J.evol.equ (2007), 649–667 © 2007 Birkh¨auser Verlag, Basel 1424-3199/07/040649-19, published online August 09, 2007 DOI 10.1007/s00028-007-0326-7 Exponential attractors for a class of reaction-diffusion problems with time delays ´ Maurizio Grasselli and Dalibor Praˇzak Abstract We consider a reaction-diffusion system subject to homogeneous Neumann boundary conditions on a given bounded domain The reaction term depends on the population densities as well as on their past histories in a very general way This class of systems is widely used in population dynamics modelling Due to its generality, the longtime behavior of the solutions can display a certain complexity Here we prove a qualitative result which can be considered as a common denominator of a large family of specific models More precisely, we demonstrate the existence of an exponential attractor, provided that a bounded invariant region exists and the past history decays exponentially fast This result will be achieved by means of a suitable adaptation of the -trajectory method coming back to the seminal paper of M´alek and Neˇcas Introduction A large number of mathematical models in population dynamics have the following form (see, e.g., [1, 2, 3, 12, 18, 19, 20, 21, 26, 37, 38, 42, 43, 44, 45] and references therein) u, u t ), ∂t u − D u = F (u in × (0, ∞) , (1.1) where is a bounded, open and connected subset of R , h ∈ N Here u = (u1 , , uM ) : × R → [0, ∞)M represents the population density vector and D = diag[d1 , , dM ] is a diffusion matrix (di > 0, i = 1, , M) Moreover, F is a reaction function which depends not only on u (x, t), but also includes a general functional dependence on the past history up to t, denoted by h u t (·, s) = u (·, t + s), s ∈ (−∞, 0] (1.2) We suppose that the populations are isolated, so that system (1.1) is subject to the Neumann homogeneous boundary conditions ∂nu = 0, on ∂ × (0, ∞) (1.3) Mathematics Subject Classifications (2000): 35B41, 45K05, 92D25 Key words: Reaction-diffusion equations, nonlocal effects, invariant regions, -trajectory method, exponential attractors The first author was partially supported by the Italian PRIN 2006 research project Problemi a frontiera libera, ˇ 0021620839 transizioni di fase e modelli di isteresi The second author was supported by the research project MSM and by the project LC06052 (Jindˇrich Neˇcas Center for Mathematical Modeling) 650 ´ M Grasselli and D Praˇzak J.evol.equ In addition, in view of (1.2), the initial datum is the past history up to t = 0, i.e., u = η, in × (−∞, 0] (1.4) Among the main goals in the investigation of such kind of systems there are, in particular, the stability of the equilibria and the existence of (stable) periodic solutions However, if M is very large even the determination of the equilibria might be an extremely difficult task Therefore, dealing with very general situations, it seems convenient to interpret the model as a dissipative dynamical system in a suitable phase space, trying to prove the existence of sufficiently small (i.e., finite fractal dimensional) compact invariant sets which characterize the longtime dynamics of the model itself Typical mathematical objects possessing the mentioned features are the global attractor (see, e.g., [9, 23, 40]) and the exponential attractor (cf [13, 14, 15]) It is also worth seeing [31] for an updated review of such notions The key step of such an analysis is a suitable choice of the underlying phase space (cf the pioneering contribution [24]) One possibility is to introduce a new variable, the so-called summed past history, which accounts for the memory effects, and obeys certain first order dissipative equation (see [22], or [6, 7] for equations like (1.1)) This approach is useful when one wants to analyze the stability of the attractor with respect to certain parameters of the given system (e.g., in [6], the relaxation times) The limitation of the summed past history approach is that the delay has to be described in terms of a convolution with rather restricted class of kernels Another approach is the construction of the so-called trajectory attractor (see [9], cf also [8].) Here the phase space consists of negative semi-trajectories and the dynamical system