Asymptotic behavior for the long-range nonlinear Schrödinger equation on star graph with the Kirchhoff boundary condition
Kazuki Aoki, Takahisa Inui, Hayato Miyazaki, Haruya Mizutani, Kota Uriya
aa r X i v : . [ m a t h . A P ] J un ASYMPTOTIC BEHAVIOR FOR THE LONG-RANGENONLINEAR SCHR ¨ODINGER EQUATION ON STAR GRAPHWITH THE KIRCHHOFF BOUNDARY CONDITION
KAZUKI AOKI, TAKAHISA INUI, HAYATO MIYAZAKI, HARUYA MIZUTANI,AND KOTA URIYA
Abstract.
We consider the cubic nonlinear Schr¨odinger equation on the stargraph with the Kirchhoff boundary condition. We prove modified scatteringfor the final state problem and the initial value problem. Moreover, we alsoconsider the failure of scattering for the Schr¨odinger equation with power-typelong-range nonlinearities. These results are extension of the results for NLSon the one dimensional Euclidean space.
Contents
1. Introduction 11.1. Background 11.2. Setting and notations 32. Main results 53. Preliminalies 64. Proof of the Main results 84.1. Final state problem 84.2. Initial value problem 114.3. Failure of the scattering for 1 ≤ p ≤ Introduction
Background.
We mainly consider the following cubic nonlinear Schr¨odingerequation on the star graph G with n -edges: i∂ t u + ∆ K u + λ | u | u = 0 , ( t, x ) ∈ I × G , (1.1)where I is a time interval, λ = ±
1, and ∆ K is the Laplacian with the Kirchhoffboundary condition on the star graph G . Recently, researches of dispersive equa-tions on metric graphs have attracted much attention. Roughly, the star graph is ametric graph such as in Figure 1 below. We give precise definition of the star graphand setting of the problem later. Date : June 26, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Schr¨odinger equation, star graph, long-range nonlinearity, modifiedscattering, failure of scattering.
O O
Figure 1. i∂ t u + ∆ u + λ | u | p u = 0 , ( t, x ) ∈ R × R , where p > λ = ±
1. It is well-known that p = 2 is the critical exponent inthe sense of the asymptotic behavior of the solutions. More precisely, if p >
2, thesolution to (1.2) scatters to a solution of the free Schr¨odinger equation [17]. In thiscase, the nonlinearity is called “short-range”. On the other hand, if 0 < p ≤
2, thesolutions to (1.2) do not scatter to free solutions. See for instance [4,5,13,15] and thereferences given there. In the critical case, i.e. p = 2, it is known that the solutionof (1.2) scatters to a free solution with a phase modification. This phenomena iscalled modified scattering. In the case of final state problem, Ozawa [14] showedthe modified scattering. In the case of initial value problem, Hayashi–Naumkin [8]showed the modified scattering. Several alternative proofs of the result [8] weregiven by Lindbald–Soffer [12], Kato–Pusateri [10], and Ifrim–Tataru [9]. Moreover,the modified scattering phenomena for the nonlinear Schr¨odinger equation on thehigher dimensional Euclidean space R d ( d = 2 ,
3) was also proved by e.g. Ginibre–Ozawa [6] and Hayashi–Naumkin [8]. (Note that the critical exponent is given by p = 2 /d in this case.)We consider the Schr¨odinger equation on the star graph with power-type non-linearity: i∂ t u + ∆ K u + λ | u | p u = 0 , t ∈ R , x ∈ G , (1.3)where λ = ± p >
0. Since the star graph is the connected half-lines (the stargraph with 2-edges is just a line), the critical exponent is expected to be p = 2 as thecase of the line R . Yoshinaga [18] proved that the solutions of (1.3) scatter to thefree solution when p >
2, whose argument is based on [17]. (See also [3, Remark3] for the precise statement.) Recently, the first, second, and fourth authors [3]proved the failure of scattering when 0 < p < p = 2. We will give modified scattering results for the final stateproblem and the initial value problem when p = 2. Our proofs are based on theargument of [8, 14]. Namely, we use the factorization formula, which is also calledthe Dollard decomposition, of the propagator e it ∆ K . To derive the factorizationformula, we apply the Fourier transform F with respect to ∆ K derived by Weder[16], which is an extension of the usual Fourier transform on the line R . We also havean interest in the asymptotic behavior of the solutions in the case of 1 ≤ p ≤
2. We
SYMPTOTIC BEHAVIOR OF THE SOLUTIONS TO NLS ON STAR GRAPH 3 will show that the scattering to the free solution fails when 1 ≤ p ≤ Setting and notations.
Before stating the main results, we give some nota-tions used in the main results and their proofs. A finite graph is a 4-tuple ( V, I , E , ∂ ),where V is the finite set of the vertices , I is the finite set of internal edges, E is thefinite set of external edges. A map ∂ is from I ∪ E to the set of vertices and orderedpairs of two vertices which satisfies ∂ ( i ) = ( v i , v i ) (possibly v i = v i , v i , v i ∈ V )for i ∈ I and ∂ ( e ) = v for e ∈ E . We call v i =: ∂ − ( i ) and v i =: ∂ + ( i ) initialand final vertex of the internal edge i ∈ I , respectively. We endow the graphwith the metric structure. We assume that for any internal edge i ∈ I there ex-ist a i > i [0 , a i ] corresponding ∂ − ( i ) to 0 and ∂ + ( i ) to a i andthat for any external edge e ∈ E there exists a map e [0 , ∞ ). We call a i thelength of the internal edge i ∈ I . The graph endowed with such metric structureis called metric graph. For given n ∈ N , a star graph with n -edges is a metricgraph ( { } , ∅ , { e j } nj =1 , ∂ : { e j } nj =1 → { } ). See Figure 1 for typical examples ofstar graph. Throughout the paper, let G = ( { } , ∅ , { e j } nj =1 , ∂ : { e j } nj =1 → { } ) bea star graph.A function f on G is given by a vector f = ( f , f , · · · , f n ) T , where each f j is a complex-valued function defined on e j = (0 , ∞ ) and f T denotes the trans-pose of f . We emphasize that the notation | f | for a function f on G does notmean ( P nj =1 | f j | ) / . In the paper, we regard | f | as ( | f j | ) ≤ j ≤ n . Moreover f g =( f j g j ) ≤ j ≤ n for functions f, g on G . That is, our calculation is component-wise.Especially, the nonlinearity | f | f is ( | f j | f j ) ≤ j ≤ n .The Lebesgue measure on G is naturally induced by the Lebesgue measure onhalf-lines e , · · · , e n . The function space L ( G ) is defined as the set of measurablefunctions which are square-integrable on each external edge of G . Namely,(1.4) L ( G ) = n M j =1 L ( e j )and the inner product and the norm are defined by h f, g i = h f, g i L ( G ) := n X j =1 h f j , g j i e j = n X j =1 Z e j f j ( x ) g j ( x ) dx, (1.5) k f k L ( G ) := h f, f i = n X j =1 Z e j | f j ( x ) | dx = n X j =1 k f j k L ( e j ) , (1.6)where f = ( f j ) Tj =1 , ··· ,n , g = ( g j ) Tj =1 , ··· ,n with f j , g j ∈ L ( e j ) for each j = 1 , · · · , n .Then L ( G ) is a Hilbert space. For 1 ≤ p ≤ ∞ , L p ( G ) can be defined similarly, i.e. f ∈ L p ( G ) if each component of f is L p function. The norm is defined by(1.7) k f k L p = k f k L p ( G ) := n X j =1 k f j k pL p ( e j ) p (1 ≤ p < ∞ ) , sup ≤ j ≤ n k f j k L ∞ ( e j ) ( p = ∞ ) . The norm of the weighted L space, which is denoted by H , ( G ), is defined by k f k H , = k f k H , ( G ) := k f k L ( G ) + k Xf k L ( G ) , K. AOKI, T. INUI, H. MIYAZAKII, H. MIZUTANI, AND K. where Xf = ( xf j ) Tj =1 , ··· ,n .For m = 1 ,
