Global existence and compact attractors for the discrete nonlinear Schrödinger equation

Transcription

1 J. Differential Equations 7 (005) Global existence and compact attractors for the discrete nonlinear Schrödinger equation Nikos I. Karachalios a,, Athanasios N. Yannacopoulos b a Department of Mathematics, University of the Aegean, Karlovassi 8300, Samos, Greece b Department of Statistics and Actuarial Science, University of the Aegean, Karlovassi 8300, Samos, Greece Received 8 February 004 Available online August 005 Abstract We study the asymptotic behavior of solutions of discrete nonlinear Schrödinger-type (DNLS) equations. For a conservative system, we consider the global in time solvability and the question of existence of standing wave solutions. Similarities and differences with the continuous counterpart (NLS-partial differential equation) are pointed out. For a dissipative system we prove existence of a global attractor and its stability under finite-dimensional approximations. Similar questions are treated in a weighted phase space. Finally, we propose possible extensions for various types of DNLS equations. 005 Elsevier Inc. All rights reserved. MSC: 37L60; 35B4; 35Q55. Introduction The discrete nonlinear Schrödinger equation (DNLS), is a very popular model with a great variety of applications, ranging from physics to biology. To name but a few, the DNLS has been successfully applied to the modelling of localized pulse propagation in optical fibers and wave guides, to the study of energy relaxation in solids, the behavior of amorphous material, to the modelling of self-trapping of vibrational energy Corresponding author. addresses: karan@aegean.gr (N.I. Karachalios), ayannaco@aegean.gr (A.N. Yannacopoulos) /$ - see front matter 005 Elsevier Inc. All rights reserved. doi:0.06/j.jde

2 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) in proteins or studies related to the denaturation of the DNA double strand (for an account of possible applications, see, e.g. the recent review [3]). The DNLS serves both as a model in its own right (modelling cases where the nature of the problem is inherently discrete), or as an approximation of the continuous nonlinear Schrödinger through a numerical scheme. The study of the dynamics of the DNLS, has been an active theme of research in the past decade. The major part of the activity, has focused on the study of localized excitations and nonlinear waves (solitons and breathers), in the conservative DNLS. This activity has started with the ingenious construction introduced in [6], of localized in space time periodic or time-quasiperiodic solutions of lattice systems (which include the DNLS as special case), using the so-called anti-integrable limit or anti-continuous limit. Such solutions are essentially found for the weak coupling limit between the lattice sites. Other constructions are available using dynamical systems methods, such as the reduction to a symplectic map and the homoclinic trajectory method. The stability of such solutions, has been studied using both analytic and numerical techniques, and they have been shown in many cases to be robust under perturbations. They can be applied to the study of diverse physical and biological phenomena, such as lattice dynamics of solids, selective bond excitations, localization of electromagnetic waves in photonic crystals with nonlinear response, etc. For detailed reviews on this theme, we refer to [4,4,8]. There have also been recent interesting works on more complicated solutions arising in higher dimensions than one, such as discrete vortex solutions (see e.g. [] or [7]). In this work, we mainly consider the asymptotic behavior of solutions for the following complex lattice dynamical system (LDS) i u n + ε (u n u n + u n+ ) + iδu n + F(u n ) = g n, n Z, ε > 0, () u n (0) = u n,0, n Z. () The real parameter δ > 0 introduces weak damping, while the case δ = 0 and g n = 0, for all n Z, corresponds to a conservative (NLS) lattice system. Using techniques from the theory of infinite-dimensional systems, we are able to show the global in time existence of solutions to both the conservative and the dissipative DNLS equation. This ensures the well posedness of the model. This result is interesting, as it provides a rigorous proof of the fact, that blow up is not expected to occur in the discrete models, as it may happen in the continuous model, a fact that has been observed numerically in [7]. Furthermore, for a finite lattice (assuming Dirichlet boundary conditions), and by using a discrete version of the mountain pass theorem, we prove the existence of non-trivial standing waves, in the conservative case. For the infinite-dimensional case, we show non-existence of non-trivial standing waves for certain values of energy, by using a fixed point argument. For the dissipative case we may further prove a stronger result; the existence of a global attractor, that attracts all bounded sets of the infinite-dimensional phase space. It can further be shown that the dynamics on this global attractor, may be approximated by the dynamics of

3 90 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 finite-dimensional systems. Results of this kind, for LDS of first and second order are presented in the recent works [,9,38,39,4]. Many of the physically interesting solutions of the DNLS equation present strong spatial localization properties. This is true for instance for soliton solutions or breathers. To cover such situations, we study the DNLS equation in weighted spaces, with properly chosen weight functions. For the dissipative case, we may prove global existence of solutions in weighted spaces as well as the existence of a global attractor. Finally, we comment on possible generalizations of the above results in higher spatial dimensions and for more general models.. Formulation of the problem Problem () () is an infinite system of ordinary differential equations (infinitedimensional dynamical system). We will formulate the problem in an appropriate functional setting. In what follows, a complex Hilbert space X with the sesquilinear form B X (, ) and the norm X, will be considered as a real Hilbert space with the scalar product (, ) X = Re B X (, ). Let T : D(T) X X a C-linear, self-adjoint 0 operator with dense domain D(T) on the Hilbert space X, equipped with the scalar product (, ) X. The space X T is the completion of D(T) in the norm u T = u X (Tu, u) X, for u X T, and we denote by X T its dual and by T the extension of T to the dual of D(T), denoted by D(T). A function F : X X is Lipschitz continuous on bounded subsets of X, if for all M>0 there exists a constant L(M) such that F (u) F(v) X L(R) u v X, for all u, v B M the closed ball of X, of center 0 and of radius R. For the treatment of () (), we shall use complexifications of the usual real sequence spaces denoted by ( ) /p l p := u = (u n) C : u l p := u n p <. (3) Let us recall that between l p, spaces the following elementary embedding relation [9, p. 45] holds: l q l p, u l p u l q, q p. (4) We will see in the sequel that relation (4) has important effects concerning the global in time solvability of DNLS equation, especially in the conservative case. For p = we get the usual Hilbert space of square-summable (complex) sequences, endowed with the real scalar product (u, v) l = Re u n v n, u, v l. (5)

