Attractors for differential equations with variable delays

Attractors for Differential Equations with Variable Delays Tom´as Caraballo, Jos´e A. Langa Dpto. Ecuaciones Diferenciales y An´alisis Num´erico. Universidad de Sevilla. Apdo. de Correos 1160. 41080-Sevilla (Spain) E-mail: [email protected] ; [email protected] and James C. Robinson Mathematics Institute. University of Warwick. Coventry, CV4 7AL. UK E-mail: [email protected]arwick.ac.uk Using the relatively new concept of a pullback attractor, we present some results on the existence of attractors for differential equations with variable delay. We give a variety of examples to which our result applies. 1. INTRODUCTION The theory of global attractors for autonomous systems as developed by Hale in [7] owes much to examples arising in the study of retarded functional differential equations [8] (for slightly different approaches see Babin and Vishik [1], Ladyzhenskaya [12], or Temam [15]). Although the classical theory can be extended in a relatively straightforward manner to deal with time-periodic equations, general non-autonomous equations such as ˙x(t) = F(t, x(t), x(t−ρ(t))) (1) fall outside its scope. Recently, a theory of ‘pullback attractors’ has been developed (see section 2) which allows many of the ideas for the autonomous theory to be extended to deal with such examples. However, until now this has only been applied to ordinary and partial differential equations. It is our intention here to show how pullback attractors can be used to investigate the behaviour of non-autonomous delay equations. In particular, we are able to compare the dynamics of systems of ordinary differential equations with that of the same system with a small delay, and show that these are ‘close’ in some global sense. 1 2TOM´ AS CARABALLO, JOS´ E A. LANGA AND JAMES C. ROBINSON 2. DELAY DIFFERENTIAL EQUATIONS AS DYNAMICAL SYSTEMS We take as our canonical example of a non-autonomous delay equation a system with one, time-varying delay, ρ(t) where ρ:R→[0, h] is a continuous function and h > 0, d dtx(t) = F(t, x(t), x(t−ρ(t))) xs=ψ, ψ ∈ C.(2) The initial condition xsis specified in C, the space C0([−h, 0]; Rn) of continuous functions from [−h, 0] into Rn, and, for a function x∈C0([−h, T]; Rn), the notation xsdenotes the function in Cgiven by xs(θ) = x(s+θ) for all θ∈[−h, 0] (and so makes sense for any 0 ≤s≤T). This equation can be written in a more general framework, which allows one to consider a larger set of problems in a unified way. Rather than make the delay explicit, we write f(t, xt) = F(t, x(t), x(t−ρ(t))), and so can rewrite (2) as ˙x(t) = f(t, xt)xs=ψ, ψ ∈ C.(3) In what follows we concentrate on this form of the equation, assuming that f:R×C → Rnis continuous and ‘a bounded map’ (i.e. maps bounded sets into bounded sets). We note here that this formulation immediately includes examples other than the single, time-varying delay of (2). For example, the integrodifferential equation (see Kuang [11] for more details) ˙x(t) = Z0 −h g(t, s, x(t+s)) ds also fits into this framework, although we do not develop this theory here. It is known (Hale [7]) that for any (s, ψ)∈R×C there exists a unique solution x(t;s, ψ) for (3) defined on [s−h, αs,ψ). We assume that αs,ψ = +∞, for all s∈R,since we are interested in long-time behaviour of solutions. We define a solution operator φ(t, s) which gives the solution (in C) at time twhen xs=ψ, via φ(t, s)ψ=xt(·;s, ψ).