Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation
Roberto de A. Capistrano Filho, Márcio Cavalcante, Fernando A. Gallego
aa r X i v : . [ m a t h . A P ] A ug FORCING OPERATORS ON STAR GRAPHS APPLIED FOR THE CUBIC FOURTHORDER SCHR ¨ODINGER EQUATION
ROBERTO A. CAPISTRANO–FILHO*, M ´ARCIO CAVALCANTE, AND FERNANDO A. GALLEGO
Abstract.
In a recent article “Lower regularity solutions of the biharmonic Schr¨odinger equation in aquarter plane”, to appear on Pacific Journal of Mathematics [15], the authors gave a starting point of thestudy on a series of problems concerning the initial boundary value problem and control theory of BiharmonicNLS in some non-standard domains. In this direction, this article deals to present answers for some questionsleft in [15] concerning the study of the cubic fourth order Schr¨odinger equation in a star graph structure G .Precisely, consider G composed by N edges parameterized by half-lines (0 , + ∞ ) attached with a commonvertex ν . With this structure the manuscript proposes to study the well-posedness of a dispersive modelon star graphs with three appropriated vertex conditions by using the boundary forcing operator approach .More precisely, we give positive answer for the Cauchy problem in low regularity Sobolev spaces. Wehave noted that this approach seems very efficient, since this allows to use the tools of Harmonic Analysis,for instance, the Fourier restriction method, introduced by Bourgain, while for the other known standardmethods to solve partial differential partial equations on star graphs are more complicated to capture thedispersive smoothing effect in low regularity. The arguments presented in this work have prospects to beapplied for other nonlinear dispersive equations in the context of star graphs with unbounded edges. Introduction
Quantum and metric graphs.
In mathematics and physics, a quantum graph is a linear network-shaped structure of vertices connected on edges (i.e., a graph), where a differential (or pseudo-differential)equation is posed on each edge, while in the case of each edge is equipped with a natural metric the graphis denoted as a metric graph. An example would be a power network consisting of power lines (edges)connected at transformer stations (vertices); the differential equations would be then the voltage along eachof the line and the boundary conditions for each edge equipped at the adjacent vertices ensuring that thecurrent added over all edges adds to zero at each vertex.Quantum graphs were first studied by Linus Pauling as models of free electrons in organic molecules inthe 1930s. They also appear in a variety of mathematical contexts, e.g. as model systems in quantum chaos,in the study of waveguides, in photonic crystals and in Anderson localization - is the absence of diffusion ofwaves in a disordered medium -, or as limit on shrinking thin wires. Quantum graphs have become prominentmodels in mesoscopic physics used to obtain a theoretical understanding of nanotechnology. Another, moresimple notion of quantum graphs was introduced by Freedman et al. in [25].Aside from actually solving the differential equations posed on a quantum graph for purposes of concreteapplications, typical questions that arise are those of well-posedness, controllability (what inputs have to beprovided to bring the system into a desired state, for example providing sufficient power to all houses ona power network) and identifiability (how and where one has to measure something to obtain a completepicture of the state of the system, for example measuring the pressure of a water pipe network to determinewhether or not there is a leaking pipe).1.2.
Nonlinear dispersive models on star graphs.
In the last years, the study of nonlinear dispersivemodels in a metric graph has attracted a lot of attention of mathematicians, physicists, chemists and en-gineers, see for details [9, 10, 13, 33, 34] and references therein. In particular, the framework prototype(graph-geometry) for description of these phenomena have been a star graph G , namely, on metric graphswith N half-lines of the form (0 , + ∞ ) connecting at a common vertex ν = 0, together with a nonlinearequation suitably defined on the edges such as the nonlinear Schr¨odinger equation (see Adami et al. [1, 2]and Angulo and Goloshchapova [3, 4]). We note that with the introduction of nonlinearities in the disper-sive models, the network provides a nice field, where one can look for interesting soliton propagation andnonlinear dynamics in general. A central point that makes this analysis a delicate problem is the presence of Mathematics Subject Classification.
Key words and phrases.
Unbounded star graphs structure, Fourth order Schr¨odinger equation, Boundary forcing operatorapproach, Boundary conditions.*Corresponding author: roberto.capistranofi[email protected] ; capistranofi[email protected]. a vertex where the underlying one-dimensional star graph should bifurcate (or multi-bifurcate in a generalmetric graph).Looking at other nonlinear dispersive systems on graphs structure, we have some interesting results. Forexample, related with well-posedness theory, the second author in [17], studied the local well-posedness forthe Cauchy problem associated to Korteweg-de Vries equation in a metric star graph with three semi-infiniteedges given by one negative half-line and two positives half-lines attached to a common vertex ν = 0 (the Y -junction framework). Another nonlinear dispersive equation, the Benjamin–Bona–Mahony (BBM) equation,is treated in [11, 36]. More precisely, Bona and Cascaval [11] obtained local well-posedness in Sobolev space H and Mugnolo and Rault [36] showed the existence of traveling waves for the BBM equation on graphs.Using a different approach Ammari and Cr´epeau [6] derived results of well-posedness and, also, stabilizationfor the Benjamin-Bona-Mahony equation in a star-shaped network with bounded edges.In this aspect, regarding control theory and inverse problems, let us cite some previous works. Ignat et al. in [30] worked on the inverse problem for the heat equation and the Schr¨odinger equation on a tree.Later on, Baudouin and Yamamoto [7] proposed a unified and simpler method to study the inverse problemof determining a coefficient. Results of stabilization and boundary controllability for KdV equation onstar-shaped graphs was also proved by Ammari and Cr´epeau [5] and Cerpa et al. [20, 21].We caution that this is only a small sample of the extant work on graphs structure for partial differentialequations.1.3. Presentation of the model.
Let us now present the equation that we will study in this paper. Thefourth-order nonlinear Schr¨odinger (4NLS) equation or biharmonic cubic nonlinear Schr¨odinger equation(1.1) i∂ t u + ∂ x u − ∂ x u = λ | u | u, have been introduced by Karpman [31] and Karpman and Shagalov [32] to take into account the role ofsmall fourth-order dispersion terms in the propagation of intense laser beams in a bulk medium with Kerrnonlinearity. Equation (1.1) arises in many scientific fields such as quantum mechanics, nonlinear optics andplasma physics, and has been intensively studied with fruitful references (see [8, 23, 31, 37, 38] and referencestherein).In the past twenty years such 4NLS have been deeply studied from different mathematical points ofview. For example, Fibich et al. [24] worked various properties of the equation in the subcritical regime, withpart of their analysis relying on very interesting numerical developments. The well-posedness and existenceof the solutions has been shown (see, for instance, [37, 38, 40, 41]) by means of the energy method, harmonicanalysis, etc.Recently, in [14], the first and the second authors worked with equation (1.1) with the purpose toobtain controllability results. More precisely, they proved that on torus T , the solution of the associatedlinear system (1.1) is globally exponential stable, by using certain properties of propagation of compactnessand regularity in Bourgain spaces. This property, together with the local exact controllability, ensures thatfourth order nonlinear Schr¨odinger is globally exactly controllable, we suggest the reader to see [14] for moredetails.Considering another domain instead of the torus T , the authors, in [15], considered the cubic fourthorder Schr¨odinger equation on the right half-line(1.2) i∂ t u + γ∂ x u + λ | u | u = 0 , ( t, x ) ∈ (0 , T ) × (0 , ∞ ) ,u (0 , x ) = u ( x ) , x ∈ (0 , ∞ ) ,u ( t,
0) = f ( t ) , u x ( t,
0) = g ( t ) t ∈ (0 , T ) , for γ, λ ∈ R . When γλ < γλ >
0, is called defocusing.In [15], Capistrano-Filho et al. consider γ = − f ( t ) and g ( t ) in the equation (1.2),precisely, by assuming ( u , f, g ) ∈ H s ( R + ) × H s +38 ( R + ) × H s +18 ( R + ) , they obtained local well-posedness on the Sobolev spaces H s ( R + ) for s ∈ [0 , ). For s > /
2, by the Sobolevembedding and the energy method one can easily show the local well-posedness in H s ( R + ), giving a startingpoint of the study on a series of problems concerning of the Biharmonic NLS on bounded domains or stargraphs.Due these results presented in this recent work, naturally, we should see what happens for the system(1.2) in star graph structure given by N unbounded edges (0 , ∞ ) connected with a common vertex ν = 0,where a function on the graph G is a vector u ( t, x ) = ( u ( t, x ) , u ( t, x ) , ..., u N ( t, x )). Thus, let us consider NLS ON GRAPHS 3 the fourth order nonlinear Sch¨odinger equation on G , given by(1.3) ( i∂ t u j − ∂ x u j + λ | u j | u j = 0 , ( t, x ) ∈ (0 , T ) × (0 , ∞ ) , j = 1 , , ..., Nu j (0 , x ) = u j ( x ) , x ∈ (0 , ∞ ) , j = 1 , , ..., N with initial conditions ( u (0 , x ) , u (0 , x ) , ..., u N (0 , x )). (0 , + ∞ ) ( , + ∞ ) ( , + ∞ ) ( , + ∞ ) ( , + ∞ ) Figure 1.
Star graph with 5 edgesTherefore, the first following natural question arise.
Problem A : Which are the boundary conditions that we can impose, at least mathematically acceptable,to ensure the well-posedness result for the system (1.3)?1.4. Choosing the boundary conditions and main result.