4 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) We describe next the discrete differential operators employed. For any u, v l,we consider the linear operators A, B, B : l l, Moreover, it holds that (Bu) = u n+ u n, (B u) = u n u n, (6) (Au) = (u n u n + u n+ ). (7) (Bu, v) l = (u, B v) l, (8) (Au, v) l = (Bu, Bv) l, u,v l. (9) The functional setup will also include (see [38]) the following bilinear form and induced norm: (u, v) l := (Bu, Bv) l + (u, v) l, (0) u l := Bu l + u l. () We denote by l the Hilbert space with scalar product (0) and norm (). The usual norm of l and () are equivalent, since u l u l c u l, c > 0. () For local existence, we examine the following examples of nonlinearities F : C C: (N )F(0) = 0 and there exist constants c>0, β 0 such that F(z ) F(z ) c ( z β + z β ) z z, or alternatively. (N ) F (z) = f( z )z, where f, f : R R, are continuous. Both cases include the power-law nonlinearity F(z) = z σ z (for (N ) we must have < σ < ). Since we intend to apply the general theory of abstract Schrödinger equations [,], the following Lemma will be useful. Lemma.. Let F : C C satisfy (N ) or (N ). Then the function F defines an operator (still denoted by F) F : l l, (F (u)) := F(u n ), which is Lipschitz continuous on bounded sets of l. Proof. We consider first the case (N ). Let u B R a closed ball in l of center 0 and radius R. We have from (3) that F (u) l c u n β+ = c u β+ l β+ c u β+ l, (3) hence F : l l, is bounded on bounded sets of l.

5 9 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 For u, v B R, we observe that F (u) F(v) l c ( u n β + v n β ) u n v n (4) c sup [( u n β + v n β ) ] u n v n c 4R β u v l, (5) justifying that the map F : l l is Lipschitz continuous, on bounded sets of l with Lipschitz constant L(R) = cr β. For the case (N ) we have F (u) l = f( u n ) u n. (6) Since f : R R is continuous, there exists a monotone increasing C -function g : R R, such that f(ρ) g( ρ ) for all ρ R, (7) (see e.g [37, p. 796]). Note also that u n u n l for all n Z. Now (6) and (7) imply the inequality F (u) l g( u n ) u n g( u l ) u n {max ρ [0,R ] g(ρ)} u n K(R) u l. (8) with K(R) = g(r ). To check the Lipschitz property, we may argue as for the proof of (8). For some θ (0, ) and an appropriate C -function g : R R, we get the inequality F (u) F(v) l f( u n ) u n v n + f( u n ) f( v n ) v n, (9)

6 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) f( u n ) f( v n ) v n = f (θ u n + ( θ) v n ) ( u n + v n ) ( u n v n ) v n {max ρ [0,R ] g (ρ)} sup [( u n + v n ) v n ] u n v n. (0) A combination of (9) and (0), implies that there exists a constant L(R) = (g (R )+ 8R 4 g (R )) /, such that F (u) F(v) l L(R) u v l. () The lemma is proved. We observe by (9), that the operator A satisfies the relations (Au, u) l = Bu l 0, () (Au, v) l = (u, Av) l, (3) therefore defines a self-adjoint operator on D(A) = X = l and A 0. Note that the graph norm u D(A) = Au l + u l, is also an equivalent norm with the l -norm since u l u n+ u n + u n + u n c u l. In our case, as it is indicated by (0) () (3), we may choose X A = l equipped with the norm u A = u X (Au, u) X u l, for u l. Moreover, D(A) = X = l = D(A). Obviously A = A, and the operator ia : l l defined by (ia)u = iau for u l, is C-linear and skew-adjoint, and ia generates a group (T (t)) t R, of isometries on l. It is easy to check that the same properties hold for the operator ε A with ε > 0 (i.e. we may consider the equivalent norm ε Bu + l u l ).

7 94 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 Thus, for fixed T > 0 and (u n,0 ) := u 0 l, a function u C([0,T],l ) is a solution of () () if and only if t u(t) = T (t)u 0 + i T (t s)f (u(s)) ds, 0 (F (u)) := F (u n ) = iδu n + F(u n ) g n. (4) Our local existence result can be stated as follows: Theorem.. Let assumptions (N ) or (N ) be satisfied, assume that g := (g n ) l. Then there exists a function T : l (0, ] with the following properties: (a) For all u 0 l, there exists u C([0,T (u 0 )), l ), such that for all 0 <T < T (u 0 ), u is the unique solution of () () in C([0,T],l ) (well posedness). (b) For all t [0,T (u 0 )), T (u 0 ) t (L (R) + ) := T R, R = g l + u(t) l, L (R) = δ + L(R), (5) where L(R) is the Lipschitz constant for the map F : l l. Moreover, the following alternative holds: (i) T (u 0 ) =, or (ii) T (u 0 ) < and lim t T (u 0 ) u(t) l = (maximality). (c) T : l (0, ] is lower semicontinuous. In addition, if {u n0 } n N is a sequence in l such that u n0 u 0 and if T<T (u 0 ), then S(t)u 0n S(t)u 0 in C([0,T],l ), where S(t)u 0 = u(t), t [0,T (u 0 )), denotes the solution operator (continuous dependence on initial data). Proof. Since the result is an application of the results in [, p ], we present only an outline of the proof, for the sake of completeness. It follows from Lemma. that F : l l is Lipschitz continuous on bounded sets of l with Lipschitz constant L (R) = δ + L(R). For all u 0 l with u 0 l M we set R = M + F (0) l = M + g l. Following closely the lines of [, Lemma 4.3., Proposition 4.3.3] and using (4), we may show that the map t Φ u (t) := T (t)u 0 + i T (t s)f (u(s)) ds, 0 is a contraction on the complete metric space (X R,d) where X R := d = { } u C([0,T R ],l ) : u(t) l R, t [0,T R ], T R = L (R) +, max t [0,T R ] u(t) v(t) l.