2, the Sobolev space H m ( G ) is defined by(1.8) H m ( G ) := n M j =1 H m ( e j )and the norm is defined by(1.9) k f k H m = k f k H m ( G ) := n X j =1 k f j k H m ( e j ) . We set Σ = Σ( G ) := H ( G ) ∩ H , ( G ) . We remark that we do not assume any conditions at the vertex. When we assumethe continuity at the vertex, we use the subscript c such as H mc ( G ) and Σ c ( G ).We introduce the Laplacian on the star graph with the Kirchhoff boundary. Let(1.10) A = − · · · − · · · · · · −
10 0 0 · · · , B = · · · · · · · · · · · · . Then, we define the Laplacian ∆ K as follows: D (∆ K ) := { f ∈ H ( G ) : Af (0) + Bf ′ (0+) = 0 } , (1.11) ∆ K f := ( f ′′ , f ′′ , · · · , f ′′ n ) , (1.12)where f ′ j ( x ) = ∂f j ∂x . This ∆ K is called the Laplacian on the star graph G withthe Kirchhoff boundary. In this case, the condition Af (0) + Bf ′ (0+) = 0 impliesthat f j (0) = f k (0) for all j, k ∈ { , , · · · , n } and P nj =1 f ′ j (0+) = 0. Since thereis no external interaction at the vertex, the Laplacian ∆ K is regarded as the freeLaplacian on the star graph. (See [2, 7, 11] for the definitions of other boundaries.)The Schr¨odinger propagator U ( t ) = e it ∆ K can be defined as the unitary operatoron L ( G ) by the Stone theorem.According to Weder [16], we define the Fourier transform F with respect to ∆ K and its inverse F − by F := ( F − − F + ) I n + 2 n F + J n , F − := F ∗ = ( F + − F − ) I n + 2 n F − J n , where I n is the identity matrix, J n is the matrix whose all elements are 1,[ F ± f ]( ξ ) := (2 π ) − / Z ∞ e ± ixξ f ( x ) dx and F ∗ denotes the adjoint of F . The Fourier transform with respect to moregeneral boundary is derived by Weder [16]. SYMPTOTIC BEHAVIOR OF THE SOLUTIONS TO NLS ON STAR GRAPH 5
We define a multiplier operator M and a dilation operator D on the star graphby M = M ( t ) := M ( t ) I n , D = D ( t ) := D ( t ) I n , where [ M ( t ) ϕ ]( x ) := e i | x | t ϕ ( x ) and [ D ( t ) ϕ ]( x ) := (2 it ) − / ϕ ( x/ t ) for t >
0. Wedenote those inverse operators by M − = M − I n = M ( − t ) , D − = D − I n , where [ M − ( t ) ϕ ]( x ) := e − i | x | t ϕ ( x ), and [ D − ( t ) ϕ ]( x ) := (2 it ) / ϕ (2 tx ) for t > A . B means that there exists a positive constant C such that A ≤ CB . Suchconstants may be different from line to line.2. Main results
In the paper, we only treat the positive direction in time for simplicity. The firstresult is the modified scattering for the final state problem for (1.1).
Theorem 2.1 (Modified scattering for final state problem) . Let / < α < / .There exists ε > with the following properties: For any ϕ ∈ H c ( G ) satisfying k ϕ k L ∞ ( G ) < ε , there exists T ∈ R and a unique solution u ∈ C ([ T, ∞ ); L ( G )) ∩ L (( T, ∞ ); L ∞ ( G )) of (1.1) satisfying (cid:13)(cid:13)(cid:13)(cid:13) u j ( t ) − it ) ϕ j (cid:16) x t (cid:17) exp (cid:18) i | x | t + i λ (cid:12)(cid:12)(cid:12) ϕ j (cid:16) x t (cid:17)(cid:12)(cid:12)(cid:12) log t (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( e j ) . t − α (2.1) for t ≥ T and each j = 1 , , ..., n . The second result is the modified scattering for the initial value problem. Wehave the small data global existence.
Theorem 2.2 (Global existence for initial value problem) . There exists ε > such that the following assertion holds: For any < ε ≤ ε and u ∈ Σ c ( G ) with k u k Σ ≤ ε , there exists a unique global solution u ∈ C ([0 , ∞ ) , Σ c ∩ L ∞ ) to (1.1) with u (0) = u satisfying k u ( t ) k L ∞ ≤ Cε (1 + | t | ) − (2.2) for any t ≥ . And, we obtain the following modified scattering result.
Theorem 2.3 (Modified scattering for initial value problem) . Let u ( t ) be a globalsolution with u (0) = u given by Theorem 2.2. Then, if k u k Σ ≤ ε , there exists aunique W ∈ L ∞ ( G ) ∩ L ( G ) such that (cid:13)(cid:13)(cid:13)(cid:13) ( F U ( − t ) u ) j ( t ) exp (cid:18) − i λ Z t | ( F u ) j | dττ (cid:19) − W j (cid:13)(cid:13)(cid:13)(cid:13) L ( e j ) ∩ L ∞ ( e j ) . εt − + δ (2.3) for any t ≥ , each j = 1 , , . . . , n , where δ is sufficiently small depending on ε .Moreover, it holds that there exists a real valued function Ψ ∈ L ∞ ( G ) such that (cid:13)(cid:13)(cid:13)(cid:13) λ Z t |F w ( τ ) | dττ − λ | W | log t − Ψ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( G ) . εt − + δ log t (2.4) K. AOKI, T. INUI, H. MIYAZAKII, H. MIZUTANI, AND K. for all t ≥ . Furthermore, the estimate (cid:13)(cid:13)(cid:13)(cid:13) ( F U ( − t ) u ) j ( t ) − exp (cid:18) i λ | W j | log t + i Ψ j (cid:19) W j (cid:13)(cid:13)(cid:13)(cid:13) L ( e j ) ∩ L ∞ ( e j ) . εt − + δ log t (2.5) is valid for any t ≥ and each j = 1 , , . . . , n . Thus, the asymptotic formula u ( t ) = 1(2 it ) W + (cid:16) x t (cid:17) exp (cid:18) i | x | t + i λ (cid:12)(cid:12)(cid:12) W + (cid:16) x t (cid:17)(cid:12)(cid:12)(cid:12) log t (cid:19) + O (cid:16) εt − + δ log t (cid:17) (2.6) holds for x ∈ G , where W + = W exp( i Ψ) . The last result is the failure of scattering for (1.3) when 1 ≤ p ≤ Theorem 2.4 (Failure of scattering) . Let ≤ p ≤ . If u is a forward-globalsolution of (1.3) satisfying u ∈ C ([0 , ∞ ) : Σ c ( G )) and (cid:13)(cid:13) e − it ∆ K u ( t ) − v + (cid:13)(cid:13) Σ( G ) → t → ∞ ) for some v + ∈ Σ c ( G ) , then v + ≡ . Preliminalies
In this section, we introduce some notations and prepare some lemmas. We givetheir proofs in Appendix A.First of all, we introduce the following factorization formula of U ( t ), which isvery useful to investigate the modified scattering. Proposition 3.1.
We have U ( t ) = MDFM . This is similar to the factorization formula of the usual Schr¨odinger propagator e it ∆ = M D F R M on the Euclidean space, where F R denotes the usual Fouriertransform on R .We define the co-Fourier transform F c by F c := ( F − + F + ) I n − n F + J n and its inverse F − c is given by F − c = ( F c ) ∗ = ( F + + F − ) I n − n F − J n . Then, wehave the following relation between F and F c . Lemma 3.2. If ϕ ∈ H c ( G ) , then we have X F − ϕ = i F − c ∂ x ϕ, F c ∂ x ϕ = iX F ϕ. We note that the above lemma does not hold for general ϕ ∈ H ( G ). Namely,we need to assume the continuity at the origin. On the other hand, in the followinglemma, we do not need to assume the continuity. Lemma 3.3.
For ϕ ∈ H , ( G ) , we have ∂ x ( F ϕ ) = − i F c ( Xϕ ) . Remark 3.1.
It is worth emphasizing that ∂ x F 6 = i F X unlike the usual Fouriertransform on R . SYMPTOTIC BEHAVIOR OF THE SOLUTIONS TO NLS ON STAR GRAPH 7
It is known by [16] that the Fourier transform F and the co-Fourier transform F c are unitary operators on L ( G ). Namely, we have the following. Lemma 3.4.
