i ˜ y r +1 n + · · · + a id − r ˜ y dn ) − y in | {z } =0 by the definition of y in → ¯ y i as n → ∞ . and ˜ y jn − y jn = 0. Therefore, d { kd − d { (cid:18) ˜ y n (cid:19) − (cid:18) y n (cid:19) → ¯ y as n → ∞ . Note that ¯ y r +1 = ¯ y r +2 = · · · = ¯ y kd = 0. Define x n := Q y n by (A.1). Then, we have0 = B y n = P ( I − A ) Q y n = P ( I − A ) x n . Multiplying P − from the left, we obtain A x n = I x n . Moreover,˜ x n − x n = Q (˜ y n − y n ) → Q ¯ y ...¯ y d =: x ∞ . This completes the proof. (cid:3)
Lemma A.3.
Let f ∈ H G ′ and { x n } satisfy | x n − G x n | → ∞ as n → ∞ for G ∈ G \ G ′ . We have the following identities. (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X G∈ G G ( τ x n f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H λ = G G ′ k G ′ f k H λ + o (1) , (A.3) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X G∈ G G ( τ x n f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p = G G ′ k G ′ f k pL p + o (1)(A.4) where λ ∈ [0 , , p ≥ and o (1) → as n → ∞ . In particular, the following identityholds for any ω > . S ω X G∈ G G ( τ x n f ) ! = G G ′ S ω ( G ′ f ) + o (1) . To prove Lemma A.3, we need Refined Fatou’s lemma. See [6] and [34, Theorem1.9].
Lemma A.4 (Refined Fatou’s lemma) . Let { f n } ⊂ L p ( R d ) with lim sup n →∞ k f n k L p < ∞ . If f n → f almost everywhere, then we have Z R d || f n | p − | f n − f | p − | f | p | dx → as n → ∞ . In particular, k f n k pL p − k f n − f k pL p → k f k pL p as n → ∞ . Proof of Lemma A.3.
Let {G k } G/ G ′ k =1 be the set of left coset representatives. First,we prove (A.3). (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X G∈ G G ( τ x n f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H λ = ( G ′ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) G/ G ′ X k =1 G k ( τ x n f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H λ = ( G ′ ) G/ G ′ X k =1 kG k ( τ x n f ) k H λ + G/ G ′ X k,l =1 k = l ( G k ( τ x n f ) , G l ( τ x n f )) ˙ H λ = G G ′ k G ′ f k H λ + ( G ′ ) G/ G ′ X k,l =1 k = l ( G k ( τ x n f ) , G l ( τ x n f )) ˙ H λ The second term tends to 0 as n → ∞ since | x n −G x n | → ∞ as n → ∞ for G ∈ G \ G ′ and G − k G l
6∈ G ′ for k = l . This implies (A.3). Next, we prove (A.4). Without loss ofgenerality, we may assume that G = (0 , I d ). (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X G∈ G G ( τ x n f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p = ( G ′ ) p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) G/ G ′ X k =1 G k ( τ x n f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p = ( G ′ ) p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ x n f + G/ G ′ X k =2 G k ( τ x n f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p = ( G ′ ) p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f + G/ G ′ X k =2 τ − x n + G k x n G k f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p . We set f n := f + P G/ G ′ k =2 τ − x n + G k x n G k f . Then, sup n ∈ N k f n k L p < ∞ and f n → f almost everywhere since | x n − G x n | → ∞ as n → ∞ for G ∈ G \ G ′ . Therefore,Refined Fatou’s lemma gives us (cid:13)(cid:13) f n (cid:13)(cid:13) pL p − (cid:13)(cid:13) f n − f (cid:13)(cid:13) pL p − k f k pL p → . LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 31
Here, f n − f = P G/ G ′ k =2 τ − x n + G k x n G k f and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) G/ G ′ X k =2 τ − x n + G k x n G k f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) G/ G ′ X k =2 τ G k x n G k f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ G x n G f + G/ G ′ X k =3 τ G k x n G k f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) G ( τ x n f ) + G/ G ′ X k =3 τ G k x n G k f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f + G/ G ′ X k =3 τ − x n + G − G k x n G − G k f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p Let f n := f + P G/ G ′ k =3 τ − x n + G − G k x n G − G k f . Then, sup n ∈ N k f n k L p < ∞ and f n → f almost everywhere since | x n − G x n | → ∞ as n → ∞ for G ∈ G \ G ′ . Therefore, byRefined Fatou’s lemma again, we get (cid:13)(cid:13) f n (cid:13)(cid:13) pL p − (cid:13)(cid:13) f n − f (cid:13)(cid:13) pL p − k f k pL p → . By repeating this procedure, we get (cid:13)(cid:13) f n (cid:13)(cid:13) pL p − (cid:13)(cid:13)(cid:13) f G/ G ′ − n − f (cid:13)(cid:13)(cid:13) pL p − ( G/ G ′ − k f k pL p → , where f G/ G ′ − n := f + τ − x n + G − G/ G ′− ···G − G G/ G ′ x n G − G/ G ′ − · · · G − G k f . Now, wehave (cid:13)(cid:13)(cid:13) f G/ G ′ − n − f (cid:13)(cid:13)(cid:13) pL p = (cid:13)(cid:13)(cid:13) τ − x n + G − G/ G ′− ···G − G G/ G ′ x n G − G/ G ′ − · · · G − G k f (cid:13)(cid:13)(cid:13) pL p = k f k pL p . Thus, we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X G∈ G G ( τ x n f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p = ( G ′ ) p (cid:13)(cid:13) f n (cid:13)(cid:13) pL p → ( G ′ ) p G G ′ k f k pL p . This completes the proof. (cid:3)
Lemma A.5.