is defined by means of a translation semigroup Note that this is a natural setting for the problems with delay, and no additional variables are needed A detailed comparison of the past history approach and the concept of trajectory attractor is given in [8] It turns out that if both methods are applicable, they are equivalent regarding the notion of global attractors, but the advantages of the former stand out Nonetheless, there are models to which the past history approach does not apply, while the trajectory one does For instance, in the case of discrete state-dependent delays, a recent result about the existence of the global trajectory attractor is proven in [36] (see also references therein) There the author also treats the case of distributed (but finite) delays by means of a more conventional approach which regards the delays as additional variables In the case of infinite delays, a meaningful example can be easily given by exploiting the generality of (1.1) Indeed, hereditary effects can be modelled, for instance, as follows (see, e.g., [39]) u(t + s), s)ds K(u −∞ In the present paper, we want to cover such general classes of models; hence the summed past history approach is not applicable As a phase space, we take the set of the histories η, Vol 7, 2007 Reaction-diffusion problems with time delays 651 defined on (−∞, 0] An additional requirement is that η solves (1.1) on [− , 0], where > is some fixed number This has certain similarity to the trajectory approach discussed above, but also goes back to the ideas of [28], see also [29] or [34], where a similar approach was applied to the class of problems with bounded delay The main advantage of such a choice is that the phase space can be endowed with a more convenient L2 -norm rather then the L∞ -norm; see also the discussion below Theorem 1.1 In particular, we easily establish the smoothing property of the dynamics, yielding the existence of an exponential attractor The main requirement of our analysis is the existence of a bounded invariant region This is a natural assumption for the reaction-diffusion systems (without delay), and has been used by many authors; see e.g [5, 10, 11, 30] However, in presence of delays, things become more delicate As shown in [16, 27], the simple problem ∂t u(x, t) − µ u(x, t) = u(x, t) − u(x, t − τ ) admits solutions that grow exponentially, despite a natural sign condition of the right-hand side Nonetheless, there are numerous models governed by systems of type (1.1), for which the invariant regions exist (cf., for instance, [6, 7, 19, 32, 33, 37, 38, 39, 42, 43, 44, 45]) Some meaningful examples are considered in the last section To be more specific, we assume that there exist ≥ and bi > 0, i = 1, , M, such that, setting I = M for any s ∈ (−∞, 0], then i=1 [ai , bi ], if η ∈ I almost everywhere in u(x, t) ∈ I for t ≥ as well Here u is a suitable concept of solution, defined on a maximal time interval [0, Tmax ) The immediate consequence is that Tmax = +∞, and, trivially, the asymptotic boundedness of the dynamics At our abstract level of presentation, we will assume that there exists a closed, bounded set K ⊂ L2 ( ; RM ), such that, if η ∈ C((−∞, 0]; K), then u(·, t) ∈ K for all t ≥ In concrete examples, however, one always has K = v ∈ L2 ( ; RM ) : v (x) ∈ I, a.e x ∈ , thus the solutions are even uniformly bounded As a phase space of our problem, we choose B = C((−∞, 0]; K) Before introducing a metric in B, we first indicate by · the standard norm in L2 ( ; RM ) induced by the natural scalar product (·, ·), and then we state the basic assumption on F , namely, there exist > and γ > such that  F (u u1 , η1 ) − F (u u2 , η2 ) 2 ≤  u1 − u2  2 + −∞ η1 (s) − η2 (s) γs 2e ds  , (1.5) 652 J.evol.equ ´ M Grasselli and D Praˇzak for any u , u ∈ K and η1 , η2 ∈ B Observe that, the only requirement on the past history dependence is the exponential decay This generality allows us to consider a wide class of memory kernels in concrete cases (see, e.g., Section 6, compare also with [12]) We can now endow B with the norm given by η X = η(s) γs 2e (1.6) ds , −∞ where γ > is the same as in (1.5) Observe that B is not complete (in fact it is not closed) with respect to the X-norm The basic