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Pullback exponential attractors for differential equations with delay

We show the existence of an exponential attractor for non-autono-mous dynamical system with bounded delay. We considered the case of strong dissipativity then prove that the result remains for the weak dissipativity. We conclude then the existence of the global attractor and ensure the boundedness of its fractal dimension.


1.
Introduction. Delayed dynamical systems generated by retarded differential equations are of obvious interest in applied mathematics due to physical reasons specially in biological motivations like species' growth [7]. The asymptotic behavior of such models has meaningful interpretation like permanence of species in a given domain or their possible extinction.
The theory of attractors for retarded dynamical systems was developed in the global sense by Hale [12]. Then, the notion of pullback attractor was introduced to generalize the autonomous case to the general non autonomous framework where the study of the long time behavior of infinite dimensional dynamical systems can be usually reduced to the description of the compact invariant set attracting all bounded subsets of the phase space called the global pullback attractor. This uniquely determined object has frequently a finite dimension but the attraction to it is typically unknown, it may be arbitrarily slow. The need to overcome this motivated the creation of the notion of exponential pullback attractors which was first introduced in [8]. Pullback exponential attractors are families of compact subsets of the phase space whose fractal dimensions are uniformly bounded for all times and that pullback attract each bounded subset of the phase space at an exponential rate which makes them more robust under perturbation and they contain the global pullback attractor. In particular, to show the existence of a pullback exponential attractor is one way of proving the existence and finite dimensionality of the global pullback attractor.
Technical theorems were established in [5,4] to prove the existence of attractors for asymptotically compact evolution processes. The existence of pullback attractors for differential equation with one variable delay was studied in [2]. Then Caraballo et al. [1] proved a more general result for the existence of pullback attractors for autonomous and non-autonomous retarded systems with one varying and distributed delay, where the variable delay was differentiable and its derivative is bounded by 1. In paper [13] the authors improved that result by showing the existence of exponential attractors for differential equations with one varying delay which, in particular, means the existence of a pullback attractor with bounded fractal dimension. With the analysis we carry out in our current paper, in particular we extend the previous results obtained for variable delay to a much wider class of delay differential equations (which, in particular, includes the cases of variable and distributed delay). The main feature for our research is to impose a Lipschitz-like integral condition on the term containing the delays.
Consider the following delay differential equation where f : R × C → R n is continuous and maps bounded sets into bounded sets, where C = C([−h, 0], R n ) denotes the Banach space of continuous functions with the sup-norm, which is the usual space when we deal with delay equations. For a given continuous function x(·) : R → R n and t ∈ R, we denote by x t (·) the element in C given by When x(·) is a solution of (1), then x t (·) is said to be the solution segment at time t. It is known that for any (s, u) ∈ R × C, there exists a solution x(t, s, u) for (1) defined on [s − h, α s,u ). We assume that α s,u = +∞ , for all s ∈ R since we are interested in the long-time behavior of solutions. For more details we invite the readers to see [12].
Remark 1.1. Uniqueness holds straightforwardly if, for instance f is locally Lipschitz with respect to its second variable, see [12] for more details.
We define a solution operator which gives the solution (inC) at time t when where x(·, s, u) denotes the solution of (1). The article is organized as follows. In Section 2 we recall main definitions and notations for the theory of non-autonomous dynamical systems that will be needed. Next, in Section 3, we state two main results for the existence of exponential attractors when the absorbing set is uniform and when it is time-dependent. Our main result is proved in Section 4 where we develop an abstract concept for the existence of an exponential attractor for a class of differential equation with finite delay and, finally in Section 5, we prove first the existence of an exponential attractor when the dependence in time is uniform, and then prove that the result remains true when the dissipativity is time dependent.

2.
Preliminaries. We recall some basic notions from the theory of dynamical systems that will be needed in the subsequent sections.
Definition 2.1. A two-parameter family {U (t, s)| t, s ∈ R, t ≥ s} of continuous operators from C into itself is called an evolution process in C if it satisfies the following properties As we look for attracting sets, we introduce first the Hausdorff semi-distance between subsets A and B in a metric space (X, d) as (iii) It is minimal, that is, for any closed attracting set Y at time t, we have We recall now the definition of an absorbing set which is crucial when we deal with global and exponential attractors. Definition 2.3. {B(t)} t∈R is said to be absorbing with respect to the process U if, for all t ∈ R and all D ⊂ X bounded, there exists T D (t) > 0 such that for all Theorem 2.4. Suppose that U (t, s) maps bounded sets into bounded sets and there exists a family {B(t)} t∈R of bounded absorbing sets for U . Then there exists a pullback attractor for problem (1).
