Existence of Pullback Attractors for Pullback Asymptotically Compact Processes

EXISTENCE OF PULLBACK ATTRACTORS FOR PULLBACK ASYMPTOTICALLY COMPACT PROCESSES TOM´ AS CARABALLO1, ALEXANDRE N. CARVALHO2, JOS´ E A. LANGA3, AND FELIPE RIVERO4 Abstract. This paper is concerned with the existence of pullback attractors for evolution processes. Our aim is to provide results that extend the following results for autonomous evolution processes (semigroups) i) An autonomous evolution process which is bounded dissipative and asymptotically compact has a global attractor. ii) An autonomous evolution process which is bounded, point dissipative and asymptotically compact has a global attractor. The extension of such results requires the introduction of new concepts and brings up some important differences between the asymptotic properties of autonomous and nonautonomous evolution processes. An application is considered to damped wave problem with non-autonomous damping. 1. Introduction One of the central problems in dynamical systems is the study of the asymptotic behavior of evolution processes associated to the modeling of real world phenomena. When the model under study is an autonomous differential equation, the study of its asymptotic behavior is rather well established and many references on the subject are available (cf. Temam [17], Hale [8], Ladyzhenskaya [10], Babin-Vishik [1], Robinson [13] for example). However, if the evolution process comes from a non-autonomous differential equation, even though some nice references are already available (cf. Cheban [4], Chepyzhov-Vishik [6], Kloeden [9], Sell-You [15]), much is yet to be done. The subject of this paper is the existence of pullback attractors for evolution processes. As we will see, even this fundamental question is not as well developed as its autonomous counterpart. Our aim is to give results that allow us to conclude the existence of pullback 1Partially supported by Ministerio de Ciencia e Innovaci´on grant # MTM2008-0088 and Junta de Andaluc´ıa grants # P07-FQM-02468, # FQM314, Spain. 2Partially supported by CNPq 305447/2005-0 and 451761/2008-1, CAPES/DGU 267/2008 and FAPESP 2008/53094-4, Brazil. 3Partially supported by Ministerio de Ciencia e Innovaci´on grants # MTM2008-0088, # PBH2006-0003- PC, and Junta de Andaluc´ıa grants # P07-FQM-02468, # FQM314, Spain. 4Partially supported by Ministerio de Ciencia e Innovaci´on grant # MTM2008-0088 and Junta de Andaluc´ıa grants # P07-FQM-02468, # FQM314, Spain. 1 2 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND F. RIVERO attractors for pullback asymptotically compact and pullback point (or bounded) dissipative processes. We also point out the features of non-autonomous evolution processes must be considered to obtain results which are analogues to those for the autonomous evolution processes emphasizing the distinction between the autonomous and non-autonomous cases. To better explain the results in the paper we introduce some basic terminology. Let Xbe a metric space and d(·,·) : X×X→[0,∞) be its metric. Denote by C(X) the set of all continuous transformations from Xinto itself. An evolution process in Xis a family of maps {S(t, s) : t⩾s} ⊂ C(X) with the following properties 1) S(t, t) = I, for all t∈R, 2) S(t, s) = S(t, τ)S(τ, s),for all t⩾τ⩾s, 3) {(t, s)∈R2:t⩾s} × X3(t, s, x)7→ S(t, s)x∈Xis continuous. The processes {S(t, s) : t⩾s}for which S(t, s) = S(t−s, 0) for all t⩾sare called autonomous evolution process and the family {T(t) : t⩾0}given by T(t) = S(t, 0), t⩾0 is called a semigroup; that is, a family {T(t) : t≥0}which satisfies 1) T(0) = I, 2) T(t+s) = T(t)T(s),for all t, s ⩾0, 3) [0,∞)×X3(t, x)7→ T(t)x∈Xis