existence result we need is the following THEOREM 1.1 Let (1.5) hold For any given η ∈ B and any fixed T > 0, there exists a unique u solution to (1.1), (1.3), (1.4) on [0, T ] such that u ∈ C((−∞, T ]; L2 ( ; RM )) ∩ L2 (0, T ; W 1,2 ( ; RM )), ∂t u ∈ L2 (0, T ; (W 1,2 ( ; RM ))∗ ) (1.7) Moreover, u (t) ∈ K for all t ∈ [0, T ] and there exists K0 > 0, independent of T , such that, for any fixed r ∈ (0, T ), t+r ∂t u sup t∈[0,T −r] t −1,2 + u 1,2 ≤ K0 (1.8) Proof Using the Lipschitz continuity in (1.5), it is easy to construct u on some small interval [0, T0 ], via the standard fixed point argument Note that, since u(t) ∈ K for all t ∈ (−∞, T0 ], it follows that T0 ∂t u −1,2 + u 1,2 ≤K Observe that K and T0 only depend on K and the constants in (1.5), but are independent of η = u |(−∞,0] This fact allows us to repeat the argument on the time interval [T0 , 2T0 ] and so on, reaching eventually the given time T Uniqueness follows by the usual Gronwall-type argument (see Lemma 3.2 below) As a consequence, we can now define a solution operator S(t) : B → B by S(t)η = u t , where u is given by the previous theorem (recall the notation (1.2).) Our aim is to construct an exponential attractor for a suitable interpretation of the dynamical system (S(t), B) Problem (1.1), (1.3), (1.4) has a unique solution for any initial datum in B However, note that we not have continuity of the solution with respect to the X-norm of the initial datum (compare with [6, 7]) One would need to use the sup-norm of B, which, however, Vol 7, 2007 Reaction-diffusion problems with time delays 653 is much less convenient for the further analysis To circumwent this difficulty, we follow the -trajectory approach (see [28, 29]), and set B− = S( )B , where > is an arbitrary fixed number Simply said, B− consists of those elements of B that are solutions on [− , 0] Obviously S(t)B− ⊂ B− for all t ≥ From the point of view of the large time dynamics it is irrelevant if we restrict our study to B− The advantage is that S(t) is Lipschitz continuous on B− with respect to the norm X, as we shall prove below (see Lemma 3.2) Note that also B− is not complete (even less compact) in X, but we will show that it is asymptotically compact This is enough to apply the standard tools from the theory of attractors REMARK 1.2 By the same scheme one can cover the more general case where there are also discrete delays (see, e.g., [32]), i.e., u(t), u (t − τ1 ), , u (t − τk ), u t ), ∂t u − D u = F (u with some τi > Moreover, the linear diffusion operator might also be replaced by a nonu) satisfying appropriate assumptions linear elliptic operator in divergence form − ∇· A (∇u Preliminaries from the theory of attractors Let X be a bounded metric space By Nχ (A, ρ) we denote the smallest number of sets with diameter ≤ 2ρ that cover A ⊂ X The fractal dimension is defined by f dX (A) = lim sup ε→0+ ln Nχ (A, ε) − ln ε Let (t) : X → X be a continuous semigroup of operators For the reader’s convenience, we recall that A ⊂ X is called a global attractor for ( (t), X ) if (1) A is compact; (2) (t)A = A, ∀t > 0; B , A) → as t → ∞, for any B ⊂ X bounded (3) distX ( (t)B Of course, if a global attractor exists, then it is unique The set E ⊂ X is called exponential attractor if (1) (2) (3) (4) E is compact; (t)E ⊂ E, ∀t > 0; f dX (E) < ∞; B , E) ≤ C(B B ) e−σ t ∃ σ > such that, for any B ⊂ X bounded, distX ( (t)B 654 J.evol.equ ´ M Grasselli and D Praˇzak It is customary to work in a setting where the underlying phase space X is either compact or, at least, complete space (typically, a closed subset of a Banach space with some higher regularity) In the present case it is convenient to relax this assumption to the asymptotic compactness, see the definition in Theorem 2.1 here below The construction of an exponential attractor is based on several ideas that are nowadays well-known and widely used in the literature Because our setting is somewhat unusual (the phase space is not complete), we present the following theorem with the proof The key step of the construction normally consists in certain squeezing or smoothing property of the semigroup (see, e.g., [13, 14, 15]) To keep the things simple for now, we replace it by a more abstract “iterated covering property” – see (2.3) below – which will later be deduced from the smoothing property in the spirit of [15] (cf Lemma 5.1 below) Remark also that (2.3) is only slightly stronger that (2.4) Also, we recall that the latter property is equivalent to the existence of an exponential attractor in the discrete case (see [35]) THEOREM 2.1 Let X be a bounded metric space, and semigroup We assume: (t) : X → X be a continuous (1) asymptotic compactness: for arbitrary sequences {ηn } ⊂ X and tn → ∞, there is a subsequence (not relabeled) {ηn } and η0 ∈ X such that (tn )ηn → η0 (2) continuity: for all η, η˜ ∈ X and t,t˜ ∈ [0, T ] ˜ ≤ c distX (η, η) ˜ , distX ( (t)η, (t)η) (2.1) distX ( (t)η, (t˜)η) ≤ c|t − t˜| , (2.2) a with some a ∈ (0, 1] (3) iterated covering property: there exist N ≥ 1, θ ∈ (0, 1) and t ∗ > such that, for any B ⊂ X with diamX B ≤ 2ρ, one has Nχ ( (t ∗ )B, θρ) ≤ N (2.3) Then there exists an exponential attractor E for ( (t), X ) and the following estimate holds f dX (E) ≤ a ln N +1 − ln θ Proof We split the argument into two main steps I We begin with an exponential attractor E ∗ for the discrete dynamical system ( n , X ), where = (t ∗ ) Pick ρ > such that diam X ≤ 2ρ Applying (2.3) repeatedly, we arrive at Nχ ( n X , θ n ρ) ≤ N n , ∀n ∈ N (2.4) Vol 7, 2007 Reaction-diffusion problems with time delays 655 This implies the existence of E ∗ – an exponential attractor for ( n , X ) – together with the estimate (see [35, Theorem 1]) ln N f dX (E ∗ ) ≤ (2.5) − ln θ A certain care is needed, since we assume that X is neither compact nor complete However, asymptotic compactness is enough for the existence of a global compact attractor A, which is defined as the ω-limit set of X The exponential attractor is constructed as E∗ = A ∪ m En , m,n≥0 where En is a finite subset of n X such that distX ( n X , En ) ≤ 2θ n ρ The existence of En s is guaranteed by (2.4) The double union is not compact, but thanks to the asymptotic compactness, any sequence that intersects infinitely many En s has an accumulation point in A Hence E ∗ is compact II We define F : X × [0, ∞) → X by F(η, t) = (t)η F is a-H¨older continuous by (2.1)–(2.2), from which one deduces that E = F(E ∗ × [0, t ∗ ]) is the desired exponential attractor for ( (t), X ) Regarding the dimension estimate, one has f dX (E) ≤ ln N +1 − ln θ 1 f dX ×[0,∞) (E ∗ × [0, t ∗ ]) ≤ a a Continuity and the smoothing property We start with some simple estimates about the continuity of the time shift Recall the convention (1.2) LEMMA 3.1 Let u : R → L2 ( ; RM ) be a measurable function Then ut ut u t X ≤ e−γ t u X ≤ e−γ t u 2 ≤ e−γ t u X , ∀t ≤ , t X + X , ∀t ≤ u (3.1) 2, ∀t ≥ , (3.3) Proof One has u t 2X = u (t + s) γs 2e ds = −∞ = e−γ t t −∞ u (τ ) γτ 2e dτ t −∞ (3.2) u (τ ) γ (τ −t) dτ 2e 656 J.evol.equ ´ M Grasselli and D Praˇzak Hence (3.1) follows, while for t ≥ one writes X ut = e−γ t u (τ ) −∞ ≤ e−γ t u X t + t dτ + γτ 2e γ (τ −t) dτ 2e u (τ ) u (τ ) 22 dτ Similarly one proves (3.3) The following lemma is a standard estimate of the difference of solutions, but it has important consequences: the continuity of S(t) as well as the smoothing property, which is the key step toward the existence of an exponential attractor LEMMA 3.2 Let u , v be solutions on [− , T ] corresponding to initial data η, η˜ in B− Then w = u − v satisfies sup w (t) t∈[0,T ] 2 T + d0 1,2 w ≤ CT η − η˜ X (3.4) , where d0 = mini=1, ,M di and CT > depends on T Proof Testing the equation written for w by w gives M ∂t w 2 + di ∇w i 2 F (u u, u t ) − F (vv , v t ), w ) = (F i=1 Using (1.5) and adding d0 w 2 ∂t w (t) 2 to both sides give + d0 w (t) 1,2 ≤c w (t) 2 + wt X ∀t ∈ [− , T ] , Note that, from now on, the constant c may change from line to line We fix s ∈ [− , 0], τ ∈ [0, T ] and integrate on (s, τ ) to get w (τ ) 2 τ + d0 w s 1,2 ≤ w (s) 2 τ +c w s 2 τ + wt s X dt 2 Using (3.1) and (3.2), we deduce τ w s 2 τ + wt s X dt ≤ c w0 X τ + w 2 Hence w (τ ) 2 τ + d0 w 1,2 ≤ w (s) 2 +c w0 X τ + w Vol 7, 2007 Reaction-diffusion problems with time delays 657 Integrating over s ∈ (− , 0) and using (3.3) eventually yields w (τ ) 2 τ + d0 w 1,2 ≤c η − η˜ X τ + w 2 Since τ ∈ [0, T ] is arbitrary, we have the conclusion from the Gronwall lemma