Dynamics of non-local lattice systems in l1

Dynamics of non-local lattice systems in `1 Jiaohui Xu1, Tom´ as Caraballo2,3, Jos´e Valero4 1Center for Nonlinear Studies, School of Mathematics Northwest University, Xi’an 710127, P. R. China 2Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain 3Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang Province, 325035, P. R. China 4Centro de Investigaci´on Operativa, Universidad Miguel Hern´andez de Elche Avenida de la Universidad s/n, 03202-Elche, Spain E-mail addresses: [email protected] (J. Xu), [email protected] (T. Caraballo), jv[email protected] (J. Valero) Abstract. In this paper, the well-posedness and asymptotic behavior of a non-local lattice system are analyzed in the space `1. In fact, the analysis is carried out in the subspace `1 +formed by the nonnegative elements, remaining open the case of the whole space. The same problem has been analyzed recently in the space `2(see Y. Li et al., Communications on Pure and Applied Analysis, 23 (2024), 935-960). However, the latter does not allow us to consider non-local terms which are natural in the modeling of reaction-diffusion problems introduced by M. Chipot in the wide literature published on this problem. With the current analysis, it is possible to investigate these interesting situations. Keywords: Non-local lattices; Reaction-diffusion models; Asymptotic behavior. 1 Introduction and preliminaries There exists a wide literature on reaction-diffusion models with non-local diffusion coefficients. The first works were published in the 90’s by Michel Chipot and his collaborators (see, e.g., [3, 4]). Later on, several variants of those models have been analyzed over the last two decades, considering nonautonomous and stochastic versions (see, e.g., [8] and the references therein). Very recently, Li et al. [7] considered lattice versions of these models (which are spatial discretization of the partial differential equations) and studied some dynamical properties. However, the set-up used for the nonlocal diffusion does not allow us to include, in particular, the natural cases mention in the works by Chipot (see [3, 4, 8]). To be more precise, let us first recall the notation that will be used throughout this work. As usual, let `k(k∈N={1,2,3· · · }) be the space of infinite sequences v= (..., v−1, v0, v1, ...) such that Pi∈Z|vi|k<∞. The norm in `kis given by kvk`k=Pi∈Z|vi|k1 k. In `2the scalar product (u, v)l2=Pi∈Zuiviis given. `∞is the space of sequences such that supi∈Z|vi|<∞endowed with the norm kvkl∞= supi∈Z|vi|. The following continuous embedding are well known: `1⊂`2⊂... ⊂ `k⊂... ⊂`∞. Li et al. considered in [7] the following lattice problem,      dui dt −a(l(u)) (ui+1 −2ui+ui−1) + αui+f(ui) = hi, i ∈Z, u(0) = u0∈l2, (1) where α > 0, u= (ui)i∈Z, h = (hi)i∈Z∈`2,l∈`2∗,a:R→R+,f:R→R. This system can be obtained as space discretization of the following nonlocal reaction-diffusion equation,      ∂u ∂t −a(l(u))∂2u ∂x2+αu +f(u) = h, x ∈R,t > 0, u(0) = u0, where l:L2(R)→Ris a continuous linear function. One possible functional l, which appears in applications, is given by l(u) = RRu(x)dx. Taking a fixed step h > 0, this integral can be 1 approximated by the sum Pi∈Zu(xi)h, where xi=hi. If we consider the infinite sequence v= (..., v−1, v0, v1, ...), where vi=u(xi), then we set l(v) = hPi∈Zvi, which is a continuous linear functional from `1into R. We