Proof. See [2]. Where the fractal dimension of a precompact subset A ⊂ X is defined as , such that N X ε (A) denotes the minimal number of ε-balls in X with centers in A needed to cover A.
3. General existence Theorems. For the reader's convenience, we will include now some results which are crucial for our analysis and we have already published, e.g., in [13].
3.1. Strongly bounded evolution processes. We first recall an existence theorem for evolution processes that are strongly bounded.
Theorem 3.1. Let U (t, s), t ≥ s, be an evolution process in a Banach space W . Moreover, we assume that the following properties are satisfied for some t 0 ∈ R: (A 2 ) The evolution process U (t, s), t ≥ s, satisfies the smoothing property in B, i.e. there exist positive constantst > 0 and κ such that Then, for every ν ∈ (0, 1 2 ) there exists a pullback exponential attractor M ν in W , and its fractal dimension is bounded by Proof. See Theorem 3.4 and Remark 3 in [4].
Let U (t, s), t ≥ s, be an evolution process in a Banach space W . If the hypotheses (A 0 ), (A 1 ) and (A 2 ) are satisfied, then the evolution process U (t, s), t ≥ s, possesses a global pullback attractor, and its fractal dimension is bounded by Time-dependent families of pullback absorbing sets. In this section, instead of using the existence of a bounded pullback absorbing set B to prove the existence of a pullback exponential attractor, we will extend the theory to the case in which there exists a time-dependent family of absorbing sets {B(t)} t∈R with certain properties. This situation is more general and natural in the pullback context for non-autonomous dynamical systems The following result generalizes Theorem 3.1.
Let U (t, s), t ≥ s, be an evolution process in a Banach space W and (A 0 ) be satisfied. Moreover, we assume that the following properties hold for some t 0 ∈ R: There exists a family of bounded pullback absorbing sets Moreover, there existst > 0 such that ( A 3 ) The evolution process U (t, s), t ≥ s, is Lipschitz continuous in B, i.e. for all t ∈ R, t ≤ s ≤ t +t, there exists L s,t ≥ 0 such that Then, for every ν ∈ (0, 1 2 ) there exists a pullback exponential attractor M ν in W , and its fractal dimension is bounded by Proof. This is a particular case of Theorem 3.4 in [4] Remark 3.4. We recall that for a metric space (X, d) we define the diameter of a subset B in X as diamB = sup d(x, y), x, y ∈ B.
Corollary 3.5. Let U (t, s), t ≥ s, be an evolution process in a Banach space W . If the hypotheses (A 0 ), ( A 1 ) and ( A 2 ) are satisfied, then the evolution process U (t, s), t ≥ s, possesses a global pullback attractor, and its fractal dimension is bounded by . Strong dissipativity. Consider the delay differential equation in the general framework Assume that f (·, ·) : R × C → R n satisfies the following conditions • iii) A dissipative condition: The function f : R × C → R n in (1) is strongly dissipative, i.e., there exists positive constants α and β such that where ·, · denotes the scalar product in R n and Φ(h)C = {φ ∈ C|φ = U (s + h, s)ψ for some s ∈ R, ψ ∈ C} is the set of functions in C that are realizable as solutions after time h.
Theorem 4.1. Let U (t, s), t ≥ s, be the evolution process generated by problem (2). We assume that hypotheses (3), (4) and (5) are satisfied. Then, there exists a pullback exponential attractor for U (t, s), t ≥ s, in C. Moreover, the global pullback attractor exists, it is contained in the pullback exponential attractor, and its fractal dimension is finite.
Proof. We verify the hypotheses of Theorem 3.3 for the spaces W = C and V = C 1 . Certainly, (A 0 ) is satisfied due to the compact embedding C 1 → → C.
(i) Existence of a family of pullback absorbing sets: We proved in [13] the existence of a family of bounded absorbing set when the delay was time varying, following the same computation we obtain the same result under condition (5).
(ii) Smoothing property ( A 2 ): Let s ∈ R, u, v ∈ B(s) and x(t, u, s) and y(t, u, s), t ≥ s, denote the corresponding solutions of (2). In order to shorten notations, in the sequel we write x(t) = x(t, u, s) and y(t) = y(t, v, s), t ≥ s. For the difference of x and y we obtain Using (4) we have: Hence Gronwall's Lemma implies Let now θ ∈ [−h, 0] and t = s + h + θ, then using (6) |x After taking the supremum over θ ∈ [−h, 0] On the other hand, we observe that where we used (8) in the last estimate. Now taking the supremum over t ∈ [−h, 0], it follows that Finally, summing up inequalities (8) and (9) we obtain which proves the smoothing property fort = h.