(4) ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 3 3. PULLBACK ATTRACTORS We now discuss the theory of pullback attractors, as developed in Kloeden and Stonier [9], Kloeden and Schmalfuss [10], and Crauel et al. [5]). As is clear above, in the case of non-autonomous differential equations the initial time is just as important as the final time, and the classical semigroup property of autonomous dynamical systems is no longer available. Instead of a family of one time-dependent maps S(t) we need to use a two-parameter process φ(t, s), as introduced above in (4) (cf. Sell [14]); φ(t, s)ψdenotes the solution at time twhich was equal to ψat time s. The semigroup property is replaced by the process composition property φ(t, s)φ(s, r) = φ(t, r) for all t≥s≥r, and, obviously, the initial condition implies φ(s, s) =Id. As with the semigroup composition S(t)S(s) = S(t+s), this just expresses the uniqueness of solutions. [It is possible to present the theory within the more general framework of cocycle dynamical systems. In this case the second component of φis viewed as an element of some parameter space J, so that the solution can be written as φ(t, p)ψ, and a shift map θt:J→Jis defined so that the process composition becomes the cocycle property, φ(t+τ, p) = φ(t, θτp)φ(τ, p). We do not pursue this approach here, but note that it has proved extremely fruitful, particularly in the case of random dynamical systems. For various examples using this general setting, see Kloeden and Schmalfuss [10], or Sell [14]. For this reason, pullback attractors are often referred to as ‘cocycle attractors’]. As in the standard theory of attractors, we seek an invariant attracting set. However, since the equation is non-autonomous this set also depends on time. Definition 3.1. Let φbe a process on a complete metric space X. A family of compact sets {A(t)}t∈Ris said to be a (global) pullback attractor for φif, for all s∈R, it satisfies i) φ(t, s)A(s) = A(t) for all t≥s, and ii) lims→∞ dist(φ(t, t −s)D, A(t))=0, forallboundedsubsetsDofX. In the definition, dist(A, B) is the Hausdorff semidistance between Aand B, defined as dist(A, B) = sup a∈A inf b∈Bd(a, b),for A, B ⊆X. 4TOM´ AS CARABALLO, JOS´ E A. LANGA AND JAMES C. ROBINSON Property i) is a generalization of the invariance property for autonomous dynamical systems. The pullback attracting property ii) considers the state of the system at time twhen the initial time t−sgoes to −∞ (cf. Chepyzhov and Vishik [3]) The notion of an attractor is closely related to that of an absorbing set. Definition 3.2. {B(t)}t∈Ris said to be absorbing with respect to the process φif, for all t∈Rand all D⊂Xbounded, there exists TD(t)>0 such that for all τ≥TD(t) φ(t, t −τ)D⊂B(t). Indeed, just as in the autonomous case, the existence of compact absorbing sets is the crucial property in order to obtain pullback attractors. For the following result see Crauel and Flandoli [4] or Schmalfuss [13]. Theorem 3.1. Let φ(t, s)be a two-parameter process, and suppose φ(t, s) : X→Xis continuous for all t≥s. If there exists a family of compact absorbing sets {B(t)}t∈R, then there exists a pullback attractor {A(t)}t∈R, and A(t)⊂B(t)for all t∈R. Furthermore, A(t) = S SS D⊂X boundedΛD(t),where ΛD(t) = \ n∈[ s≥n φ(s, t −s)D. 4. ATTRACTORS FOR NON-AUTONOMOUS DELAY DIFFERENTIAL EQUATIONS We showed above how to define the process associated with the solution of the delay differential equation ˙x(t) = f(t, xt)xs=ψ(5) via φ(t, s)ψ=xt(·;s, ψ). ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 5 We now prove a simple general result on the existence of pullback attractors for this problem. Proof that the condition of the theorem holds is significantly more onerous than the proof of the theorem itself. Theorem 4.1. Suppose that φ(t, s)maps bounded sets into bounded sets, and that there exists a family {B(t0)}t∈Rof bounded absorbing sets for φ. Then there exists a pullback attractor for problem (5). Proof. Using theorem 3.1 it suffices to prove that there exists a family of compact absorbing sets for φ. For each t0∈R, define K(t0) = φ(t0, t0−h)B(t0−h). K(t0) is clearly absorbing, since for any bounded D⊂ C we have, for t≥TD(t0) + h, (here TD(t0) denotes the absorption time corresponding to the family {B(t0)}t∈R) φ(t0, t0−t)D=φ(t0, t0−h)φ(t0−h, (t0−h)−(t−h))D ⊂φ(t0, t0−h)B(t0−h) = K(t0). Also, K(t0) is bounded, since φmaps bounded sets into bounded sets. Finally, K(t0) is a compact subset of C. This follows using the Arzel`aAscoli theorem, since we have just shown that K(t0) is bounded, and the equicontinuity follows since, for ψ∈B(t0−h) and θ∈[−h, 0],  d dθφ(t0, t0−h)ψ(θ)= d dθx(t0+θ;t0−h, ψ)=|f(t0+θ, xt0+θ(·;t0−h, ψ)| which is bounded, using the assumption on f. 4.1. The case of strong dissipativity In this section we suppose a dissipative property for the nonlinear term of the differential equation which will lead us to the existence of a uniform (over t∈R) bounded absorbing set for the process φand hence a pullback attractor. We will suppose in this section that for some α > 0, β≥0 hf(t, ψ), ψ(0)i≤−α|ψ(0)|2+βfor all ψ∈Φ(h)C(6) where h·,·i denotes the scalar product in Rnand Φ(h)C={χ∈ C :χ=φ(s+h, s)ψ, some s∈R, ψ ∈ C}. (Note that Φ(h)Cis essentially the set of all those functions in Cwhich are realisable as solutions of the equation after a time h). 6TOM´ AS CARABALLO, JOS´ E A. LANGA AND JAMES C. ROBINSON Although this seems strange at a first view, note that (6) is a consequence of a more natural set of assumptions in various particular examples. Indeed, if we consider (2) with F:Rn→Rnuniformly bounded and uniformly continuous, i.e., for some k≥0 and some function ω:R+→R+, |F(x)| ≤ kand |F(x)−F(y)| ≤ ω(|x−y|), and dissipative in a similar sense to (6), so that, for some α0>0 and β0≥0 hF(x), xi≤−α0|x|2+β0, we recover (6). Observe that, in this case, we are assuming that f(t, xt) = F(x(t−ρ(t)) or, more generally, f(t, ψ) = F(ψ(−ρ(t)),for all ψ∈ C, t ∈R. Indeed, we have hF(x(t−ρ(t)), x(t)i ≤ hF(x(t)), x(t)i+hF(x(t−ρ(t)) −F(x(t)), x(t)i ≤ −α0|x(t)|2+β0+|x(t)| |F(x(t−ρ(t)) −F(x(t))| ≤ −α0|x(t)|2+β0+|x(t)|ω(|x(t−ρ(t)) −x(t)|) ≤ −α0|x(t)|2+β0+|x(t)|ω(kh) ≤ −α|x(t)|2+β for all t≥h, since then x(t) is a solution of (2). We now show that (6) ensures the existence of a pullback attractor. Theorem 4.2. Suppose that (6) holds. Then there exists a family of bounded absorbing sets {B(t0)}t0∈Rfor (3), and thus we can conclude the existence of a pullback attractor for this problem. Proof. We