Recalling (3.2), we have, as immediate corollary, S(t)η − S(t)η˜ X ≤ CT η − η˜ X , ∀t ∈ [0, T ], ∀η, η˜ ∈ B− , (3.5) i.e., the local Lipschitz continuity of solution operators Note that CT grows at most exponentially with respect to T A second important by-product of Lemma 3.2 is  T 1/2  w  1,2 ≤ CT η − η˜ X (3.6) In fact, we can now follow the idea of [28] (see also [29]) to employ the last estimate to obtain a smoothing property of the flow in the space of L2 (0, T ; L2 ( ; RM )) This is based on the compactness from the well-known Aubin-Lions lemma Thus we need a similar estimate for the time derivative LEMMA 3.3 Let u , v be solutions on [− , T ] corresponding to initial data η, η˜ in B− Then there holds  T 1/2  ∂t w  −1,2 ≤ CT η − η˜ X (3.7) Proof The left-hand side of (3.7) equals to T sup ϕ∈B1 (0) ∂t w , ϕ , where ·, · is the duality between W 1,2 ( ; RM ) and (W 1,2 ( ; RM ))∗ , while B1 (0) is the unit ball of L2 (0, T ; W 1,2 ( ; RM )) centered at From the equation for w we have T T ∂t w , ϕ = −a T (∇w w , ∇ϕ) + = I1 + I2 F (u u, u t ) − F (vv , v t ), w ) (F 658 J.evol.equ ´ M Grasselli and D Praˇzak By (3.6) and the H¨older inequality, it follows  T |I1 | ≤ c w 1,2 ϕ 1,2 ≤ 1/2 T w  1,2 ≤ CT η − η˜ X An analogous estimate holds for I2 , in virtue of (1.5) and (3.4) Asymptotic smoothness and compactness In order to construct an exponential attractor, we need to restrict ourselves to a smoother subset of B− In particular, we need more regularity with respect to time An interesting feature of the -trajectory approach is that the regularity of weak solutions (1.7) is enough Due to the hyperbolic nature of our problem, sets of higher regularity cannot be absorbing Instead we prove the existence of smooth sets which are exponentially attracting (compare with [7]) LEMMA 4.1 There exists B1 ⊂ B− with the following properties: (1) S(t)B1 ⊂ B1 for ∀t > 0; (2) there exists K1 > such that t ∂t η sup t≤0 t−1 −1,2 + η ≤ K1 , 1,2 ∀η ∈ B1 ; (4.1) (3) B1 attracts B− exponentially fast Proof Let η ∈ B− be arbitrary and fix T ≥ Denote by u the solution on [0, T ] with u = η Thanks to (1.8) we have t ∂t u sup t∈[1,T ] t−1 −1,2 + u 1,2 ≤ K0 (4.2) Note that K0 is independent of T , because the solution on [t − 1, t] can be seen as the solution on [0, 1] with the initial condition u t−1 ∈ B− By (4.2) there exists t0 ∈ (0, 1) such that u (t0 ) 21,2 ≤ K0 We define ψ˜ ∈ B by u (t + T ), t ∈ [t0 − T , 0] , ψ˜ (t) = u (t0 ), t ≤ t0 − T It follows that t sup t≤0 t−1 ∂t ψ˜ −1,2 + ψ˜ 1,2 ≤ 2K0 (4.3) Vol 7, 2007 Reaction-diffusion problems with time delays 659 Observe that ψ˜ equals to S(t)η, redefined by u(t0 ) on (−∞, t0 − T ) Hence S(T )η − ψ˜ X t0 −T = −∞ [S(T )η](s) − ψ˜ (s) γs 2e ds ≤ c e−γ T Therefore, the set B˜ = ψ˜ ∈ B : ψ˜ satisfies (4.3) enjoys properties (2) and (3) However, B˜ ⊂ B− and B˜ is not positively invariant To circumvent this obstacle, we introduce the operator S− : B˜ → B− by S− (ψ˜ ) = S( )(η− ) In other words, S− redefines η on [− , 0] by solving the equation Continuing our construction, we finally set ψ = S− (ψ˜ ) and, consequently, w (t) = ψ(t) − [S(T )η](t) Observe that w is a difference of two solutions on [− , 0], and w − w (− ) = Similarly as in Lemma 3.2 one deduces w0 X = ψ − S(T )η X X ≤ c e−γ T , ≤ c e−γ T Hence the set ˜ S(t)S− (B) B1 = t≥0 is exponentially attracting, positively invariant, and one has B1 ⊂ B− Regularity (4.1) ˜ follows from Theorem 1.1 and the regularity of B An immediate consequence is the following LEMMA 4.2 For any η ∈ B1 , t, t˜ ≥ 0, one has S(t)η − S(t˜)η X ≤ c|t − t˜|1/2 Proof We can write S(t)η − S(t˜)η = S(T )ψ − ψ , where t = T + t˜, ψ = S(t˜)η ∈ B1 Let u be the solution with u = ψ, hence we have S(T )ψ − ψ X = u (T + s) − u (s) −∞ γs 2e ds 660 J.evol.equ ´ M Grasselli and D Praˇzak Further, we deduce u (T + s) − u (s) = u (T + s) 2 T =2 2 − u (s) 2 u(T + s) − u (s), u (s)) − 2(u ∂t u (t + s), u (t + s) − ∂t u (t + s), u (s) dt Here we use the regularity of (4.1) Note that the second term also requires u (s) ∈ W 1,2 ( ; RM ), which is true for almost any s < Hence, we deduce S(T )ψ − ψ X =2 T ds −∞ dt ∂t u (t + s), u (t + s) − ∂t u (t + s), u (s) dt eγ s ∂t u (t + s), u (t + s) − ∂t u (t + s), u (s) T =2 eγ s −∞ ds ht (s) ≤ cT