(iii) Lipschitz continuity ( A 3 ): Using (6) (iv) Existence of the pullback exponential attractor: The existence of a family of bounded absorbing sets and the smoothing property was proved in (i). The Lipschitz continuity ( A 3 ) was shown in (ii). Consequently, all assumptions of Theorem 3.3 are fulfilled, which implies the existence of a pullback exponential attractor.

Application: (i)Differential Equation With
One Variable Delay. Consider a delay differential equation where with F : R N → R N a function such that there exists L F > 0 for which and ρ ∈ C 1 (R, [0, h]) such that sup t∈R ρ (t) = ρ < 1. • The function F is dissipative in a similar sense to (5), so that there exists some α 0 > 0 and β 0 ≥ 0 such that In this case we assumed in a more general way that f (t, ψ) = F (ψ(−ρ(t))) for all ψ ∈ C, t ∈ R. The existence of pullback exponential attractor was proved in [13] by direct computation.
Here we ensure that, in this case, condition (4) holds true by setting Hence all conditions of Theorem 4.1 are fulfilled and the existence of exponential attractor under these general conditions holds straightforwardly.
Next we prove that Theorem 4.1 can be applied even for equations containing not only variable delay but also distributed delay like the following integro-differential equation with finite delay.
(ii)Integro differential equation: To simplify the computations we will consider only the term containing the distributed delay, the rest was already done in the previous example. We consider the integro-differential equation introduced below where the functions F 0 , F 1 , and b are uniformly Lipschitz, i.e., there exist positive constants L 0 , L 1 and L b such that (H1) We suppose that F 0 is strongly dissipative, i.e. there exists α, β > 0 such that F 1 and b are sublinear and Before verifying our main result we include a result from [1] which is necessary in what follows. Assume in addition that • (2) There exist positive constants k 1 , k 2 , α and a positive function β(.) such that where t −∞ β(r)e δr dr < ∞, t ∈ R, δ > 0 (20) • (3)The variable delay function ρ is continuously differentiable and satisfies In addition, we denote Then we have the following result: 2m 1 e h < 1, and 3.
We prove that this case fulfils also condition (4). Indeed The previous two examples prove how the assumptions of our main theorem hold for equations with variable and also with distributed delay.
5. Weak dissipativity. The function f : R × C → R n in (1) is now assumed to be weakly dissipative, i.e., there exists a positive constant α and two continuous function m 1 , m 2 : R → (0, ∞) such that We consider now a more general case when the Lipschitz condition and the integral Lipschitz condition are time dependent, i.e there exists two non-negative continuous functions m 1 (·) and γ(·), where m 2 (·) is non-decreasing and m 1 (·), m 2 (·) satisfy the integrability conditions Theorem 5.1. Let U (t, s), t ≥ s, be the evolution process generated by problem (1). We assume that hypotheses (23), (22) and (24) are satisfied. Then, there exists a pullback exponential attractor for U (t, s), t ≥ s, in C. Moreover, the global pullback attractor exists, it is contained in the pullback exponential attractor, and its fractal dimension is finite.
Existence of Exponential Pullback attractor Let r ∈ R be arbitrary. We observe that the function in the smoothing property in part (ii) satisfies κ(s) ≤ κ(r) for all s ≤ r. Hence, hypothesis ( A 2 ) is satisfied witht = h and κ = κ(r). The Lipschitz continuity ( A 3 ) was shown in (ii), and the existence of a family of pullback absorbing sets satisfying ( A 1 ) was already proved in [13]. Consequently, all assumptions of Theorem 3.3 are fulfilled, which implies the existence of a pullback exponential attractor.
In other words The delay function ρ is continuously differentiable and satisfies for some constant ρ * < 1. We suppose that F 0 : R n → R n in (33) is uniformly Lipschitz i.e., there exists L > 0 such that The function F 0 : R n → R n in (33) is dissipative, i.e., there exists some positive constants α > 0 and β ≥ 0 such that F 0 (x), x ≤ −α|x| 2 + β ∀x ∈ R n .
We suppose that F 1 : R n → R n is defined by where the function m(·) : R → R + satisfies t −∞ e s m(s)ds < ∞, for all t ∈ R and ≥ 0.