We are interested to prove the well-posednessof (1.3) with appropriate boundary condition, more precisely, we will solve (1.3) with the following boundaryconditions:(1.4) Type A : ( ∂ kx u ( t,
0) = ∂ kx u ( t,
0) = · · · = ∂ kx u N ( t, , k = 0 , t ∈ (0 , T ) , P Nj =1 ∂ kx u j ( t,
0) = 0 , k = 2 , t ∈ (0 , T ) , (1.5) Type B : ( ∂ kx u ( t,
0) = ∂ kx u ( t,
0) = · · · = ∂ kx u N ( t, , k = 2 , t ∈ (0 , T ) , P Nj =1 ∂ kx u j ( t,
0) = 0 , k = 0 , t ∈ (0 , T ) , and(1.6) Type C : ( ∂ kx u ( t,
0) = ∂ kx u ( t,
0) = · · · = ∂ kx u N ( t, , k = 0 , t ∈ (0 , T ) , P Nj =1 ∂ kx u j ( t,
0) = 0 , k = 1 , t ∈ (0 , T ) . These boundary conditions are motivated by the conservation of the mass. Let us denote the mass as E ( u ( t, x ) , u ( t, x ) , · · · , u N ( t, x )) = 12 N X j =1 Z ∞ | u j ( t, x ) | dx. Multiplying (1.3) by u j , taking the imaginary part, integrating by parts and using the initial conditions of(1.3), we can obtain the most basic energy identity, namely the L –energy, satisfying E ( u ( T, x ) , u ( T, x ) , · · · , u N ( T, x )) = − N X j =1 Z T Im( ∂ x u j ( t, u j ( t, dt + N X j =1 Z T Im( ∂ x u j ( t, ∂ x u j ( t, dt − E ( u (0 , x ) , u (0 , x ) , · · · , u N (0 , x )) . (1.7)Analyzing (1.7), we are interested in boundary conditions to the Cauchy problem (1.3) such that the righthand side of (1.7) ensures the conservation of the mass. In this sense, the boundary conditions (1.4), (1.5) and(1.6) are appropriated. Assuming one of the boundary conditions (1.4), (1.5) or (1.6) the mass is conserved,i.e., E ( u ( t, x ) , u ( t, x ) , · · · , u N ( t, x )) = E ( u (0 , x ) , u (0 , x ) , · · · , u N (0 , x )) . CAPISTRANO–FILHO, CAVALCANTE, AND GALLEGO
It is important to point out that the boundary conditions of types A , B or C are coherent with thestudy of biharmonic operator on L ( G ). More precisely, a simple calculation proves that the biharmonicoperator B := i∂ x : D ( B i ) ⊂ L ( G ) → L ( G ) , i = 1 , , , with the following domains D ( B ) = { H ( G ); ∂ kx u (0) = ∂ kx u (0) = · · · = ∂ kx u N (0) , k = 0 , N X j =1 ∂ kx u j (0) = 0 , k = 2 , } , D ( B ) = { H ( G ); ∂ kx u (0) = ∂ kx u (0) = · · · = ∂ kx u N (0) , k = 2 , N X j =1 ∂ kx u j (0) = 0 , k = 0 , } , or D ( B ) = { H ( G ); ∂ kx u (0) = ∂ kx u (0) = · · · = ∂ kx u N (0) , k = 0 , N X j =1 ∂ kx u j (0) = 0 , k = 1 , } , is self-adjoint. Then, by Stone’s Theorem (see e.g. [19]), B generates a linear group, denoted by e it∂ x thatsolves the linear problem ∂ t u ( x, t ) = i∂ x u ( x, t ) ,u (0 , x ) = u ∈ D ( B i ) ,u ∈ C ( R ; D ( B i )) ∩ C ( R ; L ( G )) i = 1 , . By using the Duhamel formula together with the fact that H ( G ) is a Banach algebra it is possible to showthat problem (1.3) is well posed in high regularity, precisely, in D ( B i ), i = 1 , Remarks 1.
The following remarks are now in order. • Considering the Schr¨odinger equation on a star graph G , the vertex condition Type A , when restricton the cases k = 0 and k = 1 , coincides with the classical Kirchhoff vertex condition. For this system,these conditions are rather natural in the context of water (and other fluids) waves, correspondingto continuity of the flow and flux balance. • In this direction, we cite a very interesting work of Gregorio and Mugnolo [27] that treated the bi-laplacian on star graphs and trees with bounded edges, more precisely, they given a characterizationof complete graphs in terms of the Markovian property of the semigroup generated by L ( G ) , thesquare of the discrete Laplacian acting on a connected discrete graph G . For a complete picture aboutstar graphs in unbounded edges, in the context of the Airy equation, we cite the work of Mugnolo etal. [35] . Therefore, this work gives an answer for the Problem A , in a star graph structure, when the boundaryconditions (1.4), (1.5) or (1.6) are considered. This problem was left as an open problem in [15]. Before toenunciate the principal result of this work, we will denote the classical Sobolev space on the star graph G by H s ( G ) = N M i =1 H s (0 , + ∞ ) , for s ≥ . With this notation, the main result of this work can be read as follows.
Theorem 1.1.
Let s ∈ [0 , ) . For given initial-boundary data ( u , u , ..., u N ) ∈ H s ( G ) satisfying type A , B or C vertex conditions, there exist a positive time T depending on P Nj =1 k u j k H s ( R + ) and a distributionalsolution u = ( u i ) Ni =1 ( t, x ) ∈ C ((0 , T ); H s ( G )) to (1.3) - (1.4) (or (1.3) - (1.5) or (1.3) - (1.6) ) satisfying u j ∈ C (cid:0) R + ; H s +38 (0 , T ) (cid:1) ∩ X s,b ( R + × (0 , T )) ,∂ x u j ∈ C (cid:0) R + ; H s +18 (0 , T ) (cid:1) ,∂ x u j ∈ C (cid:0) R + ; H s − (0 , T ) (cid:1) NLS ON GRAPHS 5 and ∂ x u j ∈ C (cid:0) R + ; H s − (0 , T ) (cid:1) , for some b ( s ) < and j = 1 , , ..., N . Moreover, the map ( u , u , ..., u N ) u is locally Lipschitzcontinuous from H s ( G ) to C (cid:0) (0 , T ); H s ( G ) (cid:1) . Heuristic of the paper and further comments.
In this work we prove the existence of solution tothe problem (1.3)-(1.4) (or (1.3)-(1.5) or (1.3)-(1.6)) on star graph structure G composed by N unboundededges. The proof of Theorem 1.1 will be divided in several steps. Initially, we recast the partial differentialequation in each edge for a full line with a forcing term, more precisely(1.8) ( i∂ t u j − ∂ x u j + λ | u j | u j = T j ( x ) h j ( t ) + T j ( x ) h j ( t ) , ( t, x ) ∈ (0 , T ) × R , j = 1 , , ..., Nu j (0 , x ) = e u j ( x ) , x ∈ R , j = 1 , , ..., N where T j and T j ( j = 1 , , ..., N ) are distributions supported in the negative half-line ( −∞ , h j , h j ( j = 1 , , ..., N ) are selected to ensure that the vertex conditions are satisfied and e u j ( x ) are extensions of u j ( j = 1 , , ..., N ) on full line satisfying k e u j k H s ( R ) ≤ k u j k H s ( R + ) . Upon constructing the solution ˜ u = (˜ u , ˜ u , ..., ˜ u N ) of (1.8), we obtain the solution u = ( u , u , ..., u N ) ofproblem (1.3) with appropriate boundary condition, by restriction, as u = u ( x, t ) = ( u , u , ..., u N ) (cid:12)(cid:12) x ∈G ,t ∈ (0 ,T ) := ( u | x ∈ R + ,t ∈ (0 ,T ) , u | x ∈ R + ,t ∈ (0 ,T ) , ..., u N | x ∈ R + ,t ∈ (0 ,T ) ) . Secondly, the solution of forced Cauchy problem (1.8) satisfying the vertex types A , B or C , is constructedusing the classical Fourier restriction method due Bourgain [12]. Finally, a fixed point argument ensures theproof of the Theorem 1.1.We present some comments about the relevance of the method used in this manuscript.i. It is important to point out that, in our knowledge, this work is the first one in a star graphs structure G composed by N unbounded edges by using boundary forcing operator approach introduced firstby Colliander and Kenig [22] and improved by Holmer [29].ii. The graph structure of this article is more complex than proposed in [17] in the following sense: Totreat the extended vectorial integral equation that solves system (1.3), considering N unboundededges with appropriated vertex conditions, is more delicate since the matrices associated with thisproblem have 2 N –order (see Section 4).iii. A more delicate question concerns here is the local well-posedness for the Cauchy problem (1.3) inlow regularity. To do this we need to use a dispersive approach instead of Semigroup theory, wherethe principal difficulty is to use the restriction Fourier method in the context of star graphs. Thismotivates us to solve the problem (1.3) by using this approach, since the Semigroup theory does notguarantee the lower regularity to solutions of (1.3).1.6. Organization of the article.
To end our introduction, we present the outline of the manuscript.Section 2 is devoted to present the notations, more precisely, the Sobolev spaces, the Bourgain spaces, theRiemann-Liouville fractional integral operator and the Duhamel boundary forcing operator associated of(4NLS), which are paramount to prove the main result of the article. In the section 3, we will give anoverview of the main estimates proved by the authors in [15]. With these two sections in hand, we are ableto prove Theorem 1.1, in several steps, in the Section 4. The Section 5 is devoted to prove an auxiliarylemma, which one is used in the proof of the main result of the article, namely, Theorem 1.1. Finally, at theend of the work, we present an Appendix A, which will we given a sketch of the proof of Theorem 1.1 withvertex conditions types B and C . 2. Preliminaries
This section is devoted to presenting the main notations, introducing the functions spaces used inthis work and the Duhamel boundary forcing operator associated with the fourth order linear Schr¨odingerequation.