We have hF f, g i L ( G ) = (cid:10) f, F − g (cid:11) L ( G ) . Especially, kF f k L ( G ) = k f k L ( G ) . Moreover, similar statements hold for the co-Fourier transform F c . We have the Hausdorff–Young inequality for F and F c as follows. Lemma 3.5 (Hausdorff–Young inequality) . For ≤ p ≤ ∞ , it is valid that kF f k L p ( G ) . k f k L p ′ ( G ) , kF c f k L p ( G ) . k f k L p ′ ( G ) , where p ′ is the H¨older conjugate of p . We have the following decay estimate for
M − Lemma 3.6.
Let ≤ p ≤ ∞ and α ∈ [0 , / . For t > , we have k ( M − f k L p ( G ) . | t | − α (cid:13)(cid:13) X α f (cid:13)(cid:13) L p ( G ) , where M − means M − I n and X α := x α I n for x ≥ . Lemma 3.7.
Let f ∈ Σ c ( G ) . There exists a constant c > such that (cid:13)(cid:13) XU ( − t ) | f | f (cid:13)(cid:13) L ≤ c k f k L ∞ k XU ( − t ) f k L , (cid:13)(cid:13) ∂ x ( | f | f ) (cid:13)(cid:13) L ≤ c k f k L ∞ k ∂ x f k L . The Sobolev inequality holds on the star graph without the continuity at theorigin.
Lemma 3.8 (Sobolev embedding) . Let f ∈ H ( G ) . We have k f k L p ( G ) . k f k H ( G ) for ≤ p ≤ ∞ . By [7], we have the dispersive estimates and the Strichartz estimates of thepropagator U ( t ). We say that ( q, r ) is an admissible pair if it satisfies 2 ≤ q, r ≤ ∞ and 1 q = 12 (cid:18) − r (cid:19) . Lemma 3.9 ([7]) . The following dispersive estimate holds. k U ( t ) f k L p ′ ( G ) . | t | − p k f k L p ( G ) , where p ∈ [1 , and p ′ is the H¨older conjugate of p . Moreover, we have the Strichartzestimates: k U ( t ) f k L q ( R ; L r ( G )) . k f k L ( G ) , (cid:13)(cid:13)(cid:13)(cid:13)Z t U ( t − s ) F ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L q ( R ; L r ( G )) . k F k L ˜ q ′ ( R ; L ˜ r ′ ( G )) , where ( q, r ) and (˜ q, ˜ r ) are admissible pairs and q, r, ˜ q ′ , ˜ r ′ are the H¨older conjugateof q, r, ˜ q, ˜ r , respectively. K. AOKI, T. INUI, H. MIYAZAKII, H. MIZUTANI, AND K. Proof of the Main results
Final state problem.
In this section, we consider the final state problem.Namely, we will show Theorem 2.1. We define the function space X ρ by X ρ := { f ∈ C ([ T, ∞ ); L ( G )) : k f k X < ρ } for ρ >
0, where the norm is defined by k f k X = sup t ∈ [ T, ∞ ) t α k f ( t ) k Y ( t ) , k f ( t ) k Y ( t ) = k f ( t ) k L ( G ) + (cid:18)Z ∞ t k f ( s ) k L ∞ ( G ) ds (cid:19) , where 1 / < α < /
2. We define a function w on G for a given final data ϕ ∈ H , ( G ) by ( w ( t )) j = ϕ j exp (cid:18) i λ | ϕ j | log t (cid:19) for j = 1 , ..., n and we set u ap ( t ) := MD w ( t ). We will find the solution of the integral equation u = u ap + i Z ∞ t U ( t − τ ) { N ( u ) − N ( u ap ) } dτ (4.1) − i Z ∞ t (2 τ ) − U ( t − τ ) R N ( w ) dτ + R w =: Φ u ap ( u ) = Φ( u ) . where N ( u ) := − λ | u | u and R f := MDF ( M − F − f for a function f . To findthe solution, it is enough to show the functional Φ is a contraction mapping on X ρ . Remark 4.1.
Before starting the contraction argument, we give a rough sketchof the derivation of the functional (4.1). Now, by ∆ K U ( t ) = U ( t )∆ K and thedifferential equation (1.1), we have i∂ t ( F U ( − t ) u ( t )) = F U ( − t ) N ( u ) . (4.2)By the definition of w , we also have i∂ t w ( t ) = (2 t ) − N ( w ) , (4.3)where we note that the j -th component of the nonlinearity N is ( N ( w )) j = − λ | w j | w j .By (4.2), (4.3) and the factorization formula U ( t ) = MDFM , we have i∂ t ( F U ( − t ) u ( t ) − w ( t )) = F U ( − t ) { N ( u ) − MD (2 t ) − N ( w ) } (4.4) − (2 t ) − F U ( − t ) R N ( w ) . Now, by the vector formulation, we have ( MD (2 t ) − N ( w )) j = (2 t ) − M D ( N ( w )) j and, by the gauge invariance, we obtain (2 t ) − M D | w j | w j = | M Dw j | M Dw j .Therefore, we have MD (2 t ) − N ( w ) = N ( MD w ) . (4.5)Moreover, by the factorization formula, we also have F U ( − t ) u ( t ) − w ( t ) = F U ( − t ) { u ( t ) − MD w ( t ) } − F U ( − t ) R w , (4.6) SYMPTOTIC BEHAVIOR OF THE SOLUTIONS TO NLS ON STAR GRAPH 9
Combining (4.4), (4.5), and (4.6), we obtain i∂ t {F U ( − t )( u ( t ) − MD w ( t )) } = F U ( − t ) { N ( u ) − N ( MD w ) } (4.7) − (2 t ) − F U ( − t ) R N ( w ) + i∂ t ( F U ( − t ) R w ) . We may assume that u ( t ) − u ap ( t ) will be 0 at infinite time from the final state con-dition and F U ( − t ) R w is a remainder term from Lemma 3.6. Therefore, integrating(4.7) on [ t, ∞ ) and recalling u ap = MD w , we obtain F U ( − t )( u ( t ) − u ap ( t )) = i Z ∞ t F U ( − τ ) { N ( u ) − N ( u ap ) } dτ (4.8) − i Z ∞ t (2 τ ) − F U ( − τ ) R N ( w ) dτ − F U ( − t ) R w . Acting U ( t ) F − from the left, we obtain the integral equation v = Φ( v ).We show that Φ( v ) − u ap ∈ X ρ provided that v − u ap ∈ X ρ . We set K := Z ∞ t U ( t − τ ) { N ( v ) − N ( u ap ) } dτ,K := Z ∞ t (2 τ ) − U ( t − τ ) R N ( w ) dτ. By the triangle inequality, it is sufficient to estimate the X -norms of K , K , and R w . First, we estimate K . Now, the difference of the nonlinearity can be writtenby N ( v ) − N ( u ap ) = N ( v, u ap ) + N ( v, u ap ) such that | N ( v, u ap ) | . | v − u ap || u ap | and | N ( v, u ap ) | . | v − u ap | . Therefore, by the Strichartz estimate (see Lemma3.9), we obtain k K k Y ( t ) . (cid:13)(cid:13)(cid:13)(cid:13)Z ∞ t U ( t − τ ) N ( v, u ap ) dτ (cid:13)(cid:13)(cid:13)(cid:13) Y ( t ) + (cid:13)(cid:13)(cid:13)(cid:13)Z ∞ t U ( t − τ ) N ( v, u ap ) dτ (cid:13)(cid:13)(cid:13)(cid:13) Y ( t ) . k N ( v, u ap ) k L ( t, ∞ : L ( G )) + k N ( v, u ap ) k L ( t, ∞ : L ( G )) . (cid:13)(cid:13) | v − u ap || u ap | (cid:13)(cid:13) L ( t, ∞ : L ( G )) + (cid:13)(cid:13) | v − u ap | (cid:13)(cid:13) L ( t, ∞ : L ( G )) . It holds from the H¨older inequality, v − u ap ∈ X ρ , and k u ap k L ∞ = kD w k L ∞ . t − / k w k L ∞ that (cid:13)(cid:13) | v − u ap || u ap | (cid:13)(cid:13) L ( t, ∞ : L ( G )) . Z ∞ t k v ( τ ) − u ap ( τ ) k L ( G ) k u ap ( τ ) k L ∞ ( G ) dτ . ρ Z ∞ t τ − − α k w k L ∞ ( G ) dτ . ρ k ϕ k L ∞ ( G ) t − α . and k| v − u ap | k L ( t, ∞ : L ( G )) . (cid:18)Z ∞ t k v ( τ ) − u ap ( τ ) k L ∞ ( G ) k v ( τ ) − u ap ( τ ) k L ( G ) (cid:19) . (cid:18)Z ∞ t k v ( τ ) − u ap ( τ ) k L ∞ ( G ) dτ (cid:19) (cid:18)Z ∞ t k v ( τ ) − u ap ( τ ) k L ( G ) dτ (cid:19) . ρt − α (cid:18)Z ∞ t ρ τ − α dτ (cid:19) . ρ t − α . Thus we see that(4.9) k K k Y ( t ) . ρ k ϕ k L ∞ ( G ) t − α + ρ τ − α . Next, we estimate R w . By Lemmas 3.6 and 3.2, we have k R w k L = (cid:13)(cid:13) ( M − F − w ( t ) (cid:13)(cid:13) L . t − (cid:13)(cid:13) X F − w ( t ) (cid:13)(cid:13) L . t − (cid:13)(cid:13) F − c ∂ x w ( t ) (cid:13)(cid:13) L . t − k ∂ x w ( t ) k L . t − k ∂ x ϕ k L (1 + k ϕ k L ∞ log t ) . It holds from Lemmas 3.2 and 3.6 that k R w k L ∞ . t − (cid:13)(cid:13) ( M − F − w ( t ) (cid:13)(cid:13) L . t − t − + ε (cid:13)(cid:13)(cid:13) X − ε F − w ( t ) (cid:13)(cid:13)(cid:13) L . t − + ε (cid:13)(cid:13) (1 + X ) F − w ( t ) (cid:13)(cid:13) L . t − + ε ( k w ( t ) k L + (cid:13)(cid:13) F − c ∂ x w ( t ) (cid:13)(cid:13) L ) . t − + ε ( k ϕ k L + k ∂ x w ( t ) k L ) , where ε > (cid:18)Z ∞ t k R w k L ∞ ds (cid:19) / . (cid:18)Z ∞ t s − − ε ( k ϕ k L + k ∂ x w ( s ) k L ) ds (cid:19) / . t − + ε {k ϕ k L + k ∂ x ϕ k L (1 + k ϕ k L ∞ log t ) } . At last, we estimate K . By the Strichartz estimates, we get (cid:13)(cid:13)(cid:13)(cid:13)Z ∞ t (2 τ ) − U ( t − τ ) R N ( w ) dτ (cid:13)(cid:13)(cid:13)(cid:13) Y ( t ) . Z ∞ t τ − (cid:13)(cid:13) R N ( w ) (cid:13)(cid:13) L dτ. SYMPTOTIC BEHAVIOR OF THE SOLUTIONS TO NLS ON STAR GRAPH 11
Now, it holds from Lemmas 3.2 and 3.6 that(4.11) Z ∞ t τ − (cid:13)(cid:13) R N ( w ) (cid:13)(cid:13) L dτ = Z ∞ t τ − (cid:13)(cid:13) ( M − F − N ( w ) (cid:13)(cid:13) L dτ . Z ∞ t τ − − (cid:13)(cid:13) X F − N ( w ) (cid:13)(cid:13) L dτ . Z ∞ t τ − (cid:13)(cid:13) F − c ∂ x N ( w ) (cid:13)(cid:13) L dτ . Z ∞ t τ − k ∂ x N ( w ) k L dτ . Z ∞ t τ − k w k L ∞ k ∂ x w k L dτ . t − k ϕ k L ∞ k ∂ x ϕ k L (1 + k ϕ k L ∞ log t ) . Therefore, we see that k Φ( v ) − u ap k X . ρ k ϕ k L ∞ ( G ) + ρ T − α + T − + α k ∂ x ϕ k L (1 + k ϕ k L ∞ log T )+ T − + α k ϕ k L ∞ k ∂ x ϕ k L (1 + k ϕ k L ∞ log T ) . Choosing 1 / < α < / T large enough, and k ϕ k L ∞ sufficiently small, we see thatthe map Φ is the map onto X ρ . In the same manner, we are able to show that Φis the contraction map on X ρ . The Banach fixed point theorem implies that Φ hasa unique fixed point in X ρ , which is the solution to the final state problem. Thiscompletes the proof of Theorem 2.1.4.2. Initial value problem.
The strategy relies on the argument of Hayashi andNaumkin [8].Arguing as in the above, we have the following:
Lemma 4.1.
Let u , v ∈ Σ c ( G ) . Then it holds that (cid:13)(cid:13) XU ( − t ) (cid:0) | u | u − | v | v (cid:1)(cid:13)(cid:13) L . ( k u k L ∞ + k v k L ∞ ) ( k XU ( − t ) u k L + k XU ( − t ) v k L ) k u − v k L ∞ + (cid:16) k u k L ∞ + k v k L ∞ (cid:17) k XU ( − t )( u − v ) k L , (cid:13)(cid:13) ∂ x ( | u | u − | v | v ) (cid:13)(cid:13) L . ( k u k L ∞ + k v k L ∞ ) ( k ∂ x u k L + k ∂ x v k L ) k u − v k L ∞ + (cid:16) k u k L ∞ + k v k L ∞ (cid:17) k ∂ x ( u − v ) k L . Let us recall the local existence of solutions to (1.1). Fix t ∈ R . Set the functionspace Z ε,AT = { ϕ ∈ C ( I T , (Σ c ∩ L ∞ )( G )) ; k ϕ k Z ε,AT < ∞} equipped with k ϕ k Z = k ϕ k Z ε,AT := sup t ∈ I T (1 + | t | ) − Aε ( k ϕ k H + k U ( − t ) ϕ k H , )+ sup t ∈ I T (1 + | t | ) k ϕ k L ∞ for any ε > A >
0, where I T = [ t , t + T ]. Proposition 4.2 (Local existence of solutions) . Let ε > , A > , t ∈ R , and K > . Assume that u ∈ H c ( G ) satisfies U ( − t ) u ∈ H , ( G ) with k u k H + k U ( − t ) u k H , ≤ K . Then there exists T = T ( K ) > not depending on ε and A such that (1.1) has a unique solution u ∈ Z ε,AT with u ( t ) = u . Moreover thesolution satisfies k u k Z ε,AT ≤ B K (4.12) for some constants B > not depending on t , T , A and ε .Proof. Set a complete metric space Z M = Z ε,AT,M := { f ∈ L ∞ ( I T , (Σ c ∩ L ∞ )( G )) ; k f k Z ≤ M } equipped with the distance function d ( u, v ) = k u − v k Z , where the constant M > u ) = U ( t − t ) u − iλ Z tt U ( t − s )( | u | u )( s ) ds. We shall prove that Ψ( u ) ∈ Z M whenever u ∈ Z M . Using Lemma 3.7, one easilyhas (1 + | t | ) − Aε ( k Ψ( u ( t )) k H + k U ( − t )Ψ( u ( t )) k H , )(4.13) ≤ (1 + | t | ) − Aε ( k u k H + k U ( − t ) u k H , )+ C (1 + | t | ) − Aε Z tt k u ( s ) k L ∞ ( k u ( s ) k H + k U ( − s ) u ( s ) k H , ) ds ≤ K + C Z tt M (1 + | s | ) − Aε ds = K + CM I ε,A ( t ) , for any t ∈ I T , where I ε,A ( t ) := R tt (1 + | s | ) − Aε ds . Further, similarly to theabove, it follows from the dispersive estimate i.e. Lemma 3.9, Lemma 3.7, and, k f k L . k f k H , that | t | k Ψ( u ( t )) k L ∞ ≤ | t | k U ( t − t ) u k L ∞ + C | t | (cid:13)(cid:13)(cid:13)(cid:13) U ( t ) Z tt U ( − s )( | u | u )( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ≤ C k U ( − t ) u k L + C (cid:13)(cid:13)(cid:13)(cid:13)Z tt U ( − s )( | u | u )( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L ≤ C k U ( − t ) u k H , + C Z tt k u ( s ) k L ∞ k U ( − s ) u ( s ) k H , ds ≤ C k U ( − t ) u k H , + CM I ε,A ( t ) . By Proposition 3.8, we also estimate k Ψ( u ( t )) k L ∞ ≤ k U ( t − t ) u k L ∞ + C (cid:13)(cid:13)(cid:13)(cid:13) U ( t ) Z tt U ( − s )( | u | u )( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ≤ C k u k H + C Z tt k u ( s ) k L ∞ k u ( s ) k H ds ≤ C k u k H + CM I ε,A ( t ) . SYMPTOTIC BEHAVIOR OF THE SOLUTIONS TO NLS ON STAR GRAPH 13
These yield sup t ∈ I T (1 + | t | ) k Ψ( u ( t )) k L ∞ ≤ CK + CM I ε,A ( t + T )(4.14)Hence, by (4.13) and (4.14), we obtain k Ψ( u ) k Z ε,AT ≤ C K + CM I ε,A ( t + T ) ≤ C ε + CK I ε,A ( t + T ) , when taking M = 2 C K . Here, I ε,A ( t + T ) → T → A and ε .Hence Ψ( u ) ∈ Z M holds as long as T = T ( K ) > CK I ε,A ( t + T ) ≤ C K. By using Lemma 4.1 and similar argument to the above, it is possible to show thatΨ is a contraction mapping on Z M . Therefore, we obtain the solution to u = Ψ( u )by the contraction mapping principle. We can also obtain the continuity in timeand the uniqueness of the solution. We omit the details and complete the proof. (cid:3) Corollary 4.3.