will prove more than the existence of a family of bounded absorbing sets: in fact, there exists a uniform (in t0) bounded absorbing set for (3). Indeed, given D⊂ C bounded, there exists d≥0 such that for all ψ∈D, kψkC≤d, i.e. kψkC= sup θ∈[−h,0] |ψ(θ)| ≤ d. Take now ψ∈Dand consider |φ(t0, t0−t)ψ|= sup θ∈[−h,0] |x(t0+θ;t0−t, ψ)| = sup τ∈[t0−h,t0] |x(τ;t0−t, ψ)|. Let us write x(τ) = x(τ;t0−t, ψ), τ∈[t0−t, t0]. Then, multiplying (5) by x(τ) we get d dτ|x(τ)|2= 2 hx(τ), f(τ, xτ)i ≤ 2β−2α|x(τ)|2, ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 7 for all τ≥t0−t. Then, by Gronwall’s lemma |x(τ)|2≤ |x(t0−t)|2e−2α(τ−t0+t)+β α(1 −e−2α(τ−t0+t)) ≤ |ψ|2e−2α(τ−t0+t)+β α ≤ |ψ|2e−2α(θ+t)+β α ≤de2αhe−2αt +β α. Thus we obtain sup θ∈[−h,0] |x(t0+θ)|2≤de2αhe−2αt +β α≤1 + β α if we take t≥1 2αlog(de2αh) = TD. Note that this time TDdoes not depend on t0. 4.2. A more general case In the previous section we considered differential equations which only depended on the delay term, and had no explicit dependence of the current state (in other words, f(t, xt) = F(x(t), x(t−ρ(t))) = F(x(t−ρ(t)))). However, a dependence on both the current and retarded state is more usual in applications, and often the equation can be interpreted as a perturbation of an ordinary differential equation. In this section we will assume that Fcan be written as the following sum: F(x(t), x(t−ρ(t))) = F0(x(t)) + F1(x(t−ρ(t))). In this situation, we shall show that if a dissipative hypothesis for the term F0holds, then the assumptions on the other term can be relaxed. Let us assume that F0:Rn→Rnis a continuous functions satisfying the dissipative assumption as above: hF0(x), xi ≤ −α0|x|2+β0,for all x∈Rn.(7) Firstly, if we suppose that F1:Rn→Rnis a continuous and bounded function, i.e. there exists k≥0 such that |F1(x)| ≤ k, for all x∈Rn, 8TOM´ AS CARABALLO, JOS´ E A. LANGA AND JAMES C. ROBINSON then it is easy to prove that (6) holds. Indeed, for every ψ∈Φ(h)Cand a fixed ε<α0, hf(t, ψ), ψ(0)i=hF0(ψ(0)), ψ(0)i+hF1(ψ(−ρ(t))), ψ(0)i ≤ −α0|ψ(0)|2+β0+k|ψ(0)| ≤ −(α0−ε)|ψ(0)|2+β0+k2 4ε. Secondly, it is still possible to weaken this boundedness on F1,although now it is necessary to assume more regularity for the delay function. Instead of proving that (6) holds, we prove the existence of a family of bounded absorbing sets directly. Theorem 4.3. Assume F0satisfies (7). Assume that F1is sublinear, i.e. there exists k > 0such that |F1(x)|2≤k2(1 + |x|2),for all x∈Rn, and suppose that the delay function ρis continuously differentiable with ρ0(t)≤ρ∗<1.Then, if k2< α2 0(1 −ρ∗), there exists a family of bounded absorbing sets, {B(t0)}t0∈Rfor (3), and consequently there exists a pullback attractor for this problem. Proof. Choose a positive λ(small enough) and another positive εwhich will be fixed later. As in the last theorem, let us write x(τ) = x(τ;t0−t, ψ), τ∈[t0−t, t0],for ψin a given bounded set D⊂ C,i.e. kψkC≤d, for all ψ∈D. Then, it follows d dτeλτ |x(τ)|2=λeλτ |x(τ)|2+ 2eλτ hx(τ), f(τ, xτ)i =λeλτ |x(τ)|2+ 2eλτ hx(τ), F0(x(τ))i +2eλτ hx(τ), F1(x(τ−ρ(τ)))i ≤(λ−2α0)eλτ |x(τ)|2+ 2β0eλτ +εeλτ |x(τ)|2+eλτ ε−1|F1(x(τ−ρ(τ)))|2 ≤(λ−2α0+ε)eλτ |x(τ)|2+ (2β0+k2ε−1)eλτ +k2ε−1eλτ |x(τ−ρ(τ))|2. By integration on the interval [t0−t, τ], eλτ |x(τ)|2−eλ(t0−t)|x(t0−t)|2≤2β0+k2ε−1 λeλτ −eλ(t0−t) +(λ−2α0+ε)Rτ t0−teλs|x(s)|2ds +k2 εRτ t0−teλs|x(s−ρ(s))|2ds. (8) ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 9 Evaluating the term containing the delay function by making the change of variable s−ρ(s) = uin the integral, we obtain Rτ t0−teλs|x(s−ρ(s))|2ds ≤1 1−ρ∗Rτ t0−t−heλu+λh|x(u)|2du ≤eλh 1−ρ∗hRt0−t t0−t−heλu|x(u)|2du +Rτ t0−teλu|x(u)|2dui ≤eλh 1−ρ∗hRt0−t t0−t−heλu|ψ(u)|2du +Rτ t0−teλu|x(u)|2dui ≤eλh 1−ρ∗Rτ t0−teλu|x(u)|2du +d2eλh λ(1−ρ∗)eλ(t0−t)−eλ(t0−t−h), and, consequently, eλτ |x(τ)|2≤eλ(t0−t)d2+2β0+k2ε−1λ−1eλτ −eλ(t0−t) +d2eλhk2ε−1 λ(1−ρ∗)eλ(t0−t)−eλ(t0−t−h) +hλ−2α0+ε+eλhk2ε−1 (1−ρ∗)iRτ t0−teλs|x(s)|2ds. Now, taking ε=α0and noticing that for λsmall enough we can assure that λ−2α0+ε+eλhk2ε−1 (1−ρ∗)is negative, it immediately follows that |x(τ)|2≤d2h1 + eλh k2ε−1 λ(1−ρ∗)ieλ(t0−t−τ)+2β0+k2ε−1λ−1, and setting τ=t0+θ, for θ∈[−h, 0], |x(t0+θ)|2≤d21 + eλhk2ε−1 λ(1 −ρ∗)e−λ(t+θ)+2β0+k2ε−1λ−1, and, thus sup θ∈[−h,0] |x(t0+θ)|2≤d2h1 + eλh k2ε−1 λ(1−ρ∗)ie−λt+λh +2β0+k2ε−1λ−1 ≤1 + 2β0+k2ε−1λ−1 if t≥TD=λ−1log d21 + eλhk2ε−1 λ(1 −ρ∗)eλh. 16 TOM´ AS CARABALLO, JOS´ E A. LANGA AND JAMES C. ROBINSON Given an initial condition xs=ψ, with kψkC0≤M, consider d dt(x(t)−y(t)) = F(x(t)) −F(y(t−ρ(t))). Taking the inner product with x(t)−y(t) gives 1 2 d dt|x(t)−y(t)|2=hF(x(t)) −F(y(t)), x(t)−y(t)i +hF(y(t)) −F(y(t−ρ(t))), x(t)−y(t)i ≤L|x(t)−y(t)|2+L|y(t)−y(t−ρ(t))||x(t)−y(t)| ≤2L|x(t)−y(t)|2+2M0≤t≤ kL|ρ(t)|t≥ ≤2L|x(t)−y(t)|2+2M0≤t≤ kL t ≥. On [0, ] we can deduce that |x(t)−y(t)|2≤(eLt −1)2M L, and so, in particular, |x(t)−y(t)|2≤(eL −1)2M Lfor all t∈[0, ]. Now, starting from t=we have |x(t)−y(t)|2≤1 L[2M(eL −1) + kL]eL(t−), and, therefore, |x(t)−y(t)|2≤C(, t), where C(, t)→0 as ↓0+uniformly on bounded time intervals. In particular, it follows that sup s∈R kφ(t+s, s)ψ− S(t)ψkC0→0 for all t≥0.(20) 3. This follows immediately from (15), since the proof of the existence of an absorbing set in Rnunder this condition is simple. There is then a global attractor A ⊂Rn, and we can define Aas in (19). 4. Finally, note that the radius of the absorbing set Bin theorem 4.2 depends on αand β, and using the calculations from section 4.1.1 it follows ATTRACTORS FOR DELAY DIFFERENTIAL EQUATIONS 17 that αand βcan be taken uniform over ∈(0, 0]. It then follows that the compact absorbing set in theorem 4.2 is given by φ(t0, t0−0)B. It follows from (20) that for every ∈(0, 0] this is a subset of a fixed compact set K. Since A(t)⊂B(t) (see theorem 3.1) the final condition is satisfied, and an application of theorem 6.1 gives the result as stated. Remark 6.1. Note that one could also compare the attractors by considering the subsets of Rn A(t) = {y∈Rn:y=x(t), t ∈[−, 0], x ∈A(t)}. It then follows that dist(A(t),A)→0 as →0, where now the distance is measured in Rn. 7. CONCLUSIONS By using the cocycle attractor we have extended the classical treatment of attractors for delay differential equations to the general nonautonomous case, and in particular we have recovered results on periodic equations. Furthermore, by using the upper semicontinuity result from Carballo and Langa [2], we have shown that the introduction of a small delay has little effect on the asymptotic dynamics. Finally, we note that these ideas should be applicable to a wider class of equations. 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