The last estimate follows from the fact that, by (4.1), ht (·) is translation bounded in L1loc (−∞, 0) uniformly in t As we already noticed, we deal mostly with sets which are not compact or even complete with respect to the X-norm of our phase space However, this drawback is compensated by the asymptotic compactness of the flow as the following result shows LEMMA 4.3 Let ηn ∈ B be an arbitrary sequence and tn → ∞ Then there exists η0 ∈ B1 such that S(tn )ηn → η0 in the X-norm, as tn → ∞ , possibly up to a subsequence Moreover, η0 is a solution on (−∞, 0] Proof Let u n be a solution on [0, tn ] with the initial condition ηn , and ψn = S(tn )ηn , i.e., ψn = u tn Owing to (1.7) and (1.8), we can use the Aubin-Lions lemma to find η0 such that ψn → η0 weakly in L2loc (−∞, 0; W 1,2 ( ; RM )) , ∂t ψn → ∂t η0 weakly in L2loc (−∞, 0; (W 1,2 ( ; RM ))∗ ) , ψn → η0 strongly in L2loc (−∞, 0; L2 ( ; RM )) As {ψn } is bounded in L∞ (−∞, 0; L2 ( ; RM )), we also have ψn → η0 in the X-norm Vol 7, 2007 Reaction-diffusion problems with time delays 661 This is enough to pass to the limit in equation (1.1) Hence, since ψn is a solution on [−tn , 0], then η0 is a solution on (−∞, 0] For tn ≥ T , one has that u n fulfills estimate (4.2) Consequently, (4.3) holds for η0 This ˜ Yet η0 is a solution on (−∞, 0], means (using the notation of Lemma 4.1) that η0 ∈ B hence η0 = S− (η0 ) ∈ B1 Main theorem The key step in the construction of an exponential attractor is the “iterated covering property”, that is, assumption (3) of Theorem 2.1 In the present case, this property will be deduced from the smoothing properties (3.6) and (3.7) proven above LEMMA 5.1 There exist θ ∈ (0, 1) and N ≥ such that, for any set E ⊂ B− with diamX E ≤ 2ρ, one has Nχ (S( )E, θρ)X ≤ N (5.1) Proof Let us set Z = L2 (0, ; L2 ( ; RM )) and Y = L2 (0, ; W 1,2 ( ; RM )) ∩ H (0, ; (W 1,2 ( ; RM ))∗ ) The latter Hilbert space is endowed with the norm u Y = u 1,2 + ∂t u −1,2 Further, we define a mapping P : B → Z, P : η → u |[0, ] , where u is the solution with u = η Using this notation, we rewrite (3.2), (3.6) and (3.7) as S( )η − S( )η˜ X ≤ e−γ P η − P η˜ Y ≤ K η − η˜ η − η˜ X X + P η − P η˜ Z , (5.2) (5.3) Let E ⊂ B− with diamX E ≤ 2ρ be given From (5.3) we deduce diamY P E ≤ 2ρK Since the continuous embedding Y → Z is compact, we can cover P E with the sets Hj ⊂ B− , j = 1, N, such that diamZ Hj ≤ 2ρε The number ε is taken small enough so that θ := e−γ +ε < (5.4) Note that we cover a set E of diameter 2ρK with sets of diameter 2ρε, both in the context of normed spaces Thus by the scaling argument the number N depends on K, ε, but it is independent of E and ρ To conclude with, we set Ej = ψ ∈ B− : ∃ ψ˜ ∈ E such that ψ = S( )ψ˜ and P ψ˜ ∈ Hj 662 J.evol.equ ´ M Grasselli and D Praˇzak Obviously {Ej } covers S( )E We claim that diamX Ej ≤ 2ρθ Let ψ1 , ψ2 ∈ Ej Then there exist ψ˜ , ψ˜ ∈ Ej such that ψi = S( )ψ˜ i , P ψ˜ i ∈ Ej , for i = 1, This means P ψ˜ − P ψ˜ Z ≤ 2ρε, and (5.2) entails ψ − ψ2 X = S( )ψ˜ − S( )ψ˜ ≤ e−γ ψ˜ − ψ˜ 2 X X + P ψ˜ − P ψ˜ 2 Z ≤ e−γ (2ρ)2 + (2ρε)2 = (2ρθ)2 We can eventually prove the main theorem THEOREM 5.2 The dynamical system (S(t), B), associated with equation (1.1) subject to boundary condition (1.3), has an exponential attractor Proof We apply Theorem 2.1 with X = B1 endowed with the X-topology (cf (1.6)) and (t) = S(t) The asymptotic compactness (cf (1)) follows from Lemma 4.3 Regarding the continuity properties (see (2)), inequality (2.1) derives from (3.5), while (2.2) holds by Lemma 4.2 with a = 1/2 The iterated covering property (see (3)) has been checked in Lemma 5.1 above We have thus obtained an exponential attractor E for (S(t), B1 ) Observe now that, due to Lemma 4.1, B1 attracts B exponentially fast On the other hand, the distance of arbitrary two trajectories grows at most exponentially with respect to time (cf (3.5)) Therefore we can use the so-called transitivity property of the exponential attraction (see [17]) to conclude that E is an exponential attractor for (S(t), B) REMARK 5.3 As an immediate consequence of Theorem 5.2, we have that (S(t), B) possesses a global attractor of finite fractal dimension Applications Here we illustrate how Theorem 5.2 can be applied to three meaningful