CAPISTRANO–FILHO, CAVALCANTE, AND GALLEGO
Notations.
Let us fix a cut-off function ψ ( t ) := ψ such that ψ ∈ C ∞ ( R ), 0 ≤ ψ ≤ ψ ≡ , , ψ ≡ , for | t | ≥ , and, for T >
0, we denote ψ T ( t ) = T ψ ( tT ).Now, for s ≥
0, define the homogeneous L -based Sobolev spaces ˙ H s = ˙ H s ( R ) by natural norm k φ k ˙ H s = k| ξ | s ˆ ψ ( ξ ) k L ξ and the L -based inhomogeneous Sobolev spaces H s = H s ( R ) by the norm k φ k ˙ H s = k (1 + | ξ | ) s ˆ ψ ( ξ ) k L ξ , where ˆ ψ denotes the Fourier transform of ψ . The function f belongs to H s ( R + ), if thereexists F ∈ H s ( R ) such that f ( x ) = F ( x ) for x >
0, in this case we set k f k H s ( R + ) = inf F k F k H s ( R ) . On the other hand, for s ∈ R , f ∈ H s ( R + ) provided that there exists F ∈ H s ( R ) such that F is the extensionof f on R and F ( x ) = 0 for x <
0. In this case, we set k f k H s ( R + ) = inf F k F k H s ( R ) . For s <
0, we define H s ( R + ) as the dual space of H − s ( R + ). It is well known that H s ( R + ) = H s ( R + ) for − < s < .Finally, the sets C ∞ ( R + ) = { f ∈ C ∞ ( R ); supp f ⊂ [0 , ∞ ) } and C ∞ ,c ( R + ) are defined as the subset of C ∞ ( R + ), whose members have a compact support on (0 , ∞ ). We remark that C ∞ ,c ( R + ) is dense in H s ( R + )for all s ∈ R .2.2. Solution spaces.
Consider f ∈ S ( R ), let us denote by e f or F ( f ) the Fourier transform of f withrespect to both spatial and time variables e f ( τ, ξ ) = Z R e − ixξ e − itτ f ( t, x ) dxdt. Moreover, we use F x and F t to denote the Fourier transform with respect to space and time variables,respectively (also use b for both cases).In the 90’s Bourgain [12] established a form to show the well-posedness of some classes of dispersivesystems. Precisely, on the Sobolev spaces H s , for smaller values of s , with these new spaces, Bourgainshowed a smoothing property more suitable for solutions of these classes of dispersive equations.In our case, considering s, b ∈ R we present below the Bourgain spaces X s,b associated to the linearsystem of (1.3). The space will be a completion of S ′ ( R ) under the norm k f k X s,b = Z R h ξ i s h τ + ξ i b | e f ( τ, ξ ) | dξdτ, where h·i = (1 + | · | ) / .It is important to note that X s,b –space, with b > , is well-adapted to study the IVP of dispersiveequations. However, in the study of IBVP, the standard argument cannot be applied directly. This is due tothe lack of hidden regularity, more precisely, the control of (derivatives) time trace norms of the Duhamelparts requires to work in X s,b − type spaces for b < , since the full regularity range cannot be covered (seeLemma 3.6 inequality (3.7)).Considering the space denoted by Z s,b with the following norm k f k Z s,b ( R ) = sup t ∈ R k f ( t, · ) k H s ( R ) + X j =0 sup x ∈ R k ∂ jx f ( · , x ) k H s +3 − j ( R ) + k f k X s,b , our goal is to find solutions of the Cauchy problem (1.3).Here, we will consider the spatial and time restricted space of Z s,b ( R ) defined in the standard way asfollows Z s,b ((0 , T ) × R + ) = Z s,b (cid:12)(cid:12)(cid:12) (0 ,T ) × R + equipped with the norm k f k Z s,b ((0 ,T ) × R + ) = inf g ∈ Z s,b {k g k Z s,b : g ( t, x ) = f ( t, x ) on (0 , T ) × R + } . Riemann-Liouville fractional integral.
Before beginning our study of the Cauchy problem (1.3), inthis subsection, we just give a brief summary of the Riemann-Liouville fractional integral operator to makethe work complete. We suggest [15, 22, 29] for the reader to see the proofs and more details.Consider the function t + defined in the following way t + = t if t > , t + = 0 if t ≤ . NLS ON GRAPHS 7
The tempered distribution t α − Γ( α ) is defined as a locally integrable function by * t α − Γ( α ) , f + = 1Γ( α ) Z ∞ t α − f ( t ) dt, for Re α >
0. By integrating by parts, we have that(2.2) t α − Γ( α ) = ∂ kt t α + k − Γ( α + k ) ! , for all k ∈ N . Expression (2.2) allows to extend the definition of t α − Γ( α ) , in the sense of distributions, to all α ∈ C . For f ∈ C ∞ ( R + ), define I α f as I α f = t α − Γ( α ) ∗ f. Thus, for Re α >
0, follows that(2.3) I α f ( t ) = 1Γ( α ) Z t ( t − s ) α − f ( s ) ds and the following properties easily holds I f = f, I f ( t ) = Z t f ( s ) ds, I − f = f ′ and I α I β = I α + β . Duhamel boundary forcing operator.
Now, we present the Duhamel boundary forcing operator,which was introduced by Colliander and Kenig [22], in order to construct the solution to(2.4) i∂ t u − ∂ x u = 0 . For details about this subsection and for a well exposition about this topic, the authors suggest the followingreferences [15, 16, 18, 28] .Following [15], let us consider the oscillatory integral by(2.5) B ( x ) = 12 π Z R e ixξ e − iξ dξ, which one is the key to define the Duhamel boundary forcing operator. A change of variable and contourproves that B (0) = − i / π Γ (cid:0) (cid:1) . We will denote(2.6) M = 1 B (0)Γ(3 / . For f ∈ C ∞ ( R + ), define the boundary forcing operator L (of order 0) as(2.7) L f ( t, x ) := M Z t e i ( t − t ′ ) ∂ x δ ( x ) I − f ( t ′ ) dt ′ , where e it∂ x denotes the group associated to (2.4) given by e it∂ x ψ ( x ) = 12 π Z R e ixξ e − itξ ˆ ψ ( ξ ) dξ. By using the following properties of the convolution operator (for k = 1)(2.8) ∂ kx ( f ∗ g ) = ( ∂ lx f ) ∗ g = f ∗ ( ∂ kx g ) , k ∈ N , and the integration by parts in t ′ of (2.7), we get that(2.9) i L ( ∂ t f )( t, x ) = iM δ ( x ) I − f ( t ) + ∂ x L f ( t, x ) . Using (2.5) and, again, by change of variable, we have L f ( t, x ) = M Z t e i ( t − t ′ ) ∂ x δ ( x ) I − f ( t ′ ) dt ′ = M Z t B (cid:18) x ( t − t ′ ) / (cid:19) I − f ( t ′ )( t − t ′ ) / dt ′ . (2.10) CAPISTRANO–FILHO, CAVALCANTE, AND GALLEGO
We are now generalize the boundary forcing operator (2.7). For Re λ > − g ∈ C ∞ ( R + ),we define(2.11) L λ g ( t, x ) = " x λ − − Γ( λ ) ∗ L (cid:0) I − λ g (cid:1) ( t, · ) ( x ) , where ∗ denotes the convolution operator and x λ − − Γ( λ ) = ( − x ) λ − Γ( λ ) . In particular, for Re λ >
0, we have(2.12) L λ g ( t, x ) = 1Γ( λ ) Z ∞ x ( y − x ) λ − L (cid:0) I − λ g (cid:1) ( t, y ) dy. By using the property (2.8), for k = 4, and (2.9) give us(2.13) L λ g ( t, x ) = " x ( λ +4) − − Γ( λ + 4) ∗ ∂ x L (cid:0) I − λ g (cid:1) ( t, · ) ( x )= iM x ( λ +4) − − Γ( λ + 4) I − − λ g ( t ) + i Z ∞ x ( y − x ) ( λ +4) − Γ( λ + 4) L (cid:0) ∂ t I − λ g (cid:1) ( t, y ) dy, for Re λ > −
4, where M is defined as (2.6). From (2.9) and (2.11), we have( i∂ t − ∂ x ) L λ g ( t, x ) = iM x λ − − Γ( λ ) I − − λ g ( t ) , in the distributional sense. 3. Overview of the main estimates
With all the notations and spaces defined in Section 2, we present now the main estimates of this work,which are paramount to prove the main result of the article.3.1.
Estimates for the function spaces.
Concerning of the X s,b space, we have two properties which arepresented in the lemma below for the functions ψ ( t ) and ψ T , defined in (2.1). The first item can be foundin [39, Lemma 2.11] and the second one in Ginibre et al. [26]. The lemma can be read as follows: Lemma 3.1.
Let ψ ( t ) be a Schwartz function in time. Then, we have k ψ ( t ) f k X s,b . ψ,b k f k X s,b . Moreover, if − < b ′ < b ≤ or ≤ b ′ < b < , w ∈ X s,b and s ∈ R , thus k ψ T w k X s,b ′ ≤ cT b − b ′ k w k X s,b . Another result that state important properties of the Riemann-Liouville fractional integral operator isgiven below. The proof of this can be found in [29, Lemmas 2.1, 5.3 and 5.4].