emphasize that lis not defined for elements in `2, therefore, the nonlocal term given above by l(v) = hPi∈Zvicannot be included as nonlocal term for system (1) within the framework of the paper [7], in which l∈`2∗. It is worth also pointing out that due to the fact l∈`1∗, it is not possible to use the standard phase space `2(the usual one in lattice dynamical systems [2]). For this reason, we will study the lattice problem (1) but in a different phase space. More precisely, we need to define the semigroup in the phase space `1, as it has been done before in [1] for the discretization of a model of non-newtonian fluids. However, we found some additional difficulties to proceed in `1, since we are not able to obtain the necessary estimates for arbitrary solutions and, consequently, we have restricted our analysis to the case when solutions are non-negative, in other words, to the subspace `1 +. Actually, we consider the following lattice system,      dui dt −a(l(u)) (ui+1 −2ui+ui−1) + αui+f(ui) = hi, i ∈Z, u(0) = u0∈l1, (2) where α > 0, u= (ui)i∈Z, h = (hi)i∈Z∈`1,l∈`1∗,a:R→R+,f:R→R. The content of our paper is as follows. In Section 2, we will prove the well-posedness of problem (2), while Section 3 is devoted to proving the existence of a global attractor for our lattice problem. 2 Well-posedness of the problem Let us start by describing the assumptions that will be used in this paper: (A) The function ais locally Lipschitz, that is, there is a function L(r) such that, |a(s1)−a(s2)| ≤ L(r)|s1−s2|,∀s1, s2∈R,|sj| ≤ r, j = 1,2; (f) The function fbelongs to C1(R) and f(s)s≥0,∀s∈R; (3) (h)hi≥0 for all i∈Z. It is remarkable that our hypotheses on the function fare weaker than the ones imposed in [7]. Notice that, since fis continuous, condition (3) implies that f(0) = 0. We define the operators A, B :`1→`1by (Au)i=−ui+1 + 2ui−ui−1, (Bu)i=ui+1 −ui. It is easy to see that (Au, v)`2= (Bu, Bv)`2. Let Ψ : `1→`1be the operator defined by (Ψ (u))i=hi+a(l(u)) (ui+1 −2ui+ui−1)−αui−f(ui),for all i∈Z. Then problem (2) can be written as an abstract equation in the Banach space `1by the following way,      du dt = Ψ(u), t > 0, u(0) = u0∈`1. (4) First, we show that our model makes sense in `1. Throughout the paper we assume hypotheses (A),(f) and (h) hold. Theorem 1 The function Ψis well defined and locally Lipschitz in the space `1. Furthermore, problem (4) is well-posed, i.e., for each initial value u0∈`1, there exists a unique solution u(·;u0)to problem (4), and this solution is continuous with respect to the initial values. 2 Proof. We only need to establish that the operator F(u) given by (F(u))i=f(ui) is well defined, as for the other terms it is obvious. Since f∈C1(R) and u∈`1⊂`∞, we obtain kF(u)k`1=X i∈Z |f(ui)|=X i∈Z |f(ui)−f(0)|=X i∈Z |f0(τiui)| |ui| ≤ C1X i∈Z |ui|=C1kuk`1, where τi∈[0,1] and we have used the fact that the sequence |f0(τiui)|is bounded. Let now u, v ∈`1satisfy kuk`1,kvk`1≤r. Since |l(z)| ≤ Mkzk`1, by condition (A), we have |a(l(u)) −a(l(v))| ≤ L(Mr)|l(u)−l(v)| ≤ L(Mr)Mku−vk`1. Also, kAu −Avk`1≤4ku−vk`1and a(l(v)) ≤maxs∈[−Mr,Mr]a(s) = K(r), hence ka(l(u))Au −a(l(v))Avk`1≤ |a(l(u)) −a(l(v))| kAuk`1+a(l(v)) kAu −Avk`1 ≤4rL (Mr)Mku−vk`1+ 4K(r)ku−vk`1. On the other hand, kF(u)−F(v)k`1=X i∈Z |f(ui)−f(vi)|=X i∈Z |f0(θi)| |ui−vi| ≤ C2ku−vk`1, where θi=λiui+ (1 −λi)vi,λi∈[0,1], and we have used the fact that the sequence |f0(θi)|is bounded uniformly for