examples taken from the literature We confine ourselves to systems of two equations just for the sake of simplicity Example Following [39], we consider system (1.1) with M = 2, endowed with (1.3)–(1.4), and u, ut ) = αi ui − βi ui f (u1 , u2 ) + F i (u −∞ g(s, u1 (t + s), u2 (t + s))ds , (6.1) Vol 7, 2007 Reaction-diffusion problems with time delays 663 for i = 1, Here αi and βi are given positive constants Then we suppose that the continuous functions f : [0, ∞)2 → [0, ∞) and g : (−∞, 0] × [0, ∞)2 → [0, ∞) satisfy f (0, 0) = , (6.2) f is locally Lipschitz continuous , (6.3) f is increasing in each variable , (6.4) lim f (y , 0) = ∞, y →∞ g(t, 0, 0) = 0, lim f (0, y ) = ∞ , (6.5) y →∞ ∀ t ≥ 0, (6.6) g(s, y , y ) is non decreasing w.r.t y and y , for each s ≤ , (6.7) g(t, ·, ·) is locally Lipschitz continuous, uniformly on bounded time intervals , (6.8) ∃ γg > and Cg > s.t for all (s, y , y ), (s, y˜ , y˜ ) ∈ (−∞, 0] × [0, ∞) (6.9) 2 |g(s, y , y ) − g(s, y˜ , y˜ )| ≤ Cg eγg s |y − y˜ | + |y − y˜ | Then, recalling [39, Lemma 2.1], we first find the (unique) positive numbers such that α1 α2 , f (0, ) = , f ( , 0) = β1 β2 so that the rectangle I1 = [0, Hence, taking 1] × [0, 2] and is an invariant region for the present model K = v ∈ L2 ( ; R2 ) : v (x) ∈ I1 , a.e x ∈ , Theorem 5.2 applies (note that (6.3), (6.8) and (6.9) entail the validity of (1.5)) Example We can now consider an example of prey-predator model which accounts for self-inhibitory crowding effects (see [33]) This model reduces to system (1.1) with M = and u, ut ) = u1 α1 − β1 u1 − γ1 R1 [u2 ] , F (u (6.10) u, u t ) = u2 −α2 − β2 u2 + γ2 R2 [u1 ] F (u (6.11) Here αi , βi , γi , i = 1, 2, are given positive constants Besides, R1 and R2 are given by Ri [v](x, t) = −∞ ki (s) i (x, y)v(y, t + s)dy ds , i = 1, , (6.12) 664 J.evol.equ ´ M Grasselli and D Praˇzak for (x, t) ∈ × (0, ∞) and any v ∈ C([0, ∞); L∞ ( )), where (i = 1, 2) x→ i (x, ·) : ¯ → L1 ( ; [0, ∞)) is continuous , i (x, y)dy (6.13) ∀x ∈ ¯ , = 1, (6.14) ki ∈ L1 ((−∞, 0); (0, ∞)) , −∞ ki (s)ds = , ∃ γ0 > and C0 > s.t ki (s) ≤ C0 eγ0 s , (6.15) for a.a s ≤ (6.16) On account of [33, Section 2.2], we know that, choosing ≥ α1 , β1 ≥ max γ2 − α2 ,0 , β2 the rectangle I2 = [0, ] × [0, 4] , is an invariant region for system (1.1) with M = and F given by (6.10)–(6.11), endowed with (1.3)–(1.4) Therefore, it is clear that Theorem 5.2 also holds for this problem Note that (1.5) can be easily checked thanks to (6.13)–(6.16) Example A further interesting system with nonlocal spatio-temporal effects is the competition model analyzed in [19] More precisely, this corresponds to system (1.1), with M = 2, endowed with (1.3)–(1.4) and F given by u, u t ) = u1 α1 − β1 u1 − γ1 R1 [u2 ] , F (u (6.17) u, u t ) = u2 α2 − β2 u2 − γ2 R2 [u1 ] F (u (6.18) where αi , βi , γi , i = 1, 2, are fixed positive constants Here R1 and R2 have the following form Ri [v](x, t) = −∞ ki (s) Gi (x, y, s)v(y, t + s)dy ds , i = 1, , (6.19) for (x, t) ∈ × (0, ∞) and any v ∈ C([0, ∞); L∞ ( )) However, while ki , i = 1, still satisfy (6.15) and (6.16), the functions Gi , i = 1, 2, are solutions to the following initial and boundary value problems ∂t Gi = di ∂n G i = , × (0, ∞) , , in on ∂ × (0, ∞) , x Gi Gi (x, y, 0) = δ(x − y) , Vol 7, 2007 Reaction-diffusion problems with time delays 665 where δ is the Dirac mass at 0, for any given y ∈ We refer the reader to [20] for a detailed explanation of these assumptions about Gi (see also [21]) Using the comparison theorem for parabolic equations, it can be proved that I3 = 0, α2 α1 × 0, β1 β2 is an invariant region for our problem Then, taking advantage of [19, Lemma 2.1] and (6.16), we can prove the validity of (1.5) Hence, we conclude that Theorem 5.2 is applicable to this case as well REMARK 6.1 We believe that Theorem 5.2 also holds, for instance, for the equation considered in [4] with temporal nonlocality on a bounded spatial domain This observation is somehow connected with the third open question formulated at the end of [4] Of course, our theorem can be applied to several other models (see, e.g., the examples discussed in [6]) 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Milano Via E Bonardi, I-20133 Milano, Italy maurizio.grasselli@polimi.it and Dalibor Praˇza´ k Department of Mathematical Analysis Charles University Sokolovsk´a, 83 CZ-18675 Prague 8, Czech Republic prazak@karlin.mff.cuni.cz [...]