8 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) For u 0 l we define T (u 0 ) = sup {T >0 : S(t)u 0 = u(t) C([0,T],l ) solution of (4)}. Inequality (5) follows by the contradiction argument of [, Theorem 4.3.4]: we assume instead of (5), that for R = u(t) l, T (u 0 ) t<t R and we consider the solution v C([0,T R ],l ) of v(s) = T (s)u(t) + i s 0 T (s σ)f (v(σ)) dσ, s [0,T R ]. Then we can define a solution w C([0,t + T R ],l ) of (4) at T = t + T R, defined by { u(s), s [0,t], w(s) = v(s t), s [t,t + T R ], which is in contradiction with the assumption T (u 0 )<t+ T R. To prove (c), as in [, Proposition 4.3.7], we set R = sup t [0,T ] u(t) l and define τ n = sup {t [0, T (u 0n )) : S(s)u 0n l R, s [0,t] }. Since u 0n u 0 in l, it follows that for sufficiently large n, u 0n l <R. Therefore τ n >T R > 0. Moreover, for all t T and t τ n, we get from (4) and Gronwall s Lemma the inequality S(t)u 0n S(t)u 0 l e L (R)T u 0n u 0 l. Letting n, we get that S(t)u 0n S(t)u 0 in l and τ n >T, which implies that T (u 0n )>T. 3. Global existence for the conservative case In this section, we discuss the global in time solvability of () () for the case δ = 0 and g n = 0, for all n Z (conservative case). We consider as a model problem, the following DNLS equation with arbitrary power nonlinearity: i u n + ε (u n u n + u n+ ) + u n σ u n = 0, n Z, 0 σ <, ε > 0, (6) u n (0) = u n,0, n Z. (7) We also consider the (NLS) partial differential equation i t u + u xx + u σ u = 0, x R, t > 0, (8) u(x, 0) = u 0 (x). We will show that there is a vast difference concerning the global solvability of the discrete (NLS) equation (6) with ε > 0 and its continuum limit as ε 0 given by (8). Concerning (6), we have the following theorem.

9 96 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 Theorem 3.. Let u 0 l. For any ε > 0, and 0 σ <, there exists a unique solution of (6), such that u(t) C ([0, ), l ). Proof. The result is a consequence of discrete conservation laws satisfied by the solutions of (6) (7). We take the scalar product of (6) with iu. By using () we obtain d dt u(t) l = 0 or u(t) l = u 0 l for every t [0,T (u 0 )), (9) i.e. u(t) l is uniformly bounded on the maximal interval of existence. It follows then by Theorem.(b) (c) that T (u 0 ) = and sup { u(t) l,t [0, ) } <. Remark3.. Theorem 3. can be also established by a discrete version of the conservation of energy, satisfied by solutions of (6). This approach elucidates the role of the nonlinearity exponent, in the global in time solvability of DNLS: it can be easily checked that E(u(t)) = E(u 0 ), E(u(t)) := ε u(t) l Then, by using (4) and (30) we may derive the estimate σ + u(t) σ+ l σ+, t [0,T (u 0 )). (30) u(t) l u 0 l + ε σ + u 0 σ+. (3) l As a consequence of (3) we obtain once again that T (u 0 ) = and sup{ u(t) l,t [0, )} <. Clearly, Theorem 3., covers the case of DNLS with a nonlinearity satisfying (N ). The case of a globally Lipschitz continuous function F :[0, ) R with F(0) = 0is also included. Following [, Section 7.], [, p. 53], such a function can be extended to the complex function F : C\{0} C by setting F(z) = z z F( z ). It can be shown as in Lemma. that F : l l is locally Lipschitz and (F (u), iu) l = 0 which implies (9). Setting F (z) = z 0 F(s)ds it can be shown as in Section 4, Lemma 4. that the functional (V (u)) n := F (u n ) is a C -functional on l and V (u) = F (u). In this case, we observe conservation of energy defined by E(u(t)) := ε u(t) + V (u). l For a comparison between the discrete (NLS) equation (6) and its continuous counterpart (8), we refer to the main results concerning (8) (see [,,8]). For u 0 H (R) and 0 σ < there exists a unique maximal solution of (8), u(t) C([0,T max ), H (R)) C ([0,T max ), L (R)). In addition: if σ < then T max = and u is bounded in H (R). Let σ. Assume that u 0 H (R) such that R x u 0 dx< (initial data with finite variance) and E(u 0 )<. Then T max <.

10 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) On the other hand if u 0 H is sufficiently small, T max = and u is bounded in H (R). It follows from Theorem 3., that solutions of (6) with ε > 0, exist globally in l -norm unconditionally with respect to the degree of the nonlinearity the size of the initial data and the sign of the initial energy E(u 0 ). This is in agreement with the numerical observations in [7]: according to [7], for a discrete system of the form (6), numerical simulations provide evidence that the solution of the discrete equation still exists after localization, while the solution of the continuum system blows up in finite time. Note that the discrete version of initial data with finite variance reads as n u n,0 < and such data belong to l. 4. Existence of standing wave solutions for the conservative DNLS We conclude the discussion on the conservative DNLS, with some results on the existence of standing wave solutions. The standing wave solution for fixed ω is given by the ansatz u n (t) = e iωt n, n Z, n independent of t. (3) We do not restrict n to be real. We are focused on DNLS (6). It can be easily seen, that any standing wave solution of (6), satisfies the following equation: ε ( n n + n+ ) + ω n = n σ n, n Z. (33) We will study in this section two related problems. The first problem we will study is given by Eq. (33), with n taking finite values. We consider the finite-dimensional subspace of l C m+ := { } ψ n l : ψ (m+) = ψ m+ = 0. We consider the following nonlinear system in C m+ : ε ( n n + n+ ) + ω n = n σ n, n m, (34) (m+) = m+ = 0. (35) This is a finite-dimensional problem, and is related to the problem of existence of standing wave solutions for the DNLS equation in a finite lattice with Dirichlet boundary conditions. Although this problem is finite dimensional, it is still of interest as any numerical approximation of the continuous space NLS, or even of the infinitedimensional DNLS will necessarily lead to a finite-dimensional problem of the above type.

11 98 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 The set C m+ endowed with the discrete inner product and induced norm (, ψ) := Re n=m n= m n ψ n, ψ := n=m n= m ψ n,, ψ C m+, (36) is a (finite-dimensional) Hilbert space. We consider now the operators (A ψ) n m := ψ n ψ n + ψ n+, (B ψ) n m = ψ n+ ψ n. It can be easily checked (see also [, p. 7]) that ( A ψ, ψ) = n=m n= m ψ n+ ψ n, ( A, ψ) = (B,B ψ). (37) Hence, we may also consider the inner product and the corresponding norm in C m+ (, ψ), := (B,B ψ) + (, ψ), ψ, := n=m n= m ( ψ n+ ψ n + ψ n ). (38) The norm in (38) is equivalent with (36), and the constants in the equivalence inequality, can be chosen in order to be independent of m. The second is the problem given by Eq. (33), which is an infinite-dimensional system. This is related with the problem of existence of standing wave solutions of the conservative DNLS equation in the infinite lattice. In what follows we will use a variational principle to show existence of non-trivial standing wave solutions for (34) (35) and a fixed point argument to show non-existence of non-trivial standing waves, for certain parameter values for the infinite-dimensional problem (33). 4.. Existence of non-trivial standing wave solutions for the finite-dimensional problem We will treat this problem using a variational approach. We start with some observations and results which are common to both problems. Unless the opposite is explicitly stated, everything that follows will hold for both the finite-dimensional and the infinite-dimensional system. Solutions of (33) are critical points of the functional E( ) = ε (B ) n + ω n σ + n σ+. (39)