Let
A > and t ∈ R . Then, there exists ε > such that thefollowing assertion holds: For < ε ≤ ε and u ∈ H c ( G ) satisfying U ( − t ) u ∈ H , ( G ) with k u k H + k U ( − t ) u k H , ≤ ε , there exists T = T ( ε ) > not depend-ing on A such that (1.1) has a unique solution u ∈ Z ε,AT with u ( t ) = u . Moreoverthe solution satisfies k u k Z ε,AT < ε . (4.15) Proof.
Taking ε > B < ε − / , we have B ε < ε / for any ε ∈ (0 , ε ],where B is the constant as in Proposition 4.2. Applying Proposition 4.2 as K = ε ,we obtain the statement. (cid:3) In what follows, we fix A = c , where c is given in Lemma 3.7. We show thefollowing estimate. Proposition 4.4 (Bootstrap estimate) . There exists ε > such that the followingholds: Let T > . If < ε ≤ ε , u ∈ H c ( G ) satisfies U ( − t ) u ∈ H , ( G ) with k u k H + k U ( − t ) u k H , ≤ ε , and the solution u ∈ Z ε,AT of (1.1) on [0 , T ) with u ( t ) = u satisfies k u k Z ε,AT ≤ ε , (4.16) then there exists a constant C independent of T and ε such that k u k Z ε,AT ≤ C ε. We give the proof of this proposition later. Once we obtain this estimate, The-orem 2.2 will be proven as follows.
Proof of Theorem 2.2.
Let ε be smaller than those in Corollary 4.3 and Proposi-tion 4.4 and satisfy C < ε − / , where C is the constant in Proposition 4.4. ByCorollary 4.3, we have a time T and the unique solution satisfying k u k Z ε,AT < ε . (4.17)We here denote the maximal existence time of its solution by T max ( u ) =: T max .First, we will show k u k Z ε,AT max ≤ ε (4.18) holds. If not, there exists T > k u k Z ε,AT = ε / . It follows fromProposition 4.4 and ε < ε that k u k Z ε,AT ≤ C ε < ε . This is a contradiction. Thus, k u k Z ε,AT max ≤ ε / holds.Next, let us prove T max = ∞ by a contradiction. Suppose that T max < ∞ . Takearbitrarily small δ >
0. We set T δ = T max − δ < T max . Then, by (4.18), we have k u k Z ε,ATδ ≤ ε . and thus k u ( T δ ) k H , c + k U ( − T δ ) u ( T δ ) k H , ≤ ε (1 + T δ ) Aε ≤ ε (1 + T max ) Aε =: K. We note that K is independent of δ . By Proposition 4.2 as t = T δ , we get thesolution ˜ u ( t ) on [ T δ , T δ + T ( K )]. Taking small δ > T δ + T ( K ) = T max − δ + T ( K ) >T max , since we may take δ < T ( K ). By the uniqueness and continuity in time, u ( t ) exists on [ T max , T max − δ + T ( K )]. This is a contradiction. The proof iscompleted. (cid:3) To prove Proposition 4.4, we need the following lemmas:
Lemma 4.5.
Let u ∈ C ( R , H c ( G )) and α ∈ [0 , / . Then, it holds that k u ( t ) k L ∞ . | t | − kF U ( − t ) u ( t ) k L ∞ + | t | − − α k U ( − t ) u ( t ) k H , for any t ≥ .Proof. By the factorization property of U ( t ), we have u ( t ) = U ( t ) U ( − t ) u ( t ) = MDF U ( − t ) u ( t ) + MDF ( M − U ( − t ) u ( t ) , which implies k u ( t ) k L ∞ ≤ kMDF U ( − t ) u ( t ) k L ∞ + kMDF ( M − U ( − t ) u ( t ) k L ∞ . | t | − / kF U ( − t ) u ( t ) k L ∞ + | t | − / kF ( M − U ( − t ) u ( t ) k L ∞ . Also, one sees from Lemma 3.6 and the H¨older inequality that kF ( M − U ( − t ) u ( t ) k L ∞ . k ( M − U ( − t ) u ( t ) k L . | t | − α (cid:13)(cid:13) X α U ( − t ) u ( t ) (cid:13)(cid:13) L . | t | − α k U ( − t ) u ( t ) k H , , where we use α < / (cid:3) We define the operator J and L by J = J ( t ) := U ( t ) XU ( − t ) , L = i∂ t + ∆ K . Lemma 4.6.
We have [ J , L ] := J L − LJ = 0 . SYMPTOTIC BEHAVIOR OF THE SOLUTIONS TO NLS ON STAR GRAPH 15
Proof.
Noting that U ( t ) = e it ∆ K and i∂ t U ( t ) = e it ∆ K ( − ∆ K ), we calculate LJ f = ( i∂ t + ∆ K )( U ( t ) XU ( − t ) f )= i∂ t ( U ( t ) XU ( − t ) f ) + U ( t )∆ K XU ( − t ) f = − ( U ( t )∆ K XU ( − t ) f ) + ( U ( t ) X ( i∂ t ) U ( − t ) f ) + U ( t )∆ K XU ( − t ) f = ( U ( t ) X ( i∂ t ) U ( − t ) f )= ( U ( t ) X (∆ K ) U ( − t ) f ) + ( U ( t ) XU ( − t )( i∂ t ) f )= U ( t ) XU ( − t )( i∂ t + ∆ K ) f = J L f. The proof is completed. (cid:3)
We divide the proof of Proposition 4.4 into three parts as follows:
Proposition 4.7.
There exists ε > such that the following assertion holds: If ε , u , and u satisfy the assumption in Proposition 4.4, then the estimate (1 + | t | ) − Aε ( k u ( t ) k H + k U ( − t ) u ( t ) k H , ) ≤ ε is valid for any t ∈ [0 , T ] .Proof. By Lemma 4.6, multiplying the equation (1.1) by the operator J ,( i∂ t + ∆ K )( J u ) = − λ J ( | u | u ) . The equation is rewritten as an integral equation J ( t ) u ( t ) = U ( t ) J (0) u + iλ Z t U ( t − s ) J ( s )( | u | u )( s ) ds. We see from (4.16) and Lemma 3.7 that k XU ( − t ) u ( t ) k L = kJ ( t ) u ( t ) k L ≤ k U ( t ) J (0) u k L + Z t (cid:13)(cid:13) J ( s )( | u | u )( s ) (cid:13)(cid:13) L ds ≤ k Xu k L + c Z t k u ( s ) k L ∞ k XU ( − s ) u ( s ) k L ds ≤ k Xu k L + c ε Z t (1 + | s | ) − k XU ( − s ) u ( s ) k L ds. We here note A = c . Hence, the Gronwall inequality gives us k XU ( − t ) u ( t ) k L ≤ k Xu k L (1 + | t | ) Aε , which yields (1 + | t | ) − Aε k XU ( − t ) u ( t ) k L ≤ k Xu k L (4.19)for any t ∈ [0 , T ]. Arguing as in the above, since k u ( t ) k H ≤ k U ( t ) u k H + Z t (cid:13)(cid:13) ( | u | u )( s ) (cid:13)(cid:13) H ds ≤ k u k H + c ε Z t (1 + | s | ) − k u ( s ) k H ds, we deduce that (1 + | t | ) − Aε k u ( t ) k H ≤ k u k H . (4.20)Collecting (4.19) and (4.20), one obtains the desired estimate. (cid:3) Proposition 4.8.
