Abstract
In this paper, we are concerned with the existence and stability of pullback exponential attractors for a non-autonomous dynamical system. (ⅰ) We propose two new criteria for the discrete dynamical system and continuous one, respectively. (ⅱ) By applying the criteria to the non-autonomous Kirchhoff wave models with structural damping and supercritical nonlinearity we construct a family of pullback exponential attractors which are stable with respect to perturbations.
Highlights
It is well known that pullback attractor and pullback exponential attractor are two basic concepts to study the longtime dynamics of infinite dimensional non-autonomous dynamical system
A family of nonempty compact subsets {A(t)}t∈R in E is said to be a pullback attractor of the process {U (t, τ )} if it is invariant, i.e., U (t, s)A(s) = A(t), t ≥ s, and it pullback attracts all bounded subsets of E, i.e., for every bounded subset
The treatment for non-autonomous Eq (4) is a continuation of researches in [24], under the same assumptions as in [24] except g ∈ Hb1(R; L2), by virtue of the criterion established above, we prove the existence of pullback exponential attractor and show their stability with respect to perturbations
Summary
It is well known that pullback attractor and pullback exponential attractor are two basic concepts to study the longtime dynamics of infinite dimensional non-autonomous dynamical system. Motivated by the idea in [9, 13], we establish two new abstract criteria on the existence and stability of a family of pullback exponential attractors for the discrete dynamical system and continuous one, respectively (see Theorem 2.3 and Theorem 3.1), which are of more relaxed assumptions and applicability and are the developments of construction in [13] By applying these criteria to non-autonomous Kirchhoff wave models with structural damping and supercritical nonlinearity utt − M ( ∇u 2)∆u + (−∆)αut + f (u) = g(x, t),. (ii) By applying above criterion, we construct a family of pullback exponential attractors {Mg(t)}t∈R for a family of processes {Ug(t, τ )}, g ∈ Σ generated by the Kirchhoff wave model (4), which are stable with respect to perturbations g ∈ Σ (symbol space) (see Theorem 4.4).
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