12 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) To establish differentiability of the functional E : l R, we shall use the following discrete version of the dominated convergence Theorem, provided by [8]. Lemma 4.. Let {ψ i,k } be a double sequence of summable functions (i.e. i Z ψ i,k < ) and lim k ψ i,k = ψ i for all i Z. If there exists a summable sequence {g i } such that ψ i,k g i for all i, k s, we have that lim k i Z ψ i,k = i Z ψ i. We then have the following lemma. Lemma 4.. Let ( n ) = l σ+ for some 0 < σ <. Then the functional V( ) = n σ+, is a C (l σ+, R) functional and V ( ), ψ =(σ + )Re n σ n ψ n, ψ = (ψ n ) l σ+. (40) Proof. We assume that, ψ l σ+. Then for any n Z, 0<s<, we get from the mean value theorem V( + sψ) V(ψ) s = s Re 0 d dθ n + θsψ n σ+ dθ = (σ+)re For the rhs of (4), we have the estimate (σ + ) n + θsψ n σ+ ψ n (σ + ) The sequence (z n ) is summable since 0 n +sθψ n σ ( n +sθψ n )ψ n dθ. (4) ( n + ψ n ) σ+ ψn = z n. ( ) σ+ ( ) σ+ σ+ z n c ( n + ψ n ) σ+ ψ n σ+. Letting s 0, the existence of the Gateaux derivative (40) of the functional V : l σ+ R, follows from Lemma 4. (discrete dominated convergence).

13 00 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 We show next that the functional V : l σ+ l σ+ σ+ is continuous. For ( n ) = l σ+, we set F ( n ) = n σ n. We consider a sequence m l σ+ such that m in l σ+. We get then the inequality V ( m ) V ( ), ψ c F ( m ) F ( ) l q ψ l p, q = σ +, p = σ +. (4) σ + We denote by ( m ) n the nth coordinate of the sequence m l. Since F satisfies condition (N ) with β = σ, by setting Φ n = ( m ) n σ + n σ, we get from Hölder s inequality that F (( m ) n ) F ( n ) q c (Φ n ) q ( m ) n n q ( ) ( σ+ c ( m ) n n σ+ 0 as m. (Φ n ) σ+ σ ) σ σ+ By (), (3) and Lemma (4.), it follows that the functional E defined by (39) is C (l, R). For convenience, we recall [3, Definition 4., p. 30] (PS-condition) and [3, Theorem 6., p. 40] or [3, Theorem 6., p. 09] (Mountain Pass Theorem of Ambrosseti Rabinowitz [3]). Definition 4.. Let X be a Banach space and E : X R be C. We say that E satisfies condition (PS) if, for any sequence {u n } X such that E(u n ) is bounded and E (u n ) 0asn, there exists a convergent subsequence. If this condition is only satisfied in the region where E α > 0 (resp., E α < 0) for all α > 0, we say E satisfies condition (PS + ) (resp., (PS )). Theorem 4.. Let E : X R be C and satisfy (a) E(0) = 0, (b) ρ > 0, α > 0 : u X = ρ implies E(u) α, (c) u X : u X ρ and E(u )<α. Define } Γ = {γ C 0 ([0, ],X): γ(0) = 0, γ() = u. Let F γ ={γ(t) X : 0 t } and L ={F γ : γ Γ}. If E satisfies condition (PS), then is a critical point of the functional E. β := inf F γ L sup{e(v) : v F γ} α

14 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) We shall verify Definition 4. and the assumptions of Theorem 4. to show that problem (34) (35), has a non-trivial solution. Theorem 4.. Consider the finite-dimensional problem (34) (35). Let 0 < σ < and ε > 0. Then for any ω = 0, there exists a non-trivial solution of (34) (35). Proof (Condition (PS)). We observe that the norm l ε = ( ) / ε B l + ω l, (43) is equivalent with the norm of l defined by () since { } { } min ε, ω l l max ε ε, ω l (44) and is also equivalent with the usual norm of l, as it follows from (). Now, we let a sequence m of l be such that E( m ) <M for some M>0 and E ( m ) 0as m. By using (39) and Lemma 4., we observe that for m sufficiently large M E( m ) ( E ( σ + m ), m = ) σ + m l. (45) ε Therefore the sequence m denoted by m, such that is bounded. Thus, we may extract a subsequence, still m in l as m. (46) Focusing now on (34) (35), we observe that since in the finite-dimensional space C m+ the weak convergence coincides with the strong, m converges strongly to in C m+. Therefore, the functional E associated with the boundary value problem (34) (35), satisfies condition (PS). (Mountain Pass assumptions). We now check the conditions for the validity of the Mountain Pass Theorem for the functional E. We use the same notations for the equivalent norms in the finite-dimensional space C m+, with those of the norms of the infinite-dimensional spaces, having in mind that we deal with finite-dimensional sums. For every C m+ it holds that l σ+ κ l ε, κ = min{/ε, ω}.

15 0 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 The following inequality holds: E( ) = l ε l ε σ+ σ + l κσ+ σ + σ+. l ε ( Choosing C m+ such that l ε = r, we observe that if 0 <r< then ( ) E( ) α > 0, α = r κσ+ σ + rσ. σ+ κ σ+ ) σ, This establishes the first condition for the validity of the Mountain Pass Theorem, for the functional E. To check the behavior of E at infinity, let us consider = te where t R and e C m+, e l ε =. We have that E(te) = t σ + tσ+ e σ+ l σ+ as t +. This ensures the validity of the second condition of the Mountain Pass Theorem. For fixed = 0 and choosing t sufficiently large, we may set u = t to obtain the existence of a non-trivial solution for (34) (35). Remark4.. In Theorem 4., we have to state explicitly that we deal with the finitedimensional problem, since otherwise we may not prove the validity of the PS condition. In the infinite-dimensional case, due to the lack of the Schur property for the space l (in contrast with the space l which posses this property-weak convergence coincides with strong convergence), we may not conclude the strong convergence of the sequence, from its weak convergence. The generalization of the mountain pass argument in the case of the infinite-dimensional conservative system is a subject of future research, and will be considered elsewhere. 4.. Non-existence of non-trivial standing waves We now provide some non-existence results for non-trivial standing waves. The results that follow are valid for both problems (34) (35) and (33). We recall [37, Theorem 8.E, p. 68] (Theorem of Lax and Milgram), which will be used to establish existence of solutions for an auxiliary linear, non-homogeneous problem, related to (33).