Let { t n } ⊂ R , { x n } ⊂ R d , { f n } ⊂ H ( R d ) and ψ ∈ H ( R d ) \ { } satisfy f n ⇀ and e it n ∆ f n ( · + x n ) ⇀ ψ as n → ∞ in H ( R d ) . Then, | t n | → ∞ or | x n | → ∞ as n → ∞ taking a subsequence.Proof. If | t n | + | x n | is bounded, then f n ( · + x n ) ⇀ f n ⇀
0. This contradicts ψ = 0. See also [18, Lemma 5.3]. (cid:3) Lemma A.6 ([2, Proposition A.1]) . For j ∈ { , } , let V j ∈ C ( R : H ( R d )) ∩ L α ( R : L r ( R d )) and { ( t jn , x jn ) } n ∈ N ⊂ R × R d satisfy | t n − t n | → ∞ or | x n − x n | → ∞ as n → ∞ .Then, we have (cid:13)(cid:13) | V ( · − t n , · − x n ) | p − | V ( · − t n , · − x n ) | (cid:13)(cid:13) L β ′ ( R : L r ′ ) → . Proof.
First, we assume that | t n − t n | → ∞ . Then, since we have (cid:13)(cid:13) | V ( · − t n , · − x n ) | p − | V ( · − t n , · − x n ) | (cid:13)(cid:13) L β ′ ( R : L r ′ ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V ( · − t n ) (cid:13)(cid:13) p − L rx (cid:13)(cid:13) V ( · − t n ) (cid:13)(cid:13) L rx (cid:13)(cid:13)(cid:13) L β ′ t ( R ) , we obtain the statement. Next we assume that { t n − t n } is bounded and | x n − x n | → ∞ as n → ∞ . We note that (cid:13)(cid:13) | V ( · , · − x n ) | p − | V ( · + t n − t n , · − x n ) | (cid:13)(cid:13) L β ′ ( {| t | >T } : L r ′ ) ≤ (cid:13)(cid:13) V (cid:13)(cid:13) p − L α ( {| t | >T } : L r ) (cid:13)(cid:13) V ( · + t n − t n ) (cid:13)(cid:13) L α ( {| t | >T } : L r ) → T → ∞ . Thus, it suffices to prove that(A.5) (cid:13)(cid:13) | V ( · , · − x n ) | p − | V ( · + t n − t n , · − x n ) | (cid:13)(cid:13) L β ′ ( {| t |≤ T } : L r ′ ) → n → ∞ , for any fixed T >
0. We have, for any t , (cid:13)(cid:13) | V ( t, · − x n ) | p − | V ( t + t n − t n , · − x n ) | (cid:13)(cid:13) L r ′ = (cid:13)(cid:13) | V ( t ) | p − | V ( t + t n − t n , · + x n − x n ) | (cid:13)(cid:13) L r ′ By the Sobolev embedding, we have | V ( t ) | p − ∈ L rp − for any t . Moreover, we seethat { V ( t + t n − t n , x ) : x ∈ R d } is compact in L r ( R d ) for any t since { t n − t n } isbounded. Thus, by | x n − x n | → ∞ , we see that (cid:13)(cid:13) | V ( t ) | p − | V ( t + t n − t n , · + x n − x n ) | (cid:13)(cid:13) L r ′ → n → ∞ for any t. On the other hand, since { t n − t n } is bounded and V j belongs to C ( R : H ( R d )) for j ∈ { , } , the Sobolev embbeding gives us thatsup t ∈ [ − T,T ] (cid:13)(cid:13) | V ( t ) | p − | V ( t + t n − t n , · + x n − x n ) | (cid:13)(cid:13) L r ′ ≤ sup t ∈ [ − T,T ] (cid:0)(cid:13)(cid:13) V ( t ) (cid:13)(cid:13) L r (cid:13)(cid:13) V ( t + t n − t n ) (cid:13)(cid:13) L r (cid:1) . sup t ∈ [ − T,T ] (cid:0)(cid:13)(cid:13) V ( t ) (cid:13)(cid:13) H (cid:13)(cid:13) V ( t + t n − t n ) (cid:13)(cid:13) H (cid:1) < C < ∞ . Therefore, by the Lebesgue dominated convergence theorem, we obtain (A.5). (cid:3)
Lemma A.7 ([2, Corollary A.2], [23, Lemma 5.5]) . For j ∈ { , , · · · , J } , let V j ∈ C ( R : H ( R d )) ∩ L α ( R : L r ( R d )) and { ( t jn , x jn ) } n ∈ N ⊂ R × R d satisfy | t jn − t hn | → ∞ or | x jn − x hn | → ∞ as n → ∞ if j = h . Then, we have lim sup n →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) J X j =1 V jn (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L α ( R : L r ) ≤ J X j =1 (cid:13)(cid:13) V j (cid:13)(cid:13) L α ( R : L r ) , LOBAL DYNAMICS OF SOL.S WITH GROUP INVARIANCE FOR NLS 33 where V jn ( t, x ) := V j ( t + t jn , x − x jn ) .Proof. We have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) J X j =1 V jn (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L α ( R : L r ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J X j =1 V jn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p L β ′ ( R : L r ′ ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J X j =1 V jn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p − J X j =1 | V jn | p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L β ′ ( R : L r ′ ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) J X j =1 | V jn | p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L β ′ ( R : L r ′ ) p ≤ p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J X j =1 V jn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p − J X j =1 | V jn | p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p L β ′ ( R : L r ′ ) + 2 p J X j =1 (cid:13)(cid:13) V jn (cid:13)(cid:13) L α ( R : L r ) By the inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J X j =1 z j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p − J X j =1 | z j | p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C X j = k | z j || z k | p − , and Lemma A.6, we obtain the statement. (cid:3) Appendix B. Applications
We introduce some applications of Theorem 1.2 in this appendix. Here, we onlytreat examples in one and two dimensional cases.In the one dimensional case. We have only three subgroups of R / π Z × O (1) satis-fying the assumption (A). Namely, we have G := { (0 , } ,G even := { (0 , , (0 , − } ,G odd := { (0 , , ( π, − } . When G = G , we can classify the solutions with S ω < l ω into scattering and blow-upby the functional K by Theorem 1.1 ([18, 1]). Noting that Q ω is radially symmetric,we can classify the even solutions ( i.e. in the case of G = G even ) with S ω < l ω . Onthe other hand, by Theorem 1.2, we can classify the odd solutions ( i.e. G = G odd )with S ω < min { l ω , l G odd ω } = 2 l ω , where we have used S G ω = l ω and the fact that l G odd ω = 2 l ω . This means that we canclassify the solutions above the ground state standing waves by oddness.In the two dimensional case. Unlike the one dimensional case, we have many sub-groups in the two dimensional case. Here, we only introduce three applications.(1). We consider G = (cid:26)(cid:18) , (cid:18) (cid:19)(cid:19) , (cid:18) π, (cid:18) −
11 0 (cid:19)(cid:19) , (cid:18) π, (cid:18) − (cid:19)(cid:19) , (cid:18) , (cid:18) − − (cid:19)(cid:19)(cid:27) . The proper subgroups of G are as follows. G := (cid:26)(cid:18) , (cid:18) (cid:19)(cid:19)(cid:27) ,G := (cid:26)(cid:18) , (cid:18) (cid:19)(cid:19) , (cid:18) , (cid:18) − − (cid:19)(cid:19)(cid:27) . By Theorem 1.2, we classify the solutions with G -invariance satisfying S ω < min { l ω , l Gω } , where we note that G does not satisfy the condition ( ∗ ). Since l Gω > l ω , we canclassify the solutions above the ground state standing waves by the group invariance.(2). We consider G = (cid:26)(cid:18) , (cid:18) (cid:19)(cid:19) , (cid:18) π, (cid:18) − (cid:19)(cid:19)(cid:27) . By Theorem 1.2, we classify the solutions with G -invariance satisfying S ω < min { l ω , l Gω } = 2 l ω , where we have used the fact that l Gω = 2 l ω .(3). We consider G = (cid:26)(cid:18) θ, (cid:18) cos θ − sin θ sin θ cos θ (cid:19)(cid:19) : θ ∈ [0 , π ) (cid:27) . Then, the embedding H G ֒ → L p +1 ( R d ) is compact since Gx has infinitely manyelements for x = 0. By Theorem 1.2, we can classify the solutions such that S ω < l Gω . Acknowledgement
The authors would like to express deep appreciation to Professor Kenji Nakanishiand Doctor Masahiro Ikeda for many useful suggestions, comments and constantencouragement. The author is supported by Grant-in-Aid for JSPS Research Fellow15J02570.
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Department of Mathematics, Graduate School of Science, Kyoto University, Ky-oto, Kyoto, 606-8502, Japan
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