There exists ε ∈ (0 , such that the following assertion holds:If ε , u , and u satisfy the assumption in Proposition 4.4, then the estimate (1 + | t | ) k u ( t ) k L ∞ ≤ Cε is valid for any t ∈ [0 , T ] .Proof. Take ε < min { , α (3 A ) − } for some α ∈ (0 , / t ≤
1, by (4.20)and Proposition 3.8, we easily show(1 + | t | ) k u ( t ) k L ∞ ≤ C (1 + | t | ) + Aε k u k H ≤ C k u k H ≤ Cε.
We shall consider the case t ≥
1. Combining Lemma 4.5 with Proposition 4.7, oneobtains k u ( t ) k L ∞ ≤ C | t | − kF U ( − t ) u ( t ) k L ∞ + C | t | − − α k U ( − t ) u ( t ) k H , ≤ C | t | − kF U ( − t ) u ( t ) k L ∞ + Cε | t | − − α + Aε (4.21)Let us handle the first term in the above last line. It follows from (1.1) that i∂ t ( U ( − t ) u ) = − λU ( − t )( | u | u ) . (4.22)A computation shows U ( − t )( | u | u ) = M ( − t ) F − D − M ( − t )( | u | u )= M ( − t ) F − D − ( |M ( − t ) u | M ( − t ) u )= (2 t ) − M ( − t ) F − ( |D − M ( − t ) u | D − M ( − t ) u ) . Let v satisfy U ( t ) v = u , namely, v = U ( − t ) u . Then, D − M ( − t ) u = FM ( t ) v .Hence, we have U ( − t )( | u | u ) = (2 t ) − M ( − t ) F − ( |FM ( t ) v | FM ( t ) v )= (2 t ) − ( M ( − t ) − F − ( |FM ( t ) v | FM ( t ) v )+ (2 t ) − F − {|FM ( t ) v | FM ( t ) v − |F v | F v } + (2 t ) − F − ( |F v | F v ) . Therefore, plugging the above into (4.22) and taking F , it is deduced that i ( F v ) t + λ t − ( |F v | F v )= − λ t − F ( M ( − t ) − F − ( |FM ( t ) v | FM ( t ) v ) − λ t − {|FM ( t ) v | FM ( t ) v − |F v | F v } . We here define I ( t ) = F ( M ( − t ) − F − ( |FM ( t ) v | FM ( t ) v ) ,I ( t ) = |FM ( t ) v | FM ( t ) v − |F v | F v. Let us also introduce a phase translation as follows: w = F − ( B ( t ) F v ) , SYMPTOTIC BEHAVIOR OF THE SOLUTIONS TO NLS ON STAR GRAPH 17 where B ( t ) = B · · · B · · · · · · B n , B j ( t ) = exp (cid:18) − λ i Z t | ( F v ) j | dττ (cid:19) . Thus, we obtain i∂ t ( F w ) = − λ t − B ( t )( I + I )( t ) . Integrating the above on [1 , t ], noting that F w (1) = F v (1) = F U ( − u (1), one has F w ( t ) = F U ( − u (1) + λ i Z t B ( τ )( I + I )( τ ) dττ . (4.23)We shall estimate I . By means of Lemma 3.6 and Sobolev and Hausdorff–Younginequalities, one sees that k I ( t ) k L ∞ = (cid:13)(cid:13) F ( M ( − t ) − F − ( |FM ( t ) v | FM ( t ) v ) (cid:13)(cid:13) L ∞ . (cid:13)(cid:13) ( M ( − t ) − F − ( |FM ( t ) v | FM ( t ) v ) (cid:13)(cid:13) L . t − α (cid:13)(cid:13) (1 + X ) F − ( |FM ( t ) v | FM ( t ) v ) (cid:13)(cid:13) L . t − α kFM ( t ) v k L ∞ kFM ( t ) v k L + t − α (cid:13)(cid:13) F − c ∂ x ( |FM ( t ) v | FM ( t ) v ) (cid:13)(cid:13) L . t − α kFM ( t ) v k H kFM ( t ) v k L + t − α k ∂ x FM ( t ) v k L kFM ( t ) v k L . t − α k v k H , . (4.24)Further, the estimation of I is as follows: k I ( t ) k L ∞ = (cid:13)(cid:13) |FM ( t ) v | FM ( t ) v − |F v | F v (cid:13)(cid:13) L ∞ . (cid:16) kFM ( t ) v k L ∞ + kF v k L ∞ (cid:17) kFM ( t ) v − F v k L ∞ . k v k L k ( M ( t ) − v k L . t − α k v k H , . (4.25)Moreover, we have kF U ( − u (1) k L ∞ ≤ k U ( − u (1) k L ≤ k U ( − u (1) k H , ≤ Cε by Proposition 4.7. Combining these above with Proposition 4.7, we reach to kF U ( − t ) u ( t ) k L ∞ = kF w ( t ) k L ∞ ≤ kF U ( − u (1) k L ∞ + C Z t τ − α − k U ( − τ ) u ( τ ) k H , dτ ≤ Cε + Cε Z t τ − α − Aε dτ ≤ C ε ,α ε (4.26) for any | t | ≥
1. Therefore collecting (4.21) and (4.26), we conclude(1 + | t | ) k u ( t ) k L ∞ ≤ C (1 + t − α + Aε ) ε ≤ Cε for any t ≥
1. This completes the proof. (cid:3)
Proof of Proposition 4.4.
The consequence immediately follows from Proposition4.8 and Proposition 4.7. (cid:3)
Let us move on to the proof of Theorem 2.3.
Proof of Theorem 2.3.