16 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) Theorem 4.3. Let X be a Hilbert space and A : X X be a linear continuous operator. Suppose that there exists c > 0 such that Re(Au, u) X c u X for all u X. (47) Then for given f X, the operator equation Au = f, u X, has a unique solution Theorem 4.4. There exist no non-trivial standing waves of energy less than E c (ω, σ) := ( ω 4 4 ) /4σ. Proof. Let ω R, ω = 0, be fixed. For any ε > 0, the operator A ω : l l, defined by (A ω ) = ε ( n n + n+ ) + ω n, (48) is linear and continuous and satisfies assumption (47) of Theorem 4.3: using (44) and () we get that (A ω, ) l = ε B l + ω ω l for all l. (49) For given z l, we consider the linear operator equation (A ω ) = z n σ z n, l. (50) The function F : C C, F (z) = z n σ z n satisfies Lemma., with β = σ. Therefore assumptions of Theorem 4.3 are satisfied, and (50) has a unique solution l. For some R>0, we consider the closed ball of l, B R := {z l : z l R}, and we define the map P : l l,byp(z) := where is the unique solution of the operator equation (50). Clearly the map P is well defined. Let z, ξ B R such that = P(z), ψ = P(ξ). The difference χ := ψ satisfies the equation (A ω χ) = F (z) F (ξ). (5) Taking now the scalar product of (5) with χ and using (4), we derive ε Bχ l + ω χ l F (z) F (ξ) l χ l R σ z ξ l χ l ω χ l + ω R4σ z ξ l. (5)

17 04 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 From (5), (), (44), we obtain the inequality χ l = P(z) P(ξ) l 4 ω 4 R4σ z ξ l. (53) Since P(0) = 0, from inequality (53), we derive that for R<(ω 4 /4) /4σ := E c (ω, σ), the map P : B R B R and is a contraction. Therefore P, satisfies the assumptions of Banach Fixed Point Theorem and has a unique fixed point, the trivial one. Hence, for R<E c (ω, σ) the only standing wave is the trivial. Remark4.. The above theorem is interesting when viewed in the following way: the l norm corresponds to the energy of the excitations of the lattice. If the energy of the excitation is less that E c (ω, σ) the lattice may not support a standing wave of frequency ω. This relation may be seen as some kind of dispersion relation of frequency vs. energy for the standing wave solutions. See also the results of [,3,4] for results on existence, stability and nonexistence of non-trivial standing waves, for the continuous and discrete NLS equations, respectively. It contains information on the type of nonlinearity, through its dependence on the nonlinearity exponent σ. Let us note that a similar computation to (5) shows that P maps also the ball B R with R (ω 4 ) /4σ := R, to itself and obviously E c <R, but within this range, the map is not essentially a contraction. Remark4.3. Arguments of the same type as above, may be used for the proof of existence of unique steady states for the case of nonlinearities that do not vanish when = 0 (see the end of Remark 3.). 5. Existence of global attractor in l for the dissipative case In this section, we study the asymptotic behavior of solutions of () () in the dissipative case, δ > 0 and g n = 0, i.e. taking into account the effect of a weak dissipation and of an external excitation. This time, as a model problem we shall employ the dissipative case of (6) (7) i u n + ε (u n u n + u n+ ) + iδu n + u n σ u n = g n, n Z, 0 σ <, ε > 0, (54) u n (0) = u n,0, n Z. (55) We also consider the weakly damped driven NLS partial differential equation i t u + u xx + iδu + u σ u = g, x R, t > 0, (56) u(x, 0) = u 0 (x).

18 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) It is well known that the dissipation and forcing terms have important effects in the long-time behavior of (56). As an example, we refer to the case σ =, for which it has been observed numerically the existence of chaotic attractors [0,9], in contrast with the conservative case which is completely integrable by the inverse scattering theory, for sufficiently smooth initial data. We refer to [5,6,,5,34] for results on the existence, finite dimensionality, and regularity of global attractors for (56). 5.. Existence of absorbing set This subsection is devoted to the existence of an absorbing set in l.wehavethe following lemma. Lemma 5.. Let (u 0,n ) = u 0,(g n ) = g l. For any ε > 0 and 0 σ <, the dynamical system defined by (54) (55) S(t) : u 0 l u(t) l, (57) possesses a bounded absorbing set B 0 in l : for every bounded set B of l, there exists t 0 (B, B 0 ) such that for all t t 0 (B, B 0 ), it holds S(t)B B 0. Proof. We take the scalar product of (54) with iu(t) and we get d dt u l + δ u l = Im u n g n δ u l + δ g l, (58) and therefore d dt u l + δ u l δ g l. (59) From (59) we derive that u L ([0, ), l ): Gronwall s Lemma implies that u(t) l u 0 l exp( δt) + δ g l { exp( δt)}. (60) Letting t we infer that lim sup u(t) l t δ g l. Setting ρ = g l /δ, it follows that for any number ρ > ρ the ball B 0 of l centered at 0 of radius ρ is an absorbing set for the semigroup S(t): ifb is a bounded set of

19 06 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 l included in a ball of l centered at 0 of radius R, then for t t 0 (B, B 0 ) where t 0 = δ log R ρ ρ, (6) it holds u(t) l ρ, i.e. S(t)B B 0. Note that in the absence of external excitation, the dynamical system exhibits trivial dynamics, in the sense that lim sup t u(t) l = 0, as (60) clearly shows. Remark5.. The existence of the absorbing set, can be shown again by working in a different way, in the equivalent norm of l : By taking the scalar product of (54) with u δu we may derive the energy equation d J (u(t)) + δj (u(t)) = Λ(u(t)), dt J (u) := u l ε { } σ + u σ+ (g, u), (6) l { σ Λ(u) := εδ σ + u σ+ + (g, u) l l } + Im u n g n. (63) From (4) and (6) we obtain the following estimate for Λ(u): { } σ Λ(u) εδ σ + ρσ+ + g l ρ + g l ρ = λ. (64) Gronwall s inequality applied in (6) implies that J (u(t)) J(u 0 ) exp( δt) + λ δ { exp( δt)}. (65) Letting t we get lim sup t J (u(t)) λ /δ. Hence, for t t 0 u(t) l ερ σ+ + 3ε g l ρ + δ g l ρ := ρ. (66) Relation (66), gives an estimate of the radius of the absorbing ball, involving the degree of the nonlinearity and lattice spacing. Note for example, that for sufficiently small we get that ρ < ρ. 0 < ε < ρ g l δ ρ σ+, + 3 g l Throughout the rest of the paper, for simplicity reasons we set ε =.