continuous. Reciprocally, given a semigroup {T(t) : t⩾0}the family {T(t−s) : t⩾s}is an evolution process. For an autonomous evolution process {S(t, s) = S(t−s, 0) : t⩾s}, the behavior of solutions as t→ ∞, which is called the forwards dynamics, is the same as the behavior of solutions as s→ −∞, which is called the pullback dynamics. For general processes, these two dynamical limits are totally unrelated and can produce entirely different qualitative properties (see [3]). The pullback orbit of a subset Bof Xat time t∈R, is defined by γp(B, t) :=Ss⩽tS(t, s)B, where S(t, s)B:= {S(t, s)x:x∈B}.{S(t, s) : t⩾s}is said pullback bounded if γp(B, t) is bounded whenever Bis a bounded subset of Xand for each t∈R. A set K(t)⊂Xpullback attracts a set Cat time tunder {S(t, s) : t⩾s}if lim s→−∞ dist(S(t, s)C, K(t)) = 0. A family {K(t) : t∈R}pullback attracts bounded subsets of Xunder {S(t, s) : t⩾s} if K(t) pullback attracts all bounded subsets at tunder {S(t, s) : t⩾s}, for each t∈R. Given t∈R,B(t)⊂Xpullback absorbs bounded subsets of Xat time tif, for each bounded subset Bof X, there exists T=T(t, B)⩽tsuch that S(t, s)B⊂B(t),for all s⩽T(t, B). EXISTENCE OF PULLBACK ATTRACTORS FOR PULLBACK ASYMPTOTICALLY COMPACT PROCESSES3 A family {B(t) : t∈R}pullback absorbs bounded subsets of Xif B(t) pullback absorbs bounded sets at time t, for each t∈R. If there exists a family {B(t) : t∈R}of bounded sets which pullback absorbs bounded subsets of Xthen we say that the evolution process {S(t, s) : t⩾s}is pullback bounded dissipative. In a completely similar way we define the notions of pullback point dissipative or pullback compact dissipative processes. Let {B(t) : t∈R}a family of subsets of X. We say that this family is invariant with respect to the process {S(t, s) : t⩾s}if S(t, s)B(s) = B(t),for all t⩾s. Definition 1.1. A family {A(t) : t∈R}of compact subsets of Xis said a pullback attractor for the evolution process {S(t, s) : t⩾s}if it is invariant, pullback attracts bounded subsets of Xand is minimal; that is, if there is another invariant family {C(t) : t∈ R}which pullback attracts bounded subsets of X, then A(t)⊆C(t),for all t∈R. An autonomous evolution process {S(t−s) : t≥s}has a pullback attractor {A(t) : t∈R} if and only if the semigroup {S(t) : t≥0}has a global attractor A(in the sense of [8]) and in either case A(t) = Afor all t∈R. We remark that the minimality requirement in Definition 1.1 is additional relative to the theory of attractors for semigroups. This minimality requirement is essential to ensure uniqueness of pullback attractors. Its inclusion is related to the weakening of the invariance property imposed by the non-autonomous nature of general evolution processes. If {S(t) : t⩾0}is a semigroup and {S(t−s) : t⩾s}is the process associated to it, there may exist a family {A(t) : t∈R}of compact invariant sets that pullback attracts bounded subsets and is not minimal. Indeed, if S(t−s) = e−(t−s)x0,x0∈R,t⩾sand c⩾0 the family {[−ce−t, ce−t] : t∈R}is invariant, [−ce−t, ce−t] is compact and attracts bounded subsets of Rat time tfor each t∈R. Remark 1.2. Recall that, a global solution for an evolution process {S(t, s) : t⩾s}is a function ξ:R→Xsuch that S(t, s)ξ(s) = ξ(t)for all t≥s. We say that a global solution ξ:R→Xof an evolution process is backwards bounded if there is a τ∈Rsuch that {ξ(t) : t⩽τ}is a bounded subset of X. If an evolution process {S(t, s) : t⩾s}has a pullback attractor {A(t) : t∈R}and ξ:R→Xis a backwards bounded solution, then ξ(t)∈ A(t)for all t∈R. If we require, in addition, that the pullback attractor {A(t) : t∈R}has the property that ∪s≤τA(s)is bounded for each τ∈R, the requirement that the pullback attractor is