Lemma 3.2. If f ∈ C ∞ ( R + ) , then I α f ∈ C ∞ ( R + ) , for all α ∈ C . Moreover, we have the following: (a) If ≤ Re α < ∞ and s ∈ R , then kI − α h k H s ( R + ) ≤ c k h k H s + α ( R + ) , where c = c ( α ) . (b) If ≤ Re α < ∞ , s ∈ R and µ ∈ C ∞ ( R ) , then k µ I α h k H s ( R + ) ≤ c k h k H s − α ( R + ) , where c = c ( µ, α ) . Estimates for the boundary forcing operator.
Now, let us be precisely when the boundary forcingoperator is continuous or discontinuous. Initially, we present the well-know properties of the spatial conti-nuity, the decay of the L λ g ( t, x ) and the explicit values for L λ f ( t, Lemma 3.3.
Let g ∈ C ∞ ( R + ) and M be as in (2.6) . Then, we have (3.1) L − k g = ∂ kx L I k g, k = 0 , , , . Moreover, for fixed ≤ t ≤ , ∂ kx L f ( t, x ) , k = 0 , , , is continuous in x ∈ R and L − g ( t, x ) is continuousin x ∈ R \ { } and has a step discontinuity at x = 0 . Lemma 3.4.
For Re λ > − and f ∈ C ∞ ( R + ) , we have the following value of L λ f ( t, : (3.2) L λ f ( t,
0) = M f ( t ) e − i π (1+3 λ ) + e − i π (1 − λ ) sin( − λ π ) ! . NLS ON GRAPHS 9
Energy and trilinear estimates.
In the last part, we present four lemmas related to energy andtrilinear estimates for the solutions of the 4NLS equation in the Bourgain spaces X s,b . These results andtheir proofs can also be found in [15, Section 4]. Lemma 3.5.
Let s ∈ R and b ∈ R . If φ ∈ H s ( R ) , then the following estimates holds (3.3) k ψ ( t ) e it∂ x φ ( x ) k C t (cid:0) R ; H sx ( R ) (cid:1) . ψ k φ k H s ( R ) , (3.4) k ψ ( t ) ∂ jx e it∂ x φ ( x ) k C x (cid:18) R ; H s +3 − j t ( R ) (cid:19) . ψ,s,j k φ k H s ( R ) j ∈ N and (3.5) k ψ ( t ) e it∂ x φ ( x ) k X s,b . ψ,b k φ k H s ( R ) . Lemma 3.6.
Let < b < and j = 0 , , , , we have the following inequalities (3.6) k ψ ( t ) D w ( t, x ) k C (cid:0) R t ; H s ( R x ) (cid:1) . k w k X s, − b , for s ∈ R ; (3.7) k ψ ( t ) ∂ jx D w ( t, x ) k C (cid:18) R x ; H s +3 − j ( R t ) (cid:19) . k w k X s, − b , for − + j < s < + j ; (3.8) k ψ ( t ) ∂ jx D w ( t, x ) k X s,b . k w k X s, − b , for s ∈ R . Lemma 3.7.
Let s ∈ R . Then, (a) For s − < λ < s and λ < the following inequality holds (3.9) k ψ ( t ) L λ f ( t, x ) k C (cid:0) R t ; H s ( R x ) (cid:1) ≤ c k f k H s +380 ( R + ) ;(b) For − j < λ < j , j = 0 , , , , we have (3.10) k ψ ( t ) ∂ jx L λ f ( t, x ) k C (cid:0) R x ; H s +3 − j ( R + t ) (cid:1) ≤ c k f k H s +380 ( R + ) ;(c) If s < − b , b < , − < λ < and s + 4 b − < λ < s + yields that (3.11) k ψ ( t ) L λ f ( t, x ) k X s,b ≤ c k f k H s +380 ( R + ) . Remarks 2.
Let us present two remarks. i. The previous estimates are the so-called space traces, derivative time traces and Bourgain spacesestimates, respectively. ii.
We observe that in [15] was obtained (3.4) , (3.7) and (3.10) , for j = 0 and j = 1 , but the result forall j can be obtained directly by using the fact that ∂ jx L λ = L λ − j ( I − j ) . To close this section, let us enunciate the trilinear estimates associated to fourth order nonlinearSchr¨odinger equation. The proof of this estimate can be found in [41].
Lemma 3.8.
For s ≥ , there exists b = b ( s ) < / such that (3.12) k u u u k X s, − b ≤ c k u k X s,b k u k X s,b k u k X s,b . Proof of the main result
The aim of this section is to prove the main result announced in the introduction of this work, Theorem1.1. Here, we only consider the vertex condition (1.4) (type A ) and to make the proof easy to understand,we will split it in several steps which will be divided into subsections. Additionally, the discussion of vertexconditions types B and C will be presented on Appendix A, at the end of the article. Obtaining a integral equation in L Ni =1 R . In this first step, we are interested in finding an extendedvectorial integral equation posed in L Ni =1 R , such that the restrictions of this equation on G will solve, inthe sense of distributions, the following Cauchy problem(4.1) ( i∂ t u j − ∂ x u j + λ | u j | u j = 0 , ( t, x ) ∈ (0 , T ) × (0 , ∞ ) , j = 1 , , ..., Nu j (0 , x ) = u j ( x ) , x ∈ (0 , ∞ ) , with initial conditions ( u , u , ..., u N ) ∈ H s ( G ). Let us begin rewriting the Type A vertex conditions (1.4)in terms of matrices. In this way, note that (1.4) is equivalent to ( ∂ kx u ( t,
0) = ∂ kx u ( t, , ∂ kx u ( t,
0) = ∂ kx u ( t, , · · · , ∂ kx u N − ( t,
0) = ∂ kx u N ( t, , k = 0 , , t ∈ (0 , T ) P Nj =1 ∂ kx u j ( t,
0) = 0 , k = 2 , , t ∈ (0 , T ) . Thus, we consider the following matrices(4.2) [ C ] N × N u ( t, u ( t, u ( t, u N − ( t, u N ( t, N × = N × ; [ C ] N × N ∂ x u ( t, ∂ x u ( t, ∂ x u ( t, ∂ x u N − ( t, ∂ x u N ( t, N × = N × and(4.3) [ C ] N × N ∂ x u ( t, ∂ x u ( t, ∂ x u ( t, ∂ x u N − ( t, ∂ x u N ( t, N × = N × ; [ C ] N × N ∂ x u ( t, ∂ x u ( t, ∂ x u ( t, ∂ x u N − ( t, ∂ x u N ( t, N × = N × , where [ C ] N × N := − · · · − · · · · · · −
10 0 0 · · · · · · | {z } N columns N − N + 1 rows [ C ] N × N := · · · · · · · · · − · · · − · · · · · · −
10 0 0 · · · · · · | {z } N columns N − N − (cid:27) NLS ON GRAPHS 11 [ C ] N × N := · · · · · · · · · · · · · · · | {z } N columns N − (cid:27) and [ C ] N × N := · · · · · · · · · · · · · · · · · · | {z } N columns N − (cid:27) . On the other hand, let be e u j an extension for all line R of u j , satisfying k e u j k H s ( R ) . k u j k H s ( R + ) , j = 1 , , ..., N, respectively. Initially, we look for solutions in the form u j ( t, x ) = L λ j γ j ( t, x ) + L λ j γ j ( t, x ) + F j ( t, x ) , j = 1 , , ..., N. (4.4)Here, γ ji ( · ), j = 1 , , ..., N , i = 1 ,
2, are unknown functions and F j ( t, x ) = e it∂ x e u j + D ( ψ T | u j | u j )( t, x ) , where D ( w ( t, x )) = − i R t e i ( t − t ′ ) ∂ x w ( t ′ , x ) dt ′ . By using Lemma 3.4 we see that u j ( t,
0) = M e − i π (1+3 λ j ) + e − i π (1 − λ j ) sin( − λ j π ) ! γ j ( t ) + M e − i π (1+3 λ j ) + e − i π (1 − λ j ) sin( − λ j π ) ! γ j ( t )+ F j ( t, , := a j γ j ( t ) + a j γ j ( t ) + F j ( t, , j = 1 , , ..., N. (4.5)Now, let us calculate the traces of first derivative functions. Thanks to (2.2), (2.8), (2.12) and Lemma 3.4,we get that ∂ x u j ( t,
0) = M e − i π ( − λ j ) + e − i π (6 − λ j ) sin( − λ j π ) ! I − / γ j ( t )+ M e − i π ( − λ j ) + e − i π (6 − λ j ) sin( − λ j π ) ! I − / γ j ( t )+ ∂ x F j ( t, , := b j I − / γ j ( t ) + b j I − / γ j ( t ) + ∂ x F j ( t, , j = 1 , , ..., N. (4.6)In the same way, we can have the traces of second and third derivatives functions, giving us the following ∂ x u j ( t,
0) = M e − i π ( − λ j ) + e − i π (11 − λ j ) sin( − λ j π ) ! I − / γ j ( t )+ M e − i π ( − λ j ) + e − i π (11 − λ j ) sin( − λ j π ) ! I − / γ j ( t )+ ∂ x F j ( t, , := c j I − / γ j ( t ) + c j I − / γ j ( t ) + ∂ x F j ( t, , j = 1 , , ..., N (4.7) and ∂ x u j ( t,
0) = M e − i π ( − λ j ) + e − i π (16 − λ j ) sin( − λ j π ) ! I − / γ j ( t )+ M e − i π ( − λ j ) + e − i π (16 − λ j ) sin( − λ j π ) ! I − / γ j ( t )+ ∂ x F j ( t, , := d j I − / γ j ( t ) + d j I − / γ j ( t ) + ∂ x F j ( t, , j = 1 , , ..., N. (4.8)Observe that Lemmas 3.3 and 3.4 ensure these calculus are valid for Re λ > −
4. By substituting (4.5), (4.6),(4.7) and (4.8) into (4.2) and (4.3), yields that the functions γ ji and indexes λ ji , for j = 1 , , ..., N and i = 1 ,
2, satisfy the following equalities:[ C ] N × N a a · · · a a · · · · · · a N a N N × N γ ( t ) γ ( t ) γ ( t ) γ ( t )... γ N ( t ) γ N ( t ) N × = − [ C ] N × N F ( t, F ( t, F N ( t, N × , [ C ] N × N b b · · · b b · · · · · · b N b N N × N γ ( t ) γ ( t ) γ ( t ) γ ( t )... γ N ( t ) γ N ( t ) N × = − [ C ] N × N ∂ x I F ( t, ∂ x I F ( t, ∂ x I F N ( t, N × , [ C ] N × N c c · · · c c · · · · · · c N c N N × N γ ( t ) γ ( t ) γ ( t ) γ ( t )... γ N ( t ) γ N ( t ) N × = − [ C ] N × N ∂ x I F ( t, ∂ x I F ( t, ∂ x I F N ( t, N × and[ C ] N × N d d · · · d d · · · · · · d N d N N × N γ ( t ) γ ( t ) γ ( t ) γ ( t )... γ N ( t ) γ N ( t ) N × = − [ C ] N × N ∂ x I F ( t, ∂ x I F ( t, ∂ x I F N ( t, N × . NLS ON GRAPHS 13
It follows that,(4.9) a a − a − a · · · a a − a − a · · · · · · a ( N − a ( N − − a N − a N b b − b − b · · · b b − b − b · · · · · · b ( N − b ( N − − b N − b N c c c c c c · · · · · · · · · c N c N d d d d d d · · · · · · · · · d N d N N × N γ ( t ) γ ( t ) γ ( t ) γ ( t )... γ N ( t ) γ N ( t ) N × = − F ( t, − F ( t, F N − ( t, − F N ( t, ∂ x I F ( t, − ∂ x I F ( t, ∂ x I F N − ( t, − ∂ x I F N ( t, P Nj =1 ∂ x I F j ( t, P Nj =1 ∂ x I F j ( t, N × . To simplify the notation, let us denote the equality (4.9) by(4.10) M ( λ , λ , · · · , λ N , λ N ) γ = F , where M ( λ , λ , · · · , λ N , λ N ) is the first matrix that appears in the left hand side of (4.9), γ is the matrixcolumn given by vector ( γ , γ , · · · , γ N , γ N ) and F is the matrix in the right hand side of (4.9).4.2. Choosing the appropriate parameters and functions.