u, v ∈`1satisfying kuk`1,kvk`1≤r. It follows immediately the existence of D(r)>0, such that kΨ (u)−Ψ(v)kl1≤D(r)ku−vk`1, for any u, v ∈`1satisfying kuk`1,kvk`1≤r. As a consequence of the previous results, the existence, uniqueness and the continuity of solutions with respect to initial values follow straightforwardly from the general theory of well-posedness of initial value problems in Banach spaces (see Zeidler [9]). Let us now prove that solutions to (4) are non-negative for non-negative initial values. To this end, we first recall the definitions u+:= max{u, 0},u−:= max{−u, 0}for u∈`1, and `1 +={u∈`1:ui≥0,∀i∈Z}. Lemma 2 If u0∈`1 +, then the unique solution u(t)of (2) satisfies u(t)∈`1 +for all t∈[0, Tmax), where [0, Tmax)denotes the maximal interval where the solution is defined. Proof. A solution u(t) of (2) satisfies d dt (−u)i+a(l(u)) (A(−u))i+α(−u)i−fi(ui) = −hi.(5) We take the scalar product in `2of (5) with (−u)+.Then using Lemma 2.2 in [1], we find 1 2 d dt   (−u)+   2 `2+α  (−u)+   2 `2+a(l(u)) A(−u),(−u)+`2−X i∈Z fi(ui) (−u)+ i =−X i∈Z hi(−u)+ i. As hi≥0, −Pi∈Zhi(−u)+ i≤0. Also, by (3) we have fi(ui) (−u)+ i= 0,if ui≥0, fi(ui) (−u)+ i=−fi(ui)ui≤0,if ui<0. Therefore, −X i∈Z fi(ui) (−u)+ i≥0. 3 Further, we will prove that A(−u),(−u)+`2≥0. Indeed, A(−u),(−u)+`2=A(−u)+,(−u)+`2−A(−u)−,(−u)+`2 ≥ − X i∈R (−u)− i+1 −(−u)− i,(−u)+ i+1 −(−u)+ i`2 =X i∈R (−u)− i+1 (−u)+ i+ (−u)− i(−u)+ i+1≥0. Hence,   (−u)+(t)   2 `2≤  (−u)+(0)   2 `2= 0. This implies that u(t)≥0 for any t≥0. Next, we prove an estimate that will be crucial to ensure the existence of absorbing sets. Lemma 3 There exists a constant C0>0such that, ku(t)k`1≤C0+e−αt ku(0)k`1,∀t∈[0, Tmax),(6) for any solution of (2) with u(0) ≥0. Proof. Notice that u(t)≥0 and Pi∈Z(ui+1 −2ui+ui−1) = 0, f (s)≥0 for s≥0.Thus, d dt X i∈Z ui+αX i∈Z ui≤X i∈Z hi. Consequently, ku(t)k`1≤e−αt ku(0)kl1+1 αkhk`1,∀t∈[0, Tmax). The proof of this lemma is complete. 3 Existence of global attractors Inequality (6) implies that Tmax = +∞for any solution with non-negative initial condition. Thus, all solutions are globally defined, in this way, we can define the semigroup of operators S:R+×`1 +→`1 + given by S(t, u0) = u(t) := u(t;u0). The locally Lipschitz property of the Nemitskii operator Ψ implies that, for any R > 0, there is a constant MR>0 such that,  St, u0−St, v0 `1≤eMRt u0−v0 `1,∀u0, v0∈BR, t ≥0, where BRis the closed ball with radius Rcentered at 0 in `1 +.Using the above inequality, it is easy to see that the map t, u07→ S(t, u0) is continuous. It follows immediately from Lemma 3 that the existence of a bounded absorbing set. Corollary 4 The ball B0={u∈`1 +:kuk`1≤2C0}, is absorbing for S, which means that for any bounded set in `1 +, there exists T(B)>0such that S(t, B)⊂B0,∀t≥T. To conclude with the existence of the global attractors for this semigroup in `1 +, we need to obtain an estimate on the tails of solutions. Lemma 5 Let B⊂`1 +be a bounded set. Then for any  > 0, there exist T(, B)and K(, B), such that X |i|≥2k St, u0≤, if t≥T, k ≥K. (7) 4 Proof. Define a smooth function θsatisfying, θ(s) =        0,0≤s≤1, 0≤θ(s)≤1,1≤s≤2, 1, s ≥2. Obviously, |θ0(s)| ≤ Cfor all s∈R+. Let ρk,i := θ|i| k, where k≥1.We multiply (2) by v= (vi)i∈Z, where vi=ρk,i. Then, d dt X |i|≥k uivi+αX |i|≥k uivi+a(l(u(t))) X |i|≥k (Bu)i(Bv)i+X |i|≥k f(ui)vi=X |i|≥k hivi.