minimal in Definition 1.1 can be dropped and A(t) = {ξ(t) : ξ:R→Xis a global backwards bounded solution of {S(t, s) : t⩾s}} 4 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND F. RIVERO The notion of a pullback asymptotically compact evolution process is naturally associated with evolution processes which posses a pullback attractor. Definition 1.3. An evolution process {S(t, s) : t⩾s}in a metric space Xis said pullback asymptotically compact if, for each t∈R, sequence {sk}k∈Nin (−∞, t]and bounded sequence {xk}k∈Nin Xsuch that •sk k→∞ −→ −∞ and •{S(t, sk)xk:k∈N}is bounded, the sequence {S(t, sk)xk}k∈Nhas a convergent subsequence. Next we give two characterizations of autonomous evolution processes which posses global attractors. Both characterizations are widely used to obtain existence of pullback attractors for autonomous evolution processes. We will first formulate them in the classical way, and later we will rewrite them taking into account the framework described above. (See [8] or [10] for more details of the autonomous formulation). Theorem 1.4. If {S(t) : t≥0}is a semigroup, the following conditions are equivalent: i) {S(t) : t≥0}possesses a global attractor. ii) {S(t) : t≥0}is bounded dissipative and asymptotically compact. iii) {S(t) : t≥0}is bounded, point dissipative and asymptotically compact. Condition ii) gives the following characterization of autonomous processes which have pullback attractors Theorem 1.5. If {S(t) : t≥0}is a semigroup, the autonomous evolution process {S(t−s) : t≥s}has a pullback attractor if and only if it is pullback asymptotically compact and pullback bounded dissipative. On the other hand, condition iii) gives we have that Theorem 1.6. If {S(t) : t≥0}is a semigroup, the autonomous evolution process {S(t−s) : t≥s}has a pullback attractor if and only if {S(t−s) : t≥0}is pullback bounded, pullback point dissipative and pullback asymptotically compact. The decision about which characterization should be used in a given application depends on the properties of the model under study. The first result is widely used associated to energy estimates for asymptotically compact autonomous evolution processes and the second is mostly associated to asymptotically compact gradient autonomous evolution processes with a bounded set of equilibria. Our aim is to provide extensions of the above results to the case of general (autonomous or not) evolution processes. As we will see in Section 2, even the statement of such extensions in the general case will require the introduction of several new (natural) concepts. EXISTENCE OF PULLBACK ATTRACTORS FOR PULLBACK ASYMPTOTICALLY COMPACT PROCESSES5 The paper is organized as follows. In Section 2 we state the main results of the paper, introduce the necessary definitions and terminology to be able to state and apply them. In Section 3 we prove the analogues to Theorem 1.5 and prove a result that implies pullback asymptotic compactness. In Section 4 we prove the analogue of Theorem 1.6. 2. Statement of the results and their applications Let us start recalling the definition of pullback ω−limit sets. Definition 2.1. Let {S(t, s) : t⩾s}be an evolution process in a metric space Xand Bbe a subset of X. The pullback ω-limit of Bis defined by ω(B, t) := \ σ⩽t[ s⩽σ S(t, s)B. For each subset Bof X, it holds that ω(B, t) = {y∈X: there are sequences {sk}k∈Nin (−∞, t], sk k→∞ −→ −∞ and {xk}k∈Nin B, such that y= lim k→∞ S(t, sk)xk}. (2.1) The following theorem gives a characterization of evolution processes with a pullback attractor. So far, it is the only result available to obtain existence of pullback attractors in applications (see [8, 10, 12, 13, 17] for the