In this second step, we need to choose theparameters λ ji and the functions γ ji , with j = 1 , , ..., N , i = 1 ,
2, in such a way that we can be able to writethe solution u j ( t, x ), in a integral form.To do this, let us start by using the hypothesis of Lemma 3.7. We need, firstly, to fix parameters λ ji ,such that(4.11) max (cid:26) s − , − (cid:27) < λ ji ( s ) < min (cid:26) s + 12 , (cid:27) , j = 1 , , ..., N, i = 1 , . With this restriction in hand we choose the parameters λ ji as follows(4.12) λ = λ = · · · = λ N = −
12 and λ = λ = · · · = λ N = 14 , then, we have the equation(4.13) M (cid:18) − , , · · · , − , (cid:19) γ = F . The following lemma gives us that M (cid:0) − , , · · · , − , (cid:1) is invertible. Lemma 4.1.
The determinant of matrix M (cid:0) − , , · · · , − , (cid:1) is nonzero. We will prove Lemma 4.1 on the next section. Thus, this good choices of the parameters satisfying(4.11) together with this lemma ensures that M is invertible and, consequently, the following holds(4.14) γ = M − (cid:18) − , , · · · , − , (cid:19) F . We empathize that γ ji depends on F and F , which depend on the unknown functions u and u . Thus, bysubstituting (4.14) into (4.4), we get u j ( t, x ) in the integral form(4.15) u j ( t, x ) = L − γ j ( t, x ) + L γ j ( t, x ) + F j ( t, x ) , j = 1 , , ..., N. Defining the truncated integral operator and functional space.
Using the previous subsection,we have the solution of the Cauchy problem (4.1) with Type A boundary condition (1.4) in the integral form(4.15). In order to use the Fourier restriction method, the third step is to define a truncated version for theintegral form (4.15).Pick s ∈ [0 , / λ ji as in (4.12) and define γ = ( γ , γ , γ , γ , · · · , γ N , γ N )by (4.14). Consider b = b ( s ) < and that the estimates given in Lemmas 3.5, 3.6, 3.7 and 3.8 are valid.Now, define the operator Λ by Λ = (Λ , Λ , · · · , Λ N )where Λ j u ( t, x ) = ψ ( t ) L − γ j ( t, x ) + ψ ( t ) L γ j ( t, x ) + F j ( t, x ) , j = 1 , , ..., N. Here, F j ( t, x ) = ψ ( t )( e it∂ x e u j + λ D ( ψ T | u j | u j )( t, x )) , j = 1 , , ..., N, with D ( w ( t, x )) = − i Z t e i ( t − t ′ ) ∂ x w ( x, t ′ ) dt ′ . We consider Λ defined on the Banach space Z ( s, b ) = L Nj =1 Z j ( s, b ) by Z j ( s, b ) = n w ∈ C ( R t ; H s ( R x )) ∩ C ( R x ; H s +38 ( R t )) ∩ X s,b ; w x ∈ C ( R x ; H s +18 ( R t )) , w xx ∈ C ( R x ; H s − ( R t )) , w xxx ∈ C ( R x ; H s − ( R t )) o , for j = 1 , , ..., N , with norm(4.16) k ( u , u , · · · , u N ) k Z ( s,b ) = k u k Z ( s,b ) + k u k Z ( s,b ) + · · · + k u N k Z N ( s,b ) . Each norm of k u k Z j ( s,b ) on (4.16) is defined by k u k Z j ( s,b ) = k u k C ( R t ; H s ( R x )) + k u k C (cid:18) R x ; H s +38 ( R t ) (cid:19) + k u k X s,b + k u x k C (cid:18) R x ; H s +18 ( R t ) (cid:19) + k u xx k C (cid:18) R x ; H s − ( R t ) (cid:19) + k u xxx k C (cid:18) R x ; H s − ( R t ) (cid:19) , for j = 1 , , ..., N .4.4. Proving that the functions L − γ j and L γ j , for j = 1 , , ..., N , are well-defined. Indeed, byusing Lemma 3.7 it suffices to show that these functions are in the closure of the spaces C ∞ ( R + ). By usingexpression (4.14), we see that γ ji ( j = 1 , , ..., N and i = 1 ,
2) are linear combinations of the functions F ( t, − F ( t, , F ( t, − F ( t, , · · · , F N − ( t, − F N ( t, ,∂ x I F ( t, − ∂ x I F ( t, , ∂ x I F ( t, − ∂ x I F ( t, , · · · , ∂ x I F N − ( t, − ∂ x I F N ( t, ,∂ x I F ( t,
0) + ∂ x I F ( t,
0) + · · · + ∂ x I F N ( t, ,∂ x I F ( t,
0) + ∂ x I F ( t,
0) + · · · + ∂ x I F N ( t, . Thus, we need to show that the functions F j ( t, , ∂ x I F j ( t, , ∂ x I F j ( t,
0) are in appropriate spaces. ByLemmas 3.5, 3.7, 3.6 and 3.8 we obtain(4.17) k F j ( t, k H s +38 ( R + ) ≤ c ( k u j k H s ( R + ) + k u j k X s,b ) . NLS ON GRAPHS 15
If 0 ≤ s < we have that < s +38 < , then H s +38 ( R + ) = H s +38 ( R + ). It follows that F j ( t, ∈ H s +38 ( R + )for 0 ≤ s < . Again by using Lemmas 3.5, 3.7, 3.6 and 3.8 we get k ∂ x F j ( t, k H s +18 ( R + ) ≤ c ( k u j k H s ( R + ) + k u j k X s,b ) . Since 0 ≤ s < we have ≤ s +18 < , then the functions ∂ x F j ( t, ∈ H s +18 ( R + ). Then, thanks to Lemma3.2, we have that k ∂ x I F j ( t, k H s +380 ( R + ) ≤ c ( k u j k H s ( R + ) + k u j k X s,b ) . Therefore, this yields that(4.18) ∂ x I F j ( t, − ∂ x I F j +1 ( t, ∈ H s +38 ( R + ) , j = 1 , , ..., N. In a similar way, we can obtain k ∂ x I F j ( t, k H s +380 ( R + ) . k u j k H s ( R + ) + k u j k X s,b , k ∂ x I F i ( t, k H s +380 ( R + ) . k u j k H s ( R + ) + k u j k X s,b . It follows that ∂ x I F ( t,
0) + ∂ x I F ( t,
0) + · · · + ∂ x I F N ( t, ∈ H s +38 ( R + ) ,∂ x I F ( t,
0) + ∂ x I F ( t,
0) + · · · + ∂ x I F N ( t, ∈ H s +38 ( R + ) . (4.19)Thus, (4.17), (4.18) and (4.19) imply that the functions L − γ j and L γ j , for j = 1 , , ..., N , are well-defined.4.5. Showing that Λ is a contraction in a ball of Z . Lemmas 3.2, 3.5, 3.7, 3.6 and 3.8 guarantee that k Λ( u , · · · , u N ) − Λ( v , · · · , v N ) k Z s,b ≤ T ǫ c (cid:0) k ( u , · · · , u N ) k Z + k ( v , · · · , v N ) k Z (cid:1) k ( u , · · · , u N ) − ( v , · · · , v N ) k Z and k Λ( u , · · · , u N ) k Z s,b ≤ c (cid:0) k u k H s ( R + ) + · · · + k u N k H s ( R + ) (cid:1) + T ǫ ( k u k X s,b + · · · + k u N k X s,b ) , for ǫ adequately small.Consider in Z the ball defined by B = { ( u , · · · , u N ) ∈ Z s,b ; k ( u , · · · , u N ) k Z s,b ≤ M } , where M = 2 c (cid:0) k u k H s ( R + ) + · · · + k u N k H s ( R + ) (cid:1) . Lastly, choosing T = T ( M ) sufficiently small, such that k Λ( u , · · · , u N ) k Z s,b ≤ M and k Λ( u , · · · , u N ) − Λ( v , · · · , v N ) k Z s,b ≤ k ( u , · · · , u N ) − ( v , · · · , v N ) k Z s,b , it follows that Λ is a contraction map on B and has a fixed point ( e u , · · · , e u N ). Therefore, the restriction( u , · · · , u N ) = ( e u (cid:12)(cid:12) R − × (0 ,T ) , · · · , e u N (cid:12)(cid:12) R + × (0 ,T ) )solves the Cauchy problem (4.1) with Type A vertex boundary condition (1.4), in the sense of distributions.Thus, Theorem 1.1 is a consequence of the above steps, described in the previous subsections, finalizing sothe proof. (cid:3) Proof of lemma 4.1