We shall first show (2.3). Combining (4.23) with (4.24) and(4.25), we see from Proposition 4.7 that kF w ( t ) − F w ( s ) k L ∞ ≤ C Z ts ( k I ( τ ) k L ∞ + k I ( τ ) k L ∞ ) dττ ≤ C Z ts τ − α − k v ( τ ) k H , dτ ≤ Cε Z ts τ − α − Aε dτ ≤ Cε (cid:0) t − α +3 Aε + s − α +3 Aε (cid:1) → t , s → ∞ . Using Lemmas 3.6 and 3.7, one also estimates k I ( t ) k L . t − (cid:13)(cid:13) X F − ( |FM ( t ) v | FM ( t ) v ) (cid:13)(cid:13) L . t − (cid:13)(cid:13) F − c ∂ x ( |FM ( t ) v | FM ( t ) v ) (cid:13)(cid:13) L . t − k v k H , , k I ( t ) k L . (cid:16) kFM ( t ) v k L ∞ + kF v k L ∞ (cid:17) kF ( M ( t ) − v k L . t − k v k H , . Hence it holds that kF w ( t ) − F w ( s ) k L ≤ C Z ts ( k I ( τ ) k L + k I ( τ ) k L ) dττ ≤ C Z ts τ − k v ( τ ) k H , dτ ≤ Cε (cid:16) t − +3 Aε + s − +3 Aε (cid:17) → t , s → ∞ . Therefore, there exists W ∈ L ( G ) ∩ L ∞ ( G ) such that kF w ( t ) − W k L ∩ L ∞ → t → ∞ . Taking s → ∞ , (4.27) and (4.28) reach to kF w ( t ) − W k L ∩ L ∞ ≤ Cεt − α +3 Aε , (4.29)since α < /
4, which implies kB ( t ) F U ( − t ) u ( t ) − W k L ∩ L ∞ ≤ Cεt − α +3 Aε . (4.30)Thus we have (2.3). Let us prove (2.4). We here defineΘ( t ) := λ Z t (cid:0) |F w ( τ ) | − |F w ( t ) | (cid:1) dττ . SYMPTOTIC BEHAVIOR OF THE SOLUTIONS TO NLS ON STAR GRAPH 19
A direct computation showsΘ( s ) − Θ( t ) = λ Z st (cid:0) |F w ( τ ) | − |F w ( s ) | (cid:1) dττ + λ (cid:0) |F w ( t ) | − |F w ( s ) | (cid:1) log t (4.31)for any 1 < t < τ < s . Since we see from (4.26) and (4.27) that (cid:12)(cid:12) |F w ( τ ) | − |F w ( s ) | (cid:12)(cid:12) ≤ ( |F w ( τ ) | + |F w ( s ) | ) |F w ( τ ) − F w ( s ) |≤ Cε (cid:0) τ − α +3 Aε + s − α +3 Aε (cid:1) ≤ Cετ − α +3 Aε for any τ < s , (4.31) implies that | Θ( s ) − Θ( t ) | ≤ Cε Z st τ − α +3 Aε − dτ + Cεt − α +3 Aε log t ≤ Cε (cid:0) t − α +3 Aε − s − α +3 Aε (cid:1) + Cεt − α +3 Aε log t (4.32)for any t < s . Hence there exists a real valued fonction Ψ ∈ L ∞ ( G ) such that k Θ( t ) − Ψ k L ∞ → t → ∞ . As for (4.32), taking s → ∞ , k Ψ − Θ( t ) k L ∞ ≤ Cεt − α +3 Aε log t. (4.33)By the definition of Θ( t ), we have λ Z t |F w ( τ ) | dττ − λ | W | log t − Ψ= (Θ( t ) − Ψ) + λ (cid:0) |F w ( t ) | − | W | (cid:1) log t. Thus, one sees from (4.26), (4.29), and (4.33) that (cid:13)(cid:13)(cid:13)(cid:13) λ Z t |F w ( τ ) | dττ − λ | W | log t − Ψ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ≤ k Θ( t ) − Ψ k L ∞ + C ( kF w ( t ) k L ∞ + k W k L ∞ ) kF w ( τ ) − W k L ∞ log t ≤ Cεt − α +3 Aε log t, where we note that k W k L ∞ ≤ Cε by (4.26) and (4.29). Thus (2.4) has been proven.Further, noting | − e iθ | . | θ | for any θ ∈ R , combining (4.30) with (2.4), it holdsthat (cid:13)(cid:13)(cid:13)(cid:13) ( F v ) j ( t ) − exp (cid:18) i λ | W j | log t + i Ψ j (cid:19) W j (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ∩ L ( e j ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) ( F v ) j ( t ) − exp (cid:18) λ i Z t | ( F v ) j | dττ (cid:19) W j (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ∩ L ( e j ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:26) exp (cid:18) i λ | W j | log t + i Ψ j − λ i Z t | ( F v ) j | dττ (cid:19) − (cid:27) W j (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ∩ L ( e j ) ≤ Cεt − α +3 Aε + C (cid:13)(cid:13)(cid:13)(cid:13) λ | W j | log t + i Ψ j − λ i Z t | ( F v ) j | dττ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( e j ) k W j k L ∞ ∩ L ( e j ) ≤ Cεt − α +3 Aε + Cεt − α +3 Aε log t ≤ Cεt − α +3 Aε log t for any t ≥ j = 1 , , . . . , n . Hence we conclude (2.5). In terms ofthe asymptotic formula (2.6), from (2.5) and the estimate in Proposition 4.7, it isestablished that u ( t ) = MDF v ( t ) + MDF ( M − v ( t )= MD exp (cid:18) i λ | W | log t + i Ψ (cid:19) W + MD (cid:18) F v ( t ) − exp (cid:18) i λ | W | log t + i Ψ (cid:19) W (cid:19) + MDF ( M − v ( t )= 1(2 it ) W (cid:16) x t (cid:17) exp (cid:18) i | x | t + i λ (cid:12)(cid:12)(cid:12) W (cid:16) x t (cid:17)(cid:12)(cid:12)(cid:12) log t + i Ψ (cid:16) x t (cid:17)(cid:19) + O (cid:16) εt − − α +3 Aε log t (cid:17) + O (cid:16) εt − − α + Aε (cid:17) . This completes the proof. (cid:3)
Failure of the scattering for ≤ p ≤ . In this section, we show the failureof scattering for (1.3) when 1 < p ≤
2. Our proof is based on the standard argumentby Cazenave [5].To prove Theorem 2.4, we derive a contradiction supposing v + = 0. Set F ( ϕ ) := λ | ϕ | p ϕ . Let ϕ be as in [3, Lemma 8]. Namely, there exists δ > h F ( F v + ) , F ϕ i L ( e j ) < − δ . Multiplying w := e it ∆ K ϕ to (1.3), integrating it on e j ,and taking the summation for j , we obtain the following weak formula: i ddt h u, w i L ( G ) + h F ( u ) , w i L ( G ) = 0 . (4.34) Remark 4.2.
Before taking the summation, we have the term u j ( t, ∂ x w j ( t, ∂ x u j ( t, w j ( t, n X j =1 (cid:16) u j ( t, ∂ x w j ( t, ∂ x u j ( t, w j ( t, (cid:17) = 0since u and w satisfiy the Kirchhoff boundary condition.Now, setting e u = D − M − u , and e w = D − M − w , then we have h F ( u ) , w i L ( G ) = t − p h F ( e u ) , e w i L ( G ) since F is gauge invariant. We note that e w = FM ϕ by the factorization formulaof e it ∆ K .Taking real part and integrating (4.34) on time interval [ T, τ ], where T is largeenough, Z τT Re( i∂ t h u, w i L ( G ) ) dt & − Z τT t − p Re h F ( e u ) , e w i L ( G ) dt. (4.35)The left hand side is bounded for τ since the L -norms of u and w conserve (see[7]). Therefore, we will show that the right hand side is unbounded if v + = 0. Wehave the following lemma. SYMPTOTIC BEHAVIOR OF THE SOLUTIONS TO NLS ON STAR GRAPH 21
Lemma 4.9.
Let ≤ p ≤ . If u is a forward-global solution of (1.3) satisfying (cid:13)(cid:13) e − it ∆ K u ( t ) − v + (cid:13)(cid:13) Σ( G ) → t → ∞ ) for some v + ∈ Σ ( G ) , then F ( e u ) = F ( F v + ) + o (1) in L ( G ) as t → ∞ . (4.36) Proof.
By the H¨older inequality, we get k F ( e u ) − F ( F v + ) k L ≤ C k| e u − F v + | ( | e u | p + |F v + | p ) k L ≤ C k e u − F v + k L − p (cid:0) k e u k pL + kF v + k pL (cid:1) ≤ C k e u − F v + k H (cid:16)(cid:13)(cid:13) e − it ∆ K u − v + (cid:13)(cid:13) pL + k v + k pL (cid:17) . Since e − it ∆ K = M − F − D − M − , we obtain k e u − F v + k H ≤ C (cid:13)(cid:13) (1 + ∂ x ) F (cid:0) F − D − M − u − v + (cid:1)(cid:13)(cid:13) L ≤ C (cid:13)(cid:13) ( F − i F c X ) (cid:0) F − D − M − u − v + (cid:1)(cid:13)(cid:13) L ≤ C (cid:13)(cid:13) M − F − D − M − u − M − v + (cid:13)(cid:13) H , ≤ C (cid:13)(cid:13) e − it ∆ K u − v + (cid:13)(cid:13) H , + (cid:13)(cid:13)(cid:0) − M − (cid:1) v + (cid:13)(cid:13) H , . By the assumption, we have (cid:13)(cid:13) e − it ∆ K u − v + (cid:13)(cid:13) H , → t → ∞ . We also have k (1 −M − ) v + k H , → t → ∞ by the Lebesgue dominated convergence theorem.Thus, we get k e u − F v + k H → t → . Therefore, we find k F ( e u ) − F ( F v + ) k L → t → . This completes the proof. (cid:3)
By the same method as in [3, Lemma 4], we obtain e w = F ϕ + o (1) in L ∞ ( G ) as t → ∞ . Combining this with Lemma 4.9, we obtain h F ( e u ) , e w i L ( G ) = h F ( F v + ) , F ϕ i L ( G ) + o (1) as t → ∞ . Indeed, | h F ( e u ) , e w i L ( G ) − h F ( F v + ) , F ϕ i L ( G ) |≤ k F ( e u ) − F ( F v + ) k L k e w k L ∞ + k F ( F v + ) k L k e w − F ϕ k L ∞ → t → ∞ since k e w k L ∞ = kFM ϕ k L ∞ ≤ k ϕ k L and k F ( F v + ) k L = kF v + k L p +1 ≤k v + k L /p ≤ k v + k Σ . Moreover, by the definition of ϕ , there exists δ > h F ( e u ) , e w i L ( e j ) < − δ for large t > T . Thus, (4.35) implies that C & δ Z τT t − p dt → ∞ as τ → ∞ when p ≤