20 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) Asymptotic compactness of the semigroup and existence of global attractor We now prove the asymptotic compactness of the semigroup generated by the DNLS equation (54) (55). To establish this property, we follow the approach introduced in [9] and applied also in [38], for a second-order LDS. This approach is based on the derivation of estimates on the tail ends of solutions of (54) (55). Lemma 5.. Let (u 0,n ) = u 0 B where B is a bounded set of l and (g n ) = g l. For any 0 σ <, and η > 0, there exist T(η) and K(η) such that the solution (u n ) = u of (54) (55) satisfies for all t T(η), the estimate n >M u n (t) η δ for any M>K(η). (67) Proof. Choose a smooth function θ C (R +, R) satisfying the following properties: θ(s) = 0, 0 s, 0 θ(s), s, θ(s) =, s, and θ (s) C 0, s R + (68) ( ) for some C 0 R. We shall use the shorthand notation θ n = θ n M and Re(u n ) = u,n, Im(u n ) = u,n. We now multiply the DNLS equation with the function θ n ū n, n Z, and we sum over all sites and keep the imaginary part. The effect of the function θ on u n is to cut off all sites within a ball of radius comparable to M, and to take into account only the remote sites. This results to d dt θ n u n +δ θ n u n {(Bu ) n (Bθu ) n (Bu ) n (Bθu ) n } = Im g n θ n ū n. (69) The nonlinear term contributes a purely real term, so it drops out of the above equality. We must now estimate the remaining terms. Using the Cauchy Schwartz inequality, we find for the term containing the forcing that Im g n θ n ū n δ θ n u n + δ n M g n.

21 08 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 We now estimate the last term on the lhs of Eq. (69). After some algebra we obtain I = {(Bu ) n (Bθu ) n (Bu ) n (Bθu ) n } = {(θ n+ θ n )(u,n u,n+ u,n u,n+ )}. We may further estimate I as follows, I sup θ n+ θ n C M u l C M ρ, (u,n u,n+ u,n u,n+ ) where C depends only on C 0. For the last inequality, we applied the mean value theorem and (68), and the restriction of the solution to the absorbing ball in l (ρ is the radius of the absorbing ball of Lemma 5.). Therefore we obtain the following differential inequality: d dt θ n u n + δ θ n u n C M ρ + δ n >M g n. Using the Gronwall inequality, we obtain the following estimate: θ n u n e δ(t t 0) θ n u n (t 0 ) + C δ M ρ + δ n >M g n, for t>t 0, where t 0 is the time of entry of initial data bounded in l, to the absorbing ball of radius ρ in l. Since g l, then for all η > 0, there exists K(η) such that C M ρ + δ n >M g n η, M >K(η). Therefore, for all η and for t>t 0 and M>K(η), we obtain that θ n u n e δ(t t0) ρ + δ η.

22 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) Choosing t large enough, we may then obtain n >M u n This estimate holds as long as t T(η), where T(η) = t 0 + δ ln ( δρ η θ n u n η δ. ), and M>K(η). This concludes the proof of the Lemma. Remark. The above lemma has an additional implication, since it allows us to obtain an estimate for the time scales required for the localization of the energy to the part of the lattice with n M. Proposition 5.. Let 0 σ <. The semigroup S(t) is asymptotically compact in l, that is, if the sequence n is bounded in l and t n, then S(t n ) n is precompact in l. Proof. Let us consider a bounded sequence n l, such that n l r, r>0, n =,,... By Lemma 5., there exists T(r)>0, such that S(t) n B 0, t T(r). As t n there exists an integer N (r) such that t n T(r) for n N (r) and S(t n ) n B 0, n N (r). (70) Therefore S(t n ) n is weakly relatively compact. Thus, there exists 0 B 0 and a subsequence of S(t n ) n (not relabelled), such that S(t n ) n 0, weakly in B 0.To ensure precompactness we need to show that the above convergence is strong, that is for all η > 0, there exists N(η) N such that S(t n ) n 0 l η, n N(η). From (70) and the tail estimates provided by Lemma 5., we may conclude the existence of some K (η) and T (η), such that i K (η) (S(t)S(T r ) n ) i η 8, t T (η), where by (S(t) n ) i, we denote the ith coordinate of the infinite sequence S(t) n l. Since t n we may find N (r, η) N such that t n T r + T(η) if n N (r, η).

23 0 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 Hence, i K (η) (S(t n ) n ) i = i K (η) where we used the semigroup property for S(t). Since 0 l, we have the following tail estimate: i K (η) (S(t n T r )S(T r ) n ) i η 8, ( 0 ) i η 8. Let us now choose K(η) = max(k (η), K (η)). Since S(t n ) n 0 in l,wemay see that (S(t n ) n ) i ( 0 ) i strongly in C K(η)+. To prove the strong convergence we will break l into two distinct parts, the finite-dimensional part (corresponding to C K(η)+ ) and the infinite-dimensional part corresponding to the tails. Thus, S(t n ) n 0 l = (S(t n ) n 0 ) i + (S(t n ) n 0 ) i i K(η) η + i >K(η) i >K(η) ( (S(t n ) n ) i + ( 0 ) i ) η, where the first estimate comes from the strong convergence in the finite-dimensional space C K(η)+. Using Proposition 5. and Theorem.. of Temam [33], we may arrive at the main result of this section which can be stated as follows: Theorem 5.. Let 0 σ <. The semigroup S(t) associated to (54) (55) possesses a global attractor A = ω(b 0 ) B 0 l which is compact, connected and maximal among the functional invariant sets in l. 6. Finite-dimensional approximation of the global attractor This section is devoted to a result of stability of the global attractor of the dynamical system generated by (54) (55), under its approximation by a global attractor of an appropriate finite-dimensional dynamical system. Let us note first that problem (54) (55) could be viewed as a boundary value problem satisfying the boundary condition that the solutions are decaying at infinity. Since u(t) C (R +,l ) clearly lim n u n(t) = 0, t 0.