First, we will prove the case N = 2. The vertex conditions (1.4), for this case, is given by ( ∂ kx u ( t,
0) = ∂ kx u ( t, , k = 0 , t ∈ (0 , T ) P j =1 ∂ kx u j ( t,
0) = 0 , k = 2 , t ∈ (0 , T ) . In this way, we consider the vertex conditions as the following matrices(5.1) −
10 00 00 0 (cid:20) u ( t, u ( t, (cid:21) = 0 , −
10 00 0 (cid:20) ∂ x u ( t, ∂ x u ( t, (cid:21) = 0and(5.2) (cid:20) ∂ x u ( t, ∂ x u ( t, (cid:21) = 0 , (cid:20) ∂ x u ( t, ∂ x u ( t, (cid:21) = 0 . By substituting, for N = 2, (4.5), (4.6), (4.7) and (4.8) into (5.1) and (5.2), yields that the functions γ ji and indexes λ ji , for j = 1 , i = 1 ,
2, satisfy the equality of matrices: −
10 00 00 0 (cid:20) a a a a (cid:21) γ ( t ) γ ( t ) γ ( t ) γ ( t ) = − −
10 00 00 0 (cid:20) F ( t, F ( t, (cid:21) , −
10 00 0 (cid:20) b b b b (cid:21) γ ( t ) γ ( t ) γ ( t ) γ ( t ) = − −
10 00 0 (cid:20) ∂ x I F ( t, ∂ x I F ( t, (cid:21) , (cid:20) c c c c (cid:21) γ ( t ) γ ( t ) γ ( t ) γ ( t ) = − (cid:20) ∂ x I F ( t, ∂ x I F ( t, (cid:21) and (cid:20) d d d d (cid:21) γ ( t ) γ ( t ) γ ( t ) γ ( t ) = − (cid:20) ∂ x I F ( t, ∂ x I F ( t, (cid:21) . Putting all matrices together, we have that, a a − a − a b b − b − b c c c c d d d d γ γ γ γ = − F ( t, − F ( t, ∂ x I F ( t, − ∂ x I F ( t, ∂ x I F ( t,
0) + ∂ x I F ( t, ∂ x I F ( t,
0) + ∂ x I F ( t, . In the case N = 2, the matrix M , given by (4.9), can be read as follows(5.3) M = a a − a − a b b − b − b c c c c d d d d , where a ij , b ij , c ij and d ij are given by (4.5), (4.6), (4.7) and (4.8), respectively. Claim 1. M has determinant different of zero with appropriate choice of λ ji , j = 1 , and i = 1 , . NLS ON GRAPHS 17
In fact, firstly noting that sin (cid:18) − a π (cid:19) = cos (cid:16) aπ (cid:17) and it is easy to see that(5.4) a ji = Me − iπ e − iπλji + e iπλji sin (cid:16) (1 − λji ) π (cid:17) ! , b ji = Me iπ e − iπλji − e iπλji cos (cid:16) λjiπ (cid:17) ! ,c ji = Me iπ e − iπλji + e iπλji sin (cid:16) (3 − λji ) π (cid:17) ! , d ji = Me iπ e − iπλji − e iπλji sin (cid:16) λjiπ (cid:17) ! . j = 1 , , i = 1 , M can be write as | M | = M e iπ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − iπλ + e iπλ sin (cid:16) (1 − λ π (cid:17) e − iπλ + e iπλ sin (cid:16) (1 − λ π (cid:17) − e − iπλ + e iπλ sin (cid:16) (1 − λ π (cid:17) − e − iπλ + e iπλ sin (cid:16) (1 − λ π (cid:17) e − iπλ − e iπλ cos ( λ π ) e − iπλ − e iπλ cos ( λ π ) − e − iπλ − e iπλ cos ( λ π ) − e − iπλ − e iπλ cos ( λ π ) e − iπλ + e iπλ sin (cid:16) (3 − λ π (cid:17) e − iπλ + e iπλ sin (cid:16) (3 − λ π (cid:17) e − iπλ + e iπλ sin (cid:16) (3 − λ π (cid:17) e − iπλ + e iπλ sin (cid:16) (3 − λ π (cid:17) e − iπλ − e iπλ sin ( λ π ) e − iπλ − e iπλ sin ( λ π ) e − iπλ − e iπλ sin ( λ π ) e − iπλ − e iπλ sin ( λ π ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By using the identity e ia − e ib e ia + e ib = i tan (cid:18) a − b (cid:19) , we have that | M ( λ , λ , λ , λ ) | = M e iπ × n ( e − λ π i + e λ π i )( e − λ π i + e λ π i ) o × n ( e − λ π i + e λ π i )( e − λ π i + e λ π i ) o | A | , where A is the matrix A = ( − λ π ) ( − λ π ) − ( − λ π ) − ( − λ π ) − i tan( λ π )cos( λ π ) − i tan( λ π )cos( λ π ) i tan( λ π )cos( λ π ) i tan( λ π )cos( λ π )1sin ( − λ π ) ( − λ π ) ( − λ π ) ( − λ π ) − i tan( λ π )sin( λ π ) − i tan( λ π )sin( λ π ) − i tan( λ π )sin( λ π ) − i tan( λ π )sin( λ π ) . By choosing λ = λ and λ = λ we have that the constant that appears before of the matrix A takes the form: 2 M e πi ( e − iλ π + 1) ( e − iλ π + 1) e π ( λ + λ ) i . Note that this number is zero only in the case λ = 2 k + 1 and λ = 2 l + 1 for k, l ∈ Z .Let us denote the entries of the matrix A as follows:(5.5) A = a n − a − nf g − f − gc e c ed m d m . Thus, its determinant is given by(5.6) det A = 4( de − cm )( nf − ag ) . In particular for λ = λ = − and λ = λ = matrix A can be seen as follows, A ′ = √ p − √ √ r − √ q − √ −√ p − √ −√ r − √ q − √ i √ √ i t an ( π/ π/ − i √ √ − i t an ( π/ π/ √ p √ √ r − √ − q − √ √ p √ √ r − √ − q − √ − i q √ − i tan( π/ π ) − i q √ − i tan( π/ π ) . By using determinant properties the determinant of A ′ is equivalent of the determinant of the followingmatrix: − (4 i )( √ i ) p − √ r − √ q − √ − p − √ − r − √ q − √ √ √ an ( π/ π/ − √ √ − t an ( π/ π/ p √ r − √ − q − √ p √ r − √ − q − √ p √ tan( π/ √ π ) p √ tan( π/ √ π ) . Therefore, we can rewrite matrix (5.5) as follows a n − a − n c g − c − gc e c ec m c m and its determinant is given by 4( e − m )( n − acg ) . We only need to check that e − m = 0 and n − acg = 0. An calculation proves that e − m ∼ − , n − acg ∼ , . Thus, we have that(5.7) det ( A ′ ) = − i )( √ i )( e − m )( n − acg ) ∼ − , M (cid:0) − , , − , (cid:1) given by (5.3) is nonzero, proving the Claim 1 andLemma 4.1, for the case N = 2.For a better understanding of the reader, before to do the general case, we will present briefly also theproof of Lemma 4.1 considering N = 3. For instance, vertex conditions (1.4), in this case, are given in thematrices form as follows: − −
10 0 00 0 00 0 00 0 0 u ( t, u ( t, u ( t, = 0 , − −
10 0 00 0 00 0 0 ∂ x u ( t, ∂ x u ( t, ∂ x u ( t, = 0 , and ∂ x u ( t, ∂ x u ( t, ∂ x u ( t, = 0 , ∂ x u ( t, ∂ x u ( t, ∂ x u ( t, = 0 . NLS ON GRAPHS 19
Thus, combining the above matrices and the integral form of solution (4.4), as in the case N = 2, we obtain a a − a − a a a − a − a b b − b − b b b − b − b c c c c c c d d d d d d γ γ γ γ γ γ = − F ( t, − F ( t, F ( t, − F ( t, ∂ x I F ( t, − ∂ x I F ( t, ∂ x I F ( t, − ∂ x I F ( t, ∂ x I F ( t,
0) + ∂ x I F ( t,
0) + ∂ x I F ( t, ∂ x I F ( t,
0) + ∂ x I F ( t,