(8) We see from (3) that P|i|≥kf(ui)vi≥0. Also, P|i|≥khivi≤P|i|≥khi. Let us estimate the term P∞ i=k(Bu)i(Bv)i. Observe that, ∞ X i=k (Bu)i(Bv)i= ∞ X i=k (ui+1 −ui) (ρk,i+1 −ρk,i)≤1 k ∞ X i=k |ui+1 −ui| |θ0(τi)|, where τi∈i k,i+1 k. A similar estimate holds for P∞ i=−k(Bu)i(Bv)i. Hence, using |θ0(s)| ≤ D, we find, d dt X |i|≥k uivi+αX |i|≥k uivi≤X |i|≥k hi+D ka(l(u(t))) X |i|≥k (ui+1 +ui). In view of (6), there is M1(B)>0 such that ku(t)k`1≤M1(B) for t≥0. Since l∈`∗ 1and ais continuous, there is M2(B)>0 such that a(l(u(t))) ≤M2(B) for t≥0. Hence, d dt X |i|≥k uivi+αX |i|≥k uivi≤X |i|≥k hi+M3(B) k, which implies, X |i|≥k ui(t)vi≤e−αt X |i|≥k ui(0)vi+1 α  X |i|≥k hi+M3(B) k  . It follows the existence of T(, B), K(, B) such that X |i|≥2k ui(t)≤X |i|≥k ui(t)vi≤, if k≥K, t ≥T. We finish the proof of this lemma. Proposition 6 Let Bbe bounded in `1 +. Then every sequence ξn∈S(tn, B)with tn→ ∞ is relatively compact in `1 +. Proof. By Lemma 3, we know {ξn}is bounded in `1and therefore in `2as well. Thus, up to a subsequence, ξn→ξweakly in `2. We need to show that, in fact, ξn→ξstrongly in `1 +. Let Pk:`1 +→R2k+1 be the projection operator given by Pku= (u−k,· · · , uk). The weak convergence in `2implies that Pkξn→Pkξstrongly in R2k+1 for any k. By Lemma 5, we know for any  > 0, there are K0(, B) and N0(, B) such that P|i|≥K0ξn i≤/4 if n≥N0.We then choose N1(, B)≥N0(, B) such that P|i|≤K0|ξn i−ξi| ≤ /4 if n≥N1. It follows that {ξn}is a Cauchy sequence in `1 +. Indeed, kξn−ξmk`1 +=X |i|≤K0 |ξn i−ξm i|+X |i|>K0 |ξn i−ξm i| ≤X |i|≤K0 |ξn i−ξi|+X |i|≤K0 |ξm i−ξi|+X |i|>K0 |ξn i|+X |i|>K0 |ξm i| ≤ , 5 if n, m ≥N1. It follows that {ξn}is strongly convergent in `1 +, so ξn→ξin `1 +. Let A, B ⊂`1 +.Recall that the Hausdorff semi-distance from Ato Bis defined as follows: dist(A, B) = supx∈Aninfy∈Bkx−yk`1 +o. The compact set A ⊂ `1 +is said to be the global attractor for Sif Ais invariant, i.e., A=S(t, A) for all t≥0 and Aattracts every bounded set Bof `1 +, that is, dist (S(t, B),A)→0 as t→+∞.(9) Theorem 7 The semigroup Spossesses the connected global attractor A. Proof. By means of Lemma 4 and Proposition 6, the existence of the global attractor follows from the general theory of attractors [6, Theorem 3.1]. The fact that it is connected follows from [5, p. 4]. Remark 8 We have proved the existence of the global attractor for non-negative solutions. For the semigroup defined on the whole space `1, this problem still remains open. Acknowledgements. The research has been partially supported by the National Natural Science Foundation of China (No.12301234), the Spanish Ministerio de Ciencia e Innovaci´on (MCI), Agencia Estatal de Investigaci´on (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PID2021-122991NB-C21, by the Generalitat Valenciana, project PROMETEO/2021/063. References [1] J. M. Amig´o, A. Gim´enez, F. Morillas, J. Valero, Attractors for a lattice dynamical system generated by non-newtonian fluids modelling suspensions, International Journal of Bifurcations and Chaos,20 (2010), 2681-2700. [2] P. W. Bates, K. Lu, B. Wang, Attractors for lattice dynamical systems, International Journal Bifurcations Chaos,11 (2001), 143-153. [3] M. Chipot, B. Lovat, On the asymptotic behaviour of some nonlocal problems, Positivity,3(1999), 65-81. [4] M. Chipot, I. Shafrir, V. Valente, G. V. Caffarelli, A nonlocal problem arising in the study of magneto-elastic interactions, Boll. Unione Mat. 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