autonomous case). Theorem 2.2. Let {S(t, s) : t⩾s}be an evolution process in a metric space X. Then, the following statements are equivalent •{S(t, s) : t⩾s}has a pullback attractor {A(t) : t∈R}. •There exists a family of compact sets {K(t) : t∈R}that pullback attracts bounded subsets of Xunder {S(t, s) : t⩾s}. In either case (2.2) A(t) = [{ω(B, t) : B⊂X, B bounded } and {A(t) : t∈R}is minimal in the sense that, if there exists another family of closed bounded sets {ˆ A(t) : t∈R}which pullback attracts bounded subsets of Xunder {S(t, s) : t⩾s}, then A(t)⊆ˆ A(t), for all t∈R. This result is proved in [2, 12] and we provide a short proof in Section 3 for completeness. The application of such result for processes which are not compact may be difficult because one must find a compact set K(t) which pullback attracts bounded subsets of Xfor each t∈R. We aim to provide a few alternative results to prove existence of pullback attractors in order to have a theory as complete as in the autonomous case. We prove in Section 3 that 6 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND F. RIVERO Theorem 2.3. If {S(t, s) : t⩾s}is pullback bounded dissipative and pullback asymptotically compact, then A(t)given by (2.2) is closed, invariant, pullback attracts bounded subsets of Xat time t, and the family {A(t) : t∈R}is minimal among the families {B(t) : t∈R} such that B(t)is closed and pullback attracts bounded subsets of Xat time t. Observe that the previous result does not give any compactness of A(t).This is not a restriction in the finite dimensional case, as A(t) is actually bounded and closed. However, the result shows a first difference with respect to analogous results for autonomous evolution processes in the infinite dimensional case. Getting the same kind of results requires to adopt different strategies, any of them imposing new hypotheses on the dynamics of the processes. Definition 2.4. We say that an evolution process {S(t, s) : t⩾s}is pullback strongly bounded dissipative if, for each t∈R,there is a bounded subset B(t)of Xwhich pullback attracts bounded subsets of Xat time τfor each τ⩽t; that is, given a bounded subset Bof Xand τ⩽t, there exists s0(τ, D)such that S(τ, s)B⊂B(t), for all s⩽s0(τ, D). Note that the family {B(t) : t∈R}given in the above definition does not need to have a bounded union. Nonetheless, we may choose it in such a way that, for each t∈R,Ss⩽tB(s) is bounded. The following theorem, proved in Section 3, gives a sufficient condition for the existence of a compact pullback attractor. Theorem 2.5. If an evolution process {S(t, s) : t⩾s}is pullback strongly bounded dissipative and pullback asymptotically compact, then {S(t, s) : t⩾s}has a pullback attractor {A(t) : t∈R}with the property that Ss⩽tA(s)is bounded for each t∈R. To prove (in applications) that a process is asymptotically compact, we will need to assume that the evolution process is pullback strongly bounded as defined next Definition 2.6. We say that an evolution process {S(t, s) : t⩾s}in Xis pullback strongly bounded if, for each t∈Rand bounded subset Bof X,Ss⩽tγp(B, s)is bounded. Remark 2.7. If {S(t) : t⩾0}is a semigroup, {S(t−s) : t⩾s}is pullback strongly bounded if and only if {S(t−s) : t⩾s}is pullback bounded if and only if {S(t) : t⩾0}is a bounded semigroup. For evolution processes which are pullback strongly bounded, the following result (proved in Section 3) gives sufficient conditions for pullback asymptotic compactness Theorem 2.8. Let {S(t, s) : s⩽t}be a pullback strongly bounded process such that S(t, s) = T(t, s) + U(t, s), where U(t, s)is compact and there exits a non-increasing function k:R+×R+−→ R with k(σ, r)→0when σ→ ∞, and for