0) + ∂ x I F ( t, . Let us consider M the following matrix(5.8) M = a a − a − a a a − a − a b b − b − b b b − b − b c c c c c c d d d d d d . Claim 2. M has determinant different of zero with appropriate choice of λ ji , j = 1 , , and i = 1 , . Indeed, similarly as in the case N = 2 and by using the identities (5.4), yields that a ji = Me − iπ e − iπλji + e iπλji sin (cid:16) (1 − λji ) π (cid:17) ! , b ji = Me iπ e − iπλji − e iπλji cos (cid:16) λjiπ (cid:17) ! ,c ji = Me iπ e − iπλji + e iπλji sin (cid:16) (3 − λji ) π (cid:17) ! , d ji = Me iπ e − iπλji − e iπλji sin (cid:16) λjiπ (cid:17) ! . j = 1 , , , i = 1 , . By determinant properties, we can get the determinant of M as | M | = M e − iπ ! M e iπ ! M e iπ ! M e iπ ! Y i =1 , ,j =1 , , (cid:16) e − iπλji + e iπλji (cid:17) | A | = M e iπ Y i =1 , ,j =1 , , (cid:16) e − iπλji + e iπλji (cid:17) | A | , where A is the following matrix A ′ = ( − λ π ) ( − λ π ) − ( − λ π ) − ( − λ π ) 0 00 0 ( − λ π ) ( − λ π ) − ( − λ π ) − ( − λ π ) − i tan( λ π )cos( λ π ) − i tan( λ π )cos( λ π ) i tan( λ π )cos( λ π ) i tan( λ π )cos( λ π ) − i tan( λ π )cos( λ π ) − i tan( λ π )cos( λ π ) i tan( λ π )cos( λ π ) i tan( λ π )cos( λ π )1sin ( − λ π ) ( − λ π ) ( − λ π ) ( − λ π ) ( − λ π ) ( − λ π ) − i tan( λ π )sin( λ π ) − i tan( λ π )sin( λ π ) − i tan( λ π )sin( λ π ) − i tan( λ π )sin( λ π ) − i tan( λ π )sin( λ π ) − i tan( λ π )sin( λ π ) . By choosing λ = λ = λ and λ = λ = λ , we have that the constant that appears before ofthe matrix A takes the form: M e iπ ( e − iπλ + 1) ( e − iπλ + 1) e π ( λ + λ ) i . Note that this number is zero only in the case λ = 2 n + 1 and λ = 2 m + 1 for n, m ∈ Z . Let us rewritethe entries of matrix A as follows: A = a n − a − n a n − a − nf g − f − g f g − f − gc e c e c ed m d m d m . Thus, its determinant is given by | A ′ | = 9( de − cm )( ag − nf ) . Finally, considering λ = λ = λ = − and λ = λ = λ = , thanks to the case N = 2, wehave that ( ag − f n ) = 0 and ( de − cm ) = 0, thus | A | 6 = 0. Claim 2 is thus proved and Lemma 4.5 is achieved,when N = 3.Let us now deal with the general situation, that is, when N >
3. Consider M = M ( λ , λ , · · · , λ N , λ N )defined by (4.9), namely, M N × N = a a − a − a · · · a a − a − a · · · · · · a ( N − a ( N − − a N − a N b b − b − b · · · b b − b − b · · · · · · b ( N − b ( N − − b N − b N c c c c c c · · · · · · · · · c N c N d d d d d d · · · · · · · · · d N d N | {z } N columns N − N − (cid:27) , NLS ON GRAPHS 21 where a ij , b ij , c ij and d ij are given by (4.5), (4.6), (4.7) and (4.8), respectively. As we noted in the cases N = 2 ,
3, this function of λ ji can be take the form a ji = Me − iπ e − iπλji + e iπλji sin (cid:16) (1 − λji ) π (cid:17) ! , b ji = Me iπ e − iπλji − e iπλji cos (cid:16) λjiπ (cid:17) ! ,c ji = Me iπ e − iπλji + e iπλji sin (cid:16) (3 − λji ) π (cid:17) ! , d ji = Me iπ e − iπλji − e iπλji sin (cid:16) λjiπ (cid:17) ! . j = 1 , , ..., N, i = 1 , . Thus, by using the determinant properties, we have that | M | = M e − iπ ! N − M e iπ ! N − M e iπ ! M e iπ ! Y i =1 , ,j =1 , ··· ,N (cid:16) e − iπλji + e iπλji (cid:17) | M ′ | = M N e (12+ N ) iπ N Y i =1 , ,j =1 , ··· ,N (cid:16) e − iπλji + e iπλji (cid:17) | M ′ | , where M ′ is a matrix, depending only of λ ji , given by M ′ = ¯ a ¯ a − ¯ a − ¯ a · · · a ¯ a − ¯ a − ¯ a · · · · · · ¯ a ( N − ¯ a ( N − − ¯ a N − ¯ a N ¯ b ¯ b − ¯ b − ¯ b · · · b ¯ b − ¯ b − ¯ b · · · · · · ¯ b ( N − ¯ b ( N − − ¯ b N − ¯ b N ¯ c ¯ c ¯ c ¯ c ¯ c ¯ c · · · · · · · · · ¯ c N ¯ c N ¯ d ¯ d ¯ d ¯ d ¯ d ¯ d · · · · · · · · · ¯ d N ¯ d N N × N . Here, the coefficients of matrix M ′ are given by(5.9) ¯ a ji = 1sin (cid:16) (1 − λ ji ) π (cid:17) , ¯ b ji = − i tan (cid:16) λ jiπ (cid:17) cos (cid:16) λ ji π (cid:17) , ¯ c ji = 1sin (cid:16) (3 − λ ji ) π (cid:17) , ¯ d ji = − i tan (cid:16) λ jiπ (cid:17) sin (cid:16) λ ji π (cid:17) . j = 1 , , ..., N, i = 1 , . By choosing λ = λ = · · · = λ N and λ = λ = · · · = λ N , we have that the constant that appearsbefore of the matrix M ′ takes the form: M N e (12+ N ) iπ N ( e − iπλ + 1) N ( e − iπλ + 1) N e N π ( λ + λ ) i . Note that this number is zero only in the case λ = 2 n + 1 and λ = 2 m + 1 for n, m ∈ Z . Let us denotethe entries of matrix M ′ as follows: M ′ = a n − a − n · · · a n − a − n · · · · · · a n − a − nf g − f − g · · · f g − f − g · · · · · · f g − f − gc e c e c e . . . · · · · · · c ed m d m d m . . . · · · · · · d m N × N . Moreover, by using the determinant properties, it yields that | M ′ | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a n − a − n · · · f g − f − g · · · a n − a − n · · · f g − f − g · · · · · · a n − a − n · · · f g − f − gc e c e c e . . . · · · · · · c ed m d m d m . . . · · · · · · d m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N × N . Considering the matrix A = (cid:20) a nf g (cid:21) and B = (cid:20) c ed m (cid:21) , the determinant of M ′ can be write as a block matrices, namely,(5.10) | M ′ | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A × − A × × · · · × × × A × − A × · · · × × ... ... ... ... ... ...0 × × × · · · A × − A × B × B × B × · · · B × B × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N × N . From now on, we denote 0 n × n and I n × n the null and identity matrices, respectively.Let us introduce the properties of determinants that helped us to prove Lemma 4.5 in general form.Consider a block matrix N of size ( n + m ) × ( n + m ) of the form N = (cid:20) C DF G (cid:21) , where C, D, F and G are of size n × n , n × m , m × n and m × m , respectively. If G is invertible, then(5.11) det N = det( C − DG − F ) det( G ) . In fact, this property follows immediately from the following identity (cid:20)
C DF G (cid:21) (cid:20) I − G − F I (cid:21) = (cid:20) C − DG − F D G (cid:21) . Finally, recall that the determinant of a block triangular matrix is the product of the determinants of itsdiagonal blocks.