24 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 We seek for an approximation of (54) (55), by a finite-dimensional system of ordinary differential equations. We consider the boundary-value problem in C m+, i ψ n +(ψ n ψ n +ψ n+ ) + iδψ n + ψ n σ ψ n = g n, n m, 0 σ <, (7) ψ (m+) ( ) = ψ (m+) ( ) = 0, (7) ψ n (0) = ψ n,0, n m. (73) In a similar manner to the infinite-dimensional system, we may show that for the well posedness and asymptotic behavior of solutions of (7), the following result holds. Proposition 6.. Let ψ 0 := (ψ n,0 ) n m C m+. For 0 σ <, there exists a unique solution of (7) (73) such that ψ C ([0, ), C m+ ). The dynamical system defined by (7) (73) S m (t) : ψ 0 C m+ ψ(t) C m+, (74) possesses a bounded absorbing set O 0 in C m+ and a global attractor A m O 0 C m+ : for every bounded set O of l, there exists t (O, O 0 ) such that for all t t (O, O 0 ), it holds that S m (t)o O 0 and for every t 0, S m (t)a m = A m. Note that the estimates used for the proof of Proposition 6. (similar to those used in the proof Lemma 5.) are independent of m. Following [38,39] (see also [5,0] for a similar idea applied to pdes considered in all of R N ), we observe that the C m+ -solution of (7) (73) can be extended naturally in the infinite-dimensional space l,as ( m (t)) m Z := { ψ(t) := (ψn (t)) n m, n m, 0, n >m. (75) The extension (75) will be used in order to verify that the global attractor A of the semigroup S(t) associated with (54) (55), can be approximated by the global attractor A m of S m (t) associated to (7) (73), as m. We recall that the semidistance of two non-empty compact subsets of a metric space X, endowed with the metric d X (, ), is defined as d(b, B ) = sup inf d X (x, y). x B y B Theorem 6.. For any 0 σ <, the global attractor A m converges to A in the sense of the semidistance related to l : lim m d(a m, A) = 0. Proof. We denote by U an open-neighborhood of the absorbing ball B 0 of S(t). Obviously A attracts U. For arbitrary m Z, we consider the semigroup S m (t) defined by

25 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 Proposition 6. and its global attractor A m. Exactly as in Lemma 5., it can be shown that B 0 C m+ is also an absorbing set for S m (t). Therefore A m B 0 C m+ U C m+, and A m attracts U C m+. In the light of Proposition 6. and [33, Theorem I., p. 8], it remains to verify that for every compact interval I of R +, δ m (I) := sup sup d(s m (t)ψ 0, S(t)ψ 0 ) 0 as m. (76) ψ 0 U C m+ t I We consider the corresponding solution ψ(t) = S m (t)ψ 0, ψ(0) = ψ 0,inC m+ through (7) (73). Then by Proposition 6., it follows that ψ(t) A m for any t R +. Therefore, if ρ > 0 is the m-independent radius of the absorbing ball O m in C m+, then for every t R +, ψ(t) ρ. Using (75), we clearly observe that m (t) also satisfies the estimates m (t) l ρ, m (t) l C(ρ, g ), (77) the latter derived by (7). According to [33, Theorem 0., p ] or [39, Lemma 4, p. 60], for the justification of (76) it suffices to show that m (t) converges to a solution (t) of (54) (55) in an arbitrary compact interval of R +, and 0 = (0) in a bounded set of l. Let I be an arbitrary compact interval of R +. From estimates (77), we may extract a subsequence μ of m, such that μ (t) (t) in l, as μ for every t D, (78) where D denotes a countable dense subset of I. For any t I we consider the sequence γ m (t) := ( m (t), z) l, z l, which is differentiable by (77) as a function of t and γ m (t) = ( m (t), z) l.bythe mean-value theorem, there exists ξ I such that, for fixed t,s I ( m (t) m (s), z) l = γ m (t) γ m (s) = ( ψ m (t), z) l t s sup m (ξ) l z l t s C t s. ξ I It follows then, that there exists an m-independent constant C such that m (t) m (s) l C t s, (79)

26 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) i.e. the sequence γ m is equicontinuous. Hence by Ascoli s Theorem, it follows that the convergence (78), holds uniformly on I as μ. Summarizing, we obtain that for the subsequence μ hold the convergence relations μ in C(I,l ), μ in L (I, l ), (80) μ in L (I, l ). For every z l and ω(t) C 0 (I) we consider the formula (see [, p. 59]) I (i μ (t), z) l ω(t) dt + (f ( μ (t)), z) l ω(t) dt = I I (B μ (t), Bz) l ω(t) dt + δ I I (i μ (t), z) l ω(t) dt (g, z) l ω(t) dt, (8) where f( μ ) := μ σ μ. Needless to say that any solution of (8), is in our case, a solution of (54) (55) and vice versa. Using (80) we may pass to the limit in (8). Note for example, that since f : l l is Lipschitz continuous on bounded sets of l, there exists from (77), a constant c(ρ) such that f( μ ) f( ) l c(ρ) μ l. Then from (80) we infer (f ( μ (t)) f( (t)), z)ω(t) dt f( μ (t)) f( (t)) l z l ω(t) dt I I c μ (t) (t) l z l ω(t) I c sup μ (t) (t) l z l ω(t) dt t I I 0 as μ. Now we conclude by the same arguments as in [39, Lemma 4, p. 6]: Since I is arbitrary, (8) is satisfied for all t R +, i.e. (t) solves (54) (55). Moreover by (80) we get that (t) is bounded in l for all t R +. Therefore (t) A, which implies that μ (0) (0) A. Since the convergence holds for any other subsequence having the above formulated properties, by a contradiction argument using uniqueness, we may deduce that the convergence holds for the original sequence m. Condition (76) is proved.

