Pullback exponential attractors for differential equations with variable delays

Abstract

We show how recent existence results for pullback exponential attractors can be applied to non-autonomous delay differential equations with time-varying delays. Moreover, we derive explicit estimates for the fractal dimension of the attractors. As a special case, autonomous delay differential equations are also discussed, where our results improve previously obtained bounds for the fractal dimension of exponential attractors.

Pullback exponential attractors for differential equations with delay

“…(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).…”

“…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.…”

“…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).…”

“…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].…”

“…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.…”