all s⩽tand x∈Xwith kxk⩽r,kT(t, s)xk⩽ k(t−s, r). Then, the process {S(t, s) : s⩽t}is pullback asymptotically compact. EXISTENCE OF PULLBACK ATTRACTORS FOR PULLBACK ASYMPTOTICALLY COMPACT PROCESSES7 In Section 4 we pursue a result that would enable us to conclude the existence of pullback attractors without having to prove pullback strong bounded dissipativeness but rather pullback strong point dissipativeness (see Hale [8], Raugel [12]). To that end, the notion of pullback asymptotic compactness and pullback dissipativeness associated to the elapsed time presented next is needed. Definition 2.9. We say that an evolution process {S(t, s) : t⩾s}is pullback strongly asymptotically compact if for each t∈R, bounded sequence {xk:k∈N}in X, sequences {sk:k∈N},{τk:k∈N}with t⩾τk⩾skand τk−sk k→∞ −→ ∞, then {S(τk, sk)xk:k∈N} is relatively compact. If {S(t) : t⩾0}is a semigroup, {S(t−s) : t⩾s}is pullback strongly asymptotically compact if and only if {S(t−s) : t⩾s}is pullback asymptotically compact if and only if {S(t) : t⩾0}is asymptotically compact in the sense of [10]. The process {S(t, s) : s⩽t}is called strongly compact if for each time tand B⊂Xthere exits a TB⩽0 and a compact set K⊂Xsuch that U(τ, s)B⊂Kfor all s⩽τ⩽twith τ−s⩾TB. Definition 2.10. Let {S(t, s) : t⩾s}be an evolution process in a metric space X. We say that a bounded set B(t)of Xpullback strongly absorbs points (compact subsets)of X at time tif, for each x∈X(compact subset Kof X), there exist σx>0 (σK>0) such that S(τ, s)(x)∈B(t) (S(τ, s)(K)⊂B(t)) for all s⩽τ⩽twith τ−s⩾σx(τ−s⩾σK). We say that {S(t, s) : t⩾s}is pullback strongly point dissipative (compact dissipative) if, for each t∈R, there is a bounded subset B(t)⊂Xwhich pullback strongly absorbs points (compact subsets)of Xat time t. Remark 2.11. If a set B(t)pullback strongly absorbs points (compact subsets/bounded subsets)of Xat t, then it pullback strongly absorbs points (compact subsets/bounded subsets) of Xat τfor all τ⩽t. Also, if {S(t) : t⩾0}is a semigroup then {S(t−s) : t⩾s}is pullback strongly point dissipative (compact dissipative)if and only if {S(t−s) : t⩾s}is pullback point dissipative (compact dissipative)if and only if {S(t) : t⩾0}is point dissipative (compact dissipative)in the sense of [8]. With these concepts we can prove the following result on existence of pullback attractors. Theorem 2.12. Let {S(t, s) : t⩾s}be an evolution process with the property that, for each t∈Rand τ > 0,{S(s, s −τ) : s⩽t}, is equicontinuous at xfor each x∈X. If {S(t, s) : t⩾s}is pullback strongly point dissipative, pullback strongly bounded and pullback strongly asymptotically compact, then {S(t, s) : t⩾s}is pullback strongly bounded dissipative. Consequently, {S(t, s) : t⩾s}has a pullback attractor {A(t) : t∈R}with the property that Ss⩽tA(s)is bounded for each t∈R. In applications, to prove the pullback strong asymptotic compactness property we use the following result 8 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND F. RIVERO Theorem 2.13. Let {S(t, s) : s⩽t}be a pullback strongly bounded process such that S(t, s) = T(t, s) + U(t, s), where U(t, s)is strongly compact and there exits a non-increasing function k:R+×R+−→ R with k(σ, r)→0when σ→ ∞, and for all s⩽tand x∈Xwith kxk⩽r,kT(t, s)xk⩽ k(t−s, r). Then, the process {S(t, s) : s⩽t}is pullback strongly asymptotically compact. 