NLS ON GRAPHS 23
With these two properties in hand, define
C, D, F and G , respectively, by C = A × − A × × · · · × × A × − A × · · · × ... ... ... ... ...0 × × × · · · A × N − × N − , D = × × ... − A × N − × and F = (cid:2) B × B × B × · · · B × (cid:3) × N − , G = B × . Thanks to the case N = 2, we already know that(5.12) det G = det B × = cm − de = 0 , which implies that G is invertible. Thus, the determinant (5.10) takes the form | M ′ | = (cid:12)(cid:12)(cid:12)(cid:12) C N − × N − D × N − F × N − B × (cid:12)(cid:12)(cid:12)(cid:12) N × N and by using the property (5.11), it yields that(5.13) det M ′ = det (cid:16) C N − × N − − D × N − ( B × ) − F × N − (cid:17) det B × . Claim 3. M ′ has determinant different of zero with appropriate choice of λ ji , j = 1 , , ..., N and i = 1 , . From (5.12) is enough to prove thatdet (cid:16) C N − × N − − D × N − ( B × ) − F × N − (cid:17) is nonzero. In order to analyze the above determinant, note that( B × ) − F × N − = ( B × ) − (cid:2) B × B × B × · · · B × (cid:3) × N − = (cid:2) I × I × I × · · · I × (cid:3) × N − and D × N − ( B × ) − F × N − = × × ... − A × N − × (cid:2) I × I × I × · · · I × (cid:3) × N − = × × · · · × × × · · · × ... ... · · · ...0 × × · · · × − A × − A × · · · − A × N − × N − . Therefore, we get C N − × N − − D × N − ( B × ) − F × N − = A × − A × × · · · × × A × − A × · · · × ... ... ... ... ... A × A × A × · · · A × N − × N − . Then, C N − × N − − D × N − ( B × ) − F × N − only depends of A × . Consequently, ifdet (cid:16) C N − × N − − D × N − ( B × ) − F × N − (cid:17) = 0 , we have that(5.14) dim Ker (cid:16) C N − × N − − D × N − ( B × ) − F × N − (cid:17) > . (5.14) implies that there exists a vector X N − × ∈ Ker (cid:16) C N − × N − − D × N − ( B × ) − F × N − (cid:17) such that(5.15) X N − × = ( x , x , x , x · · · , x N − , x N − ) T = 0 N − × and (cid:16) C N − × N − − D × N − ( B × ) − F × N − (cid:17) · X N − × = 0 N − × , or equivalent,(5.16) A × − A × × · · · × × A × − A × · · · × ... ... ... ... ... A × A × A × · · · A × N − × N − x x x x ... x N − x N − N − × = 0 N − × . To finalize the proof of the Claim 3, denote H × = (cid:20) x x (cid:21) , H × = (cid:20) x x (cid:21) , ..., H ( N − × = (cid:20) x ( N − x ( N − (cid:21) . Therefore, the product (5.16) can be write in the form A × − A × × · · · × × A × − A × · · · × ... ... ... ... ... A × A × A × · · · A × N − × N − H × H × ... H ( N − × N − × = 0 N − × . Thus, we have that A × (cid:0) H × − H × (cid:1) A × (cid:0) H × − H × (cid:1) ... A × (cid:0) H ( N − × − H ( N − × (cid:1) A × (cid:0) H × + H × + · · · + 2 H ( N − × (cid:1) N − × = 0 N − × . Now, let us now argue by contradiction. If there exists k ∈ { , , · · · N − } such that H k × − H ( k +1)2 × = 0 × , we obtain A × (cid:0) H k × − H ( k +1)2 × (cid:1) = 0 × , which implies that dim ker A × >
0, it means that det A × = 0. However, from the case N = 2, we knownthat det A × = ag − f n = 0 , and hence we obtain a contradiction. On the other hand, suppose that(5.17) H j × − H ( j +1)2 × = 0 × , ∀ j = 1 , , ..., N − . Thus, from (5.15) and (5.17), we deduce that H j × = 0 × for some j ∈ { , , ..., N − } and A × (cid:0) H × + H × + · · · + 2 H ( N − × (cid:1) = ( N − A × H j × = 0 N − × . Which is again a contradiction, by using the case N = 2. Hence, in the two cases, we only have thatdet (cid:16) C N − × N − − D × N − ( B × ) − F × N − (cid:17) = 0 , it implies that det M ′ = 0. Consequently, the determinant of M is nonzero, implying that the matrix M isinvertible and the Claim 3 follows. Thus, Lemma 4.1 is proved. (cid:3) NLS ON GRAPHS 25
Appendix A. Vertex conditions types B and C In this appendix, we will outline how to prove that matrices associated with vertex conditions (1.5)(type B ) and (1.6) (type C ) are invertible. We will consider the vertex conditionsType B : ( ∂ kx u ( t,
0) = ∂ kx u ( t,
0) = · · · = ∂ kx u N ( t, , k = 2 , t ∈ (0 , T ) , P Nj =1 ∂ kx u j ( t,
0) = 0 , k = 0 , t ∈ (0 , T ) , and Type C : ( ∂ kx u ( t,
0) = ∂ kx u ( t,
0) = · · · = ∂ kx u N ( t, , k = 0 , t ∈ (0 , T ) , P Nj =1 ∂ kx u j ( t,
0) = 0 , k = 1 , t ∈ (0 , T ) , which may be expressed in matrices form as follows M B γ B = F B , and M C γ C = F C , respectively. Here F B and F C are the function vectors defined by F B := − P Nj =1 F j ( t, P Nj =1 ∂ x I F j ( t, ∂ x I F ( t, − ∂ x I F ( t, ∂ x I F N − ( t, − ∂ x I F N ( t, ∂ x I F ( t, − ∂ x I F ( t, ∂ x I F N − ( t, − ∂ x I F N ( t, N × F C := − F ( t, − F ( t, F N − ( t, − F N ( t, P Nj =1 ∂ x I F j ( t, P Nj =1 ∂ x I F j ( t, ∂ x I F ( t, − ∂ x I F ( t, ∂ x I F N − ( t, − ∂ x I F N ( t, N × , γ B and γ C are the matrices column given by vectors ( γ B , γ B , · · · , γ B N , γ B N ) and ( γ C , γ C , · · · , γ C N , γ C N ),respectively.Note that choosing λ = λ = · · · = λ N and λ = λ = · · · = λ N and arguing as in the Section 5,the determinants of the matrices M B = M B ( − / , / , · · · , − / , /
4) and M C = M C ( − / , / , · · · , − / , / | M B | = M N e (12+ N ) iπ N Y i =1 , ,j =1 , ··· ,N (cid:16) e − iπλji + e iπλji (cid:17) | M ′B | , and | M C | = M N e (12+ N ) iπ N Y i =1 , ,j =1 , ··· ,N (cid:16) e − iπλji + e iπλji (cid:17) | M ′C | , where | M ′B | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a n a n a n · · · a n a nf g f g f g · · · f g f gc e − c − e · · · c e − c − e · · · · · · c e − c − ed m − d − m · · · d m − d − m · · · · · · d m − d − m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N × N , and | M ′C | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a n − a − n · · · a n − a − n · · · · · · a n − a − nf g f g f g · · · f g f gc e c e c e · · · c e c ed m − d − m · · · d m − d − m · · · · · · d m − d − m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N × N . As in the case vertex type A , we need to study the determinants of the matrices M ′B and M ′C . In orderto see the determinant of M ′B and M ′C are nonzero, we use the determinant properties together with (5.11)to observe that det M ′B = det M ′′B det (cid:20) a nf g (cid:21) and det M ′C = det M ′′C det (cid:20) c ef g (cid:21) . These two matrices, namely, M ′B and M ′C , have the following two following properties:(i) If det (cid:20) c ed m (cid:21) = cm − de = 0, then det M ′′B = 0.(ii) If det (cid:20) a nd m (cid:21) = am − dn = 0, then det M ′′C = 0. Claim 4.
The relations ag − f n = , cg − f e = 0 and cm − de = 0 , am − dn = 0 are valid. In fact, choosing λ = λ = · · · = λ N = −
12 and λ = λ = · · · = λ N = − , together with (5.6), (5.7) and (5.9), we already now that ag − f n and cm − de are nonzero. Finally, easycalculations yield that cg − f e = (cid:0) π (cid:1) ! − tan (cid:0) π (cid:1) cos (cid:0) π (cid:1) ! − (cid:0) π (cid:1) ! − tan (cid:0) − π (cid:1) cos (cid:0) − π (cid:1) ! ∼ − . = 0 ,am − dn = (cid:0) π (cid:1) ! − tan (cid:0) π (cid:1) sin (cid:0) π (cid:1) ! − (cid:0) π (cid:1) ! − tan (cid:0) − π (cid:1) sin (cid:0) − π (cid:1) ! ∼ . = 0 , showing the Claim 4, and thus the matrices M B and M C are invertible.A.1. Proof of Theorem 1.1: Vertex conditions type B and C . The analysis developed above give usthe following representations for γ B and γ C γ B = M B− F B and γ C = M C− F C , respectively. Therefore, the solution u j ( t, x ) of the Cauchy problem (4.1) with vertex conditions type B and C can be express in a integral forms(A.1) u B j ( t, x ) = L − γ B j ( t, x ) + L γ B j ( t, x ) + F j ( t, x ) , j = 1 , , ..., N and(A.2) u C j ( t, x ) = L − γ C j ( t, x ) + L γ C j ( t, x ) + F j ( t, x ) , j = 1 , , ..., N. NLS ON GRAPHS 27
Finally, in order to establish Theorem 1.1 with boundary conditions type B and C , we closely followthe same steps of subsection 4.3, 4.4 and 4.5, for use the Fourier restriction method to define a truncatedversion for (A.1) and (A.2), proving thus that L − γ B j , L − γ C j , L γ B j and L − γ C j for j = 1 , , ..., N , arewell-defined. With this in hand, a contraction mapping argument gives us the result desired. (cid:3) Acknowledgments.
Capistrano-Filho was supported by CNPq 306475/2017-0, 408181/2018-4, CAPES-PRINT 88881.311964/2018-01 and Qualis A - Propesq (UFPE). Gallego was partially supported by Facultadde Ciencias Exactas y Naturales, Nacional de Colombia Sede Manizales, under the project 45511 and CAPES-PRINT under grant number 88881.311964/2018-01.This work was carried out during some visits of the authors to the Federal University of Pernambuco,Federal University of Alagoas and Universidad Nacional de Colombia - Sede Manizales. The authors wouldlike to thank the Universities for its hospitality.
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Departamento de Matem´atica, Universidade Federal de Pernambuco (UFPE), 50740-545, Recife (PE), Brazil.
E-mail address : [email protected] ; [email protected] Instituto de Matem´atica, Universidade Federal de Alagoas (UFAL), Macei´o (AL), Brazil
E-mail address : [email protected] Departamento de Matematicas y Estad´ıstica, Universidad Nacional de Colombia (UNAL), Cra 27 No. 64-60,170003, Manizales, Colombia
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