2. This derives a contradiction. We complete the proof ofTheorem 2.4.
Appendix A. Proof of Preliminaries
The Schr¨odinger evolution group with the Kirchhoff boundary condition is ob-tained by [1] (see also [7]) as follows: U ( t ) = e it ∆ K = ( U − t − U + t ) I n + 2 n U + t J n , (A.1)where [ U ± t f ]( x ) := Z ∞ √ πit e i | x ± y | t f ( y ) dy. This formula and the definition of the Fourier transform F imply the factorizationformula in Proposition 3.1 as follows. Proof of Proposition 3.1.
First of all, we have[ U ± t f ]( x ) = e i | x | t √ it √ π Z ∞ e ± iy x t e i | y | t f ( y ) dy = [ M D F ± M f ]( x ) . Therefore, it holds from (A.1), this factorization formula, and the definition of theFourier transform F that U ( t ) = e it ∆ K = MDFM . (cid:3) We will show Lemma 3.2, which follows from the following lemmas.
Lemma A.1.
For ϕ ∈ H ( R + ) , we have x F ± ϕ = ± i √ π ϕ (0) ± i F ± ( ∂ x ϕ ) and equivalently F ± ( ∂ x ϕ ) = ∓ ix F ± ϕ − √ π ϕ (0) . Proof.
By integration by parts, we see that x F ± ϕ = ∓ i √ π Z ∞ ∂ y ( e ± ixy ) ϕ ( y ) dy = ∓ i √ π (cid:2) e ± ixy ϕ ( y ) (cid:3) y = ∞ y =0 − ∓ i √ π Z ∞ e ± ixy ( ∂ x ϕ )( y ) dy = ± i √ π ϕ (0) ± i F ± ( ∂ x ϕ ) . This completes the proof. (cid:3)
SYMPTOTIC BEHAVIOR OF THE SOLUTIONS TO NLS ON STAR GRAPH 23
Proof of Lemma 3.2.
For ϕ ∈ H c ( G ), it holds from Lemma A.1 that( X F − ϕ ) j = (cid:18) X (cid:26) ( F + − F − ) I n + 2 n F − J n (cid:27) ϕ (cid:19) j = x ( F + − F − ) ϕ j + 2 n x F − n X k =1 ϕ k ! = i √ π ϕ j (0) + i F + ( ∂ x ϕ j ) − (cid:18) − i √ π ϕ j (0) − i F − ( ∂ x ϕ j ) (cid:19) + 2 n ( − i √ π n X k =1 ϕ k (0) − i F − n X k =1 ∂ x ϕ k ) = i ( ( F + + F − ) ∂ x ϕ j − n F − n X k =1 ∂ x ϕ k ) + 2 i √ π ϕ j (0) − n i √ π n X k =1 ϕ k (0) . By the continuity at the vertex of ϕ , the summation of the last two terms is zero.Thus, we obtain the desired equality. (cid:3) Next, we will show Lemma 3.3. We do not require the continuity of ϕ at theorigin in Lemma 3.3 and the following lemma. Lemma A.2.
For ϕ ∈ H , ( R + ) , we have ∂ x F ± ϕ = ± i F ± ( yϕ ) Proof.
We have ∂ x F ± ϕ = 1 √ π Z ∞ ( ± iy ) e ± ixy ϕ ( y ) dy = ± i F ± ( yϕ ) . (cid:3) Proof of Lemma 3.3.
It follows from Lemma A.2 that ∂ x F = ∂ x ( F − − F + ) I n + 2 n ∂ x F + J n = i (cid:26) ( −F − − F + ) XI n + 2 n F + XJ n (cid:27) = − i F c X. This completes the proof of Lemma 3.3. (cid:3)
It follows from [16] that the Fourier transform and co-Fourier transform areunitary on L ( G ). Proof of Lemma 3.4.
The result for F in Lemma 3.4 follows from [16]. We note that F c is the Fourier transform with respect to the Laplacian ∆ M defined as follows: D (∆ M ) := { f ∈ H ( G ) : Bf (0) + Af ′ (0+) = 0 } , ∆ M f := ( f ′′ , f ′′ , · · · , f ′′ n ) , where A and B are in (1.10). Thus, the general theory by [16] implies the desiredstatement. (cid:3) The Hausdorff–Young inequality follows immediately as follows.
Proof of Lemma 3.5.
As seen above, F is unitary in L ( G ). Since kF ± f k L ∞ (0 , ∞ ) ≤ k f k L (0 , ∞ ) , we have kF f k L ∞ ( G ) . k f k L ( G ) . Therefore, interpolation implies the desired esti-mate. Similar estimate for F c also holds. (cid:3) We show Lemma 3.6.
Proof of Lemma 3.6.
By the H¨older continuity of e ix , we have | e i | x | t − | . | t | − α | x | α for 0 ≤ α ≤ /
2. Thus, it holds that k ( M − f k L p ( G ) = n X j =1 k ( M − f j k L p (0 , ∞ ) . | t | − α n X j =1 (cid:13)(cid:13) | x | α f j (cid:13)(cid:13) L p (0 , ∞ ) = | t | − α (cid:13)(cid:13) X α f (cid:13)(cid:13) L p ( G ) . This completes the proof. (cid:3)
Proof of Lemma 3.7.
The first inequality follows from (cid:13)(cid:13) XU ( − t ) | f | f (cid:13)(cid:13) L = (cid:13)(cid:13) X F − D − M − | f | f (cid:13)(cid:13) L = (cid:13)(cid:13) F − c ∂ x (cid:0) D − M − | f | f (cid:1)(cid:13)(cid:13) L ≤ C (cid:13)(cid:13) D − t∂ x (cid:0) |M − f | M − f (cid:1)(cid:13)(cid:13) L ≤ C (cid:13)(cid:13) M − f (cid:13)(cid:13) L ∞ (cid:13)(cid:13) t∂ x ( M − f ) (cid:13)(cid:13) L ≤ c k f k L ∞ k XU ( − t ) f k L . The second inequality holds by the Leibniz rule. (cid:3)
At last, we give the proof of the Sobolev inequality.
Proof of Proposition 3.8.
Let f be a function on the star graph. Obviously we havethe following estimate. For x ≥ | f j ( x ) | = − Z ∞ x ( | f j ( y ) | ) ′ dy ≤ Z ∞ | f j ( y ) || f ′ j ( y ) | dy ≤ k f j k L (0 , ∞ ) k f ′ j k L (0 , ∞ ) ≤ k f k H ( G ) . Thus, we get k f j k L ∞ (0 , ∞ ) ≤ k f k H ( G ) . Therefore, k f k L ∞ ( G ) = max j =1 , ··· ,n k f j k L ∞ (0 , ∞ ) ≤ √ k f k H ( G ) . SYMPTOTIC BEHAVIOR OF THE SOLUTIONS TO NLS ON STAR GRAPH 25
For 2 ≤ p < ∞ , we obtain k f k pL p ( G ) ≤ n X j =1 k f j k L (0 , ∞ ) k f j k p − L ∞ (0 , ∞ ) ≤ p − k f k p − H ( G ) k f k L ( G ) ≤ p − k f k pH ( G ) Thus, we obtain the desired estimate. (cid:3)
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E-mail address : [email protected] Department of Mathematics, Graduate School of Science, Osaka University, Toy-onaka, Osaka 560-0043, Japan
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