... equations with delays, J Math Anal Appl 281 (2003), 439–453 Wang, Z.-C., Li, W.-T and Ruan, S., Travelling wave fronts in reaction- diffusion systems with spatiotemporal delays, J Differential Equations 222 (2006), 185–232 Wu, J., Theory and applications of partial functional-differential equations, Springer-Verlag New York, 1996 Yamada, Y., On a certain class of semilinear Volterra diffusion equations,... Miranville, A and Zelik, S., Attractors for dissipative partial differential equations in bounded and unbounded domains, to appear in Handbook of Differential Equations, Evolutionary Partial Differential Equations (C.M Dafermos and M Pokorn´y, Eds.), Elsevier, Amsterdam Pao, C V., Convergence of solutions of reaction- diffusion systems with time delays, Nonlinear Anal 48 (2002), 349–362 Pozio, M A. ,... C., Nicolaenko, B and Temam, R., Exponential attractors for dissipative evolution equations, Masson Paris, 1994 Efendiev, M., Miranville, A and Zelik, S., Exponential attractors for a nonlinear reaction- diffusion 3 system in R , C R Math Acad Sci Paris 330 (2000), 713–718 Friesecke, G., Exponentially growing solutions for a delay- diffusion equation with negative feedback, J Differential Equations 98... equations, J Math Anal Appl 88 (1982), 433–451 Yamada, Y., Asymptotic stability for some systems of semilinear Volterra diffusion equations, J Differential Equations 52 (1984), 295–326 Maurizio Grasselli Dipartimento di Matematica “F Brioschi” Politecnico di Milano Via E Bonardi, 9 I-20133 Milano, Italy maurizio.grasselli@polimi.it and Dalibor Praˇza´ k Department of Mathematical Analysis Charles University... Behaviour of solutions of some abstract functional differential equations and application to predator-prey dynamics, Nonlinear Anal 4 (1980), 917–938 ´ D., Exponential attractor for the delayed logistic equation with a nonlinear diffusion, Discrete Praˇzak, Contin Dyn Syst suppl (2003), 717–726 ´ D., A necessary and sufficient condition for the existence of an exponential attractor, Centr Eur Praˇzak,... ´ ´ Avila-Vales, E., Estrella-Gonzalez, A and Sanchez-Salazar, P., The evolution of a single species food-limited population model with delay: a numerical study, WSEAS Trans on Systems 4 (2004), 1313–1317 Boushaba, K and Ruan, S., Instability in diffusive ecological models with nonlocal delay effects, J Math Anal Appl 258 (2001), 269–286 Britton, N F., Spatial structures and periodic travelling waves... retarded equations with infinite delay, Funkcial Ekvac 21 (1978), 11–41 Kuang, Y., Delay differential equations with applications in population dynamics, Academic Press, Inc Boston, 1993 Kuang, Y and Smith, H L., Global stability in diffusive delay Lotka-Volterra systems, Differential Integral Equations 4 (1991), 117–128 Luckhaus, S., Global boundedness for a delay- differential equation, Trans Amer Math Soc... V V., and Vishik, M I., Attractors for equations of mathematical physics, Amer Math Soc Providence 2002 Chueh, K N., Conley, C C and Smoller, J A. , Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ Math J 26 (1977), 373–392 Conway, E., Hoff, D and Smoller, J., Large time behavior of solutions of systems of nonlinear reactiondiffusion equations SIAM J Appl Math 35... Math 1 (2003), 411–417 Rezounenko, A V., Partial differential equations with discrete and distributed state-dependent delays, J Math Anal Appl 326 (2007), 1031–1045 Ruan, S and Wu, J., Reaction- diffusion equations with infinite delay, Canad Appl Math Quart 2 (1994), 485–550 Vol 7, 2007 [38] [39] [40] [41] [42] [43] [44] [45] Reaction- diffusion problems with time delays 667 Ruan, S and Xiao, D., Stability... 767–774 ´ ˇ Malek, J and Necas, J., A finite-dimensional attractor for three-dimensional flow of incompressible fluids, J Differential Equations 127 (1996), 498–518 ´ ´ D., Large time behavior via the method of -trajectories, J Differential Equations Malek, J and Praˇzak, 181 (2002), 243–279 Marion, M., Attractors for reaction- diffusion equations: existence and estimate of their dimension, Appl Anal 25 (1987),