2.1. A pullback strongly bounded dissipative wave equation. In this section we give an application of Theorem 2.5. Let us consider the following nonautonomous wave equation (2.3) (utt +β(t)ut= ∆u+f(u) in Ω u(x, t) = 0 in ∂Ω where Ω ⊂Rnis a bounded smooth domain Rn. For f:R→Rwe assume that: f∈C2(R),|f0(s)|⩽c(1 + |s|p−1),lim sup |s|→∞ f(s) s⩽0, (2.4) with c > 0 and p < n n−2 Assume that β:R→Ris a bounded, globally Lipschitz function and that there are (2.5) β0⩽β(t)⩽β1for some β0, β1∈(0,∞). We will prove that the nonautonomous process associated to (2.3) has a pullback attractor by applying Theorem 2.8. Let X=H1 0(Ω) ×L2(Ω), for ut=vand w= ( u v), we rewrite (2.3) as (2.6) wt=C(t)w+F(w) where (2.7) C(t) = µ0I −A−β(t)I¶and F(w) = µ0 f(u)¶ A=−∆ with Dirichlet boundary condition. For each initial value w0∈Xand initial time s∈R, system (2.6) possesses a unique solution which can be written as (2.8) S(t, s)w0=L(t, s)w0+U(t, s)w0, where L(t, s) is the solution operator for wt=C(t)w, and U(t, s)w0=Zt s L(t, τ)F(S(τ, s)w0)dτ. Let us prove that {S(t, s) : t⩾s}is pullback asymptotically compact using Theorem 2.8. EXISTENCE OF PULLBACK ATTRACTORS FOR PULLBACK ASYMPTOTICALLY COMPACT PROCESSES9 Proposition 2.14. There are positive constants K > 0,α > 0, such that: kL(t, s)k⩽Ke−α(t−s),t⩾s. Proof: Consider the following function (2.9) V(ϕ, φ) = 1 2|ϕ|2 H1 0(Ω) + 2b(ϕ, φ)L2(Ω) +1 2|φ|2 L2(Ω), (ϕ, φ)∈X For suitably small b0>0 and b≤b0we have that 1 4[|ϕ|2 H1 0(Ω) +|φ|2 L2(Ω)]⩽1 2|ϕ|2 H1 0(Ω) + 2b(ϕ, φ) + 1 2|φ|2 L2(Ω) ⩽3 4[|ϕ|2 H1 0(Ω) +|φ|2 L2(Ω)]. (2.10) If (u(t, s), ut(t, s)) is a solution of (utt +β(t)ut−∆u= 0 in Ω u(x, t) = 0 in ∂Ω, then d dtV(u, ut)=(u, ut)H1 0(Ω)+2b(ut, ut)L2(Ω)+2b(u, utt)L2(Ω)+(ut, utt)L2(Ω). Therefore, d dtV(u(t, s), ut(t, s)) =−(β(t)−2b)|ut|2 L2(Ω) −2b|u|2 H1 0(Ω) −2β(t)(u, ut)L2(Ω) ⩽−(β0−2b−bβ1 ε)|ut|2 L2(Ω) + (bβ1ε−bλ1)|u|2 L2(Ω) −b|u|2 H1 0(Ω), with ε=λ1 β1and b0>0 small enough. Then we have that, for all 0 < b ≤b0, (2.11) d dtV(u(t, s), ut(t, s)) = −αV (u(t, s)ut(t, s)). Hence, V(L(t, s)(ϕ, φ)) ⩽V(ϕ, φ)e−α(t−s) and, consequently, kL(t, s)(ϕ, φ)k2 X⩽Ke−α(t−s)k(ϕ, φ)kproving the result. Theorem 2.15. {S(t, s) : t⩾s}is pullback strongly bounded dissipative. Proof: Consider the energy functionals (2.12) V(ϕ, φ) = 1 2|ϕ|2 H1 0(Ω) + 2b(ϕ, φ) + 1 2|φ|2 L2(Ω) −ZΩ G(ϕ), 16 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND F. RIVERO such that dist(S(τ, sk)xk, ω(D, τ)) k→∞ −→ 0. Taking {rk:k∈N}with τ⩾rk⩾skand min{τ−rk, rk−sk}k→∞ −→ ∞ and using the fact that {S(t, s) : t⩾s}is pullback strongly asymptotically compact (taking subsequences if necessary), there is a z∈Xsuch that zk:= S(rk, sk)xk k→∞ −→ z. From the compactness of the set K={zk:k∈N} ∪ {z}, there is aσK∈Nsuch that S(τ, rk)K⊂B(t) whenever τ−rk⩾σK.Thus, for all suitably large k, S(τ, sk)xk=S(τ, rk)S(rk, sk)xk⊂S(τ, rk)K⊂B(t). This completes the proof that ω(D, τ)⊂B(t) for each τ⩽t. Since ω(D, τ) pullback attracts Dat time τ, it follows that B(t) pullback attracts bounded subsets of Xat time τfor each τ⩽t; that is, {S(t, s) : t⩾s}is pullback strongly bounded dissipative. As an immediate consequence of Theorem 2.5 we have that Theorem 4.3. If a process {S(t, s) : t⩾s}is pullback strongly compact dissipative and pullback strongly asymptotically compact, then {S(t, s) : t⩾s}has a pullback attractor {A(t) : t∈R}with the property that Ss⩽tA(s)is bounded for each t∈R. The proof of Theorem 2.12 is now a direct application of Lemma 4.1 and Theorem 4.3. Final Remark Although, for simplicity, the results given in this work concern the pullback attractor of bounded sets, they can be generalized in such way that other classes of sets can be attracted. In that case, the introduced definitions of dissipation and asymptotic compactness should be changed accordingly to cope with these other classes (see [2] for more details). 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Langa) Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla Spain (F. Rivero) Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla Spain