27 4 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) The case of weighted spaces: Existence of spatially exponentially localized solutions We now turn to the properties of the solutions of the DNLS (54) (55), in weighted spaces. We consider weight function w n which is an increasing function of n, satisfying for all n Z, the following condition: w n, (W) w n+ w n d w n, d w n w n+. the space l w, { l w = u n C : u l := } w n u n <. w It can easily be seen that the space l w is a Hilbert space with the norm l. w Such spaces are the discrete analogue of weighted L spaces. A choice for such a function may be the exponential function w n = exp(λ n ) for λ > 0. Existence of solutions in such spaces will provide us with the existence of (exponentially) localized solutions for the DNLS equation (54) (55). The use of such spaces is important in the study of existence of soliton solutions or breathers. An instance where weighted spaces have been used is in [6], where the existence of exponentially localized solutions has been studied in conservative lattices using a continuation argument, related to the anti-integrable limit. We also refer to the very recent work [40]. Since the operator A is not symmetric in the space l w, we cannot apply Theorem. for the local existence in weighted spaces. We thus have to resort to general existence Theorems in Banach spaces. To this end we need to show that the operators involved are Lipschitz. This time as a model case we consider () (), with a nonlinearity satisfying (N ). We have the following Lemma 7.. Let condition (N ) be fulfilled. The operator T : l w l w, defined by T(z)= f( z )z satisfies the following properties: (i) T is bounded on bounded sets of l w and (ii) T is locally Lipschitz continuous. Proof. (i) Let u B R, a closed ball of l w of radius R. It follows from condition (W) that z n z for all n Z. Since f is continuous, we may argue as for the l w proof of Lemma., in order to get the inequality T(z) l w = w n f( z n ) z n w n g( z n ) z n

28 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) { } w n g( z l w) z n max g(ρ) w n z n ρ [0,R ] c(r) z l w, for some positive constant c(r). Thus we conclude that the operator T is bounded on bounded sets of l w. (ii) Since (N ) holds, for some θ (0, ) we have T(z) T(z ) l w w n f( z n ) z n z n + w n f (θ z n +( θ) z n ) ( z n + z n ) z n z n z n. A similar inequality to (9) should be obtained: we may see that T(z) T(z ) l w w n f( z l ) z n z n w { } + max g (ρ) c(r) w n z n z ρ [0,R n ] c(r)( z z l w). This concludes the proof of the Lemma. Lemma 7.. The operator A : l w l w, defined by (Au) n = u n+ u n + u n is globally Lipschitz on l w. Proof. Let u, v B R. Then (Au) (Av) = (u n+ v n+ ) (u n v n )+(u n v n ) and it follows that Au Av l w 4 u v l w. Using the above two lemmas we may restate the local existence result, in the case of weighted spaces. Theorem 7.. Let assumption (N ) be satisfied, assume that u 0 l w. There exists T (u 0 )>0, such that for all 0 <T <T (u 0 ), there exists a unique solution of the problem () (), u(t) C ([0,T],l w ). Proof. This time, we write () (), as an ordinary differential equation in l w u(t) + Φ(u(t)) = 0, u(0) = u 0,

29 6 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 where Φ(u) = T (u) + A(u) + iδu g and u(t) lies in l w. Lemmas are sufficient for the application of standard existence and uniqueness theorems, for ordinary differential equations in Banach spaces [37, p. 78 8]. In order to prove the existence of global solutions, we need to obtain some a priori estimates for the solution. In the dissipative case, this is achieved by proving the existence of a globally attracting ball of finite radius in l w. Lemma 7.3. Assume condition (W) on the weight function and that the damping coefficient satisfies δ d d / 0. (8) Let (u 0,n ) = u 0,(g n ) = g l w and condition (N ) be satisfied. A dynamical system can be defined by () (), S(t) : u 0 l w u(t) l w, (83) possessing a bounded absorbing set B 0 in l w : For every bounded set B of l w, there exists t 0 (B, B 0 ) such that for all t t 0 (B, B 0 ), it holds S(t)B B 0. Proof. We multiply () with w n u n, n Z, add over all lattice sites and keep the imaginary part. Working similarly as in Lemma 5., we obtain d dt w n u n n+ w n )(u,n u,n+ u,n u,n+ ) + δ (w δ w n u n + w n g n. δ Using the assumptions on the weight function (W), we find that (w n+ w n )(u,n u,n+ u,n u,n+ ) d w n (u,n u,n+ u,n u,n+ ) d ( w n u n w n u,n u,n+ + ) w n u,n u,n+. (84)

30 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) We now estimate the sums in the above inequality as follows: w n u,n u,n+ = ( and similarly, ( ) / wn / wn / u,n u,n+ w n u,n ( ( w n u,n+ ) / d / w n+ u,n+ ) / d / u l w w n u,n u,n+ d / u l. w w n u,n ) / Therefore, (w n+ w n )(u,n u,n+ u,n u,n+ ) d d / u l. w Thus, we obtain the differential inequality ( ) d δ dt u l + w d d / u l w δ g l. w Using the Gronwall inequality, if δ d d / 0 we obtain the existence of the attracting ball in l w. This concludes the proof of the Proposition. Remark7.. In the case of exponential weight w n = exp(λn), it appears that d = exp(λ) and d = exp(λ). The condition for existence of attracting ball becomes 8 sinh(λ/) δ. We may further prove the existence of a global attractor attracting all bounded sets of l w. The next lemma provides us with tail estimates in the weighted space. Lemma 7.4. Let (u 0,n ) = u 0 B, where B is a bounded set of l w and (g n) = g l w. Under assumptions (N ) on the nonlinearity, (W) on the weight function and assumption (8) on the dissipation, for any η > 0, there exist T(η) and K(η) such that

31 8 N.I. Karachalios, A.N. Yannacopoulos / J. Differential Equations 7 (005) 88 3 the solution (u n ) = u of () () satisfies for all t T(η), the estimate w n u n η for any M>K(η). (85) δ n >M Proof. The proof follows closely the proof of Lemma 5. for the corresponding tail estimates in the space l, only that we now multiply the equation by w n θ n u n, instead of multiplying simply by θ n u n. We obtain the following inequality where d dt + δ w n θ n u n I + δ w n θ n u n δ w n θ n u n w n θ n g n, (86) I = {(θ n+ w n+ θ n w n )(u,n u,n+ u,n u,n+ )}. We rewrite (θ n+ w n+ θ n w n ) = (θ n+ θ n )w n+ + θ n (w n+ w n ). Using arguments similar to those used in the proof of Lemmas 5. and 7.3, we see that θ n+ θ n w n+ u,n u,n+ u,n u,n+ C ρ M, (87) where C depends only on d,d and ρ denotes the absorbing ball in l w. We also have the inequality θ n w n+ w n u,n u,n+ u,n u,n+ d θ n w n u,n u,n+ +d θ n w n u,n u,n+. (88) For the first term of the rhs of (88), holds the estimate ( ) / ( ) / d θ n w n u,n u,n+ d θ n w n u,n θ n w n u,n+ ( ) / ( ) / d d / θ n w n u,n θ n w n+ u,n+. (89)