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Abstract
In a real separable Hilbert space, we consider nonautonomous evolution equations including time‐dependent subdifferentials and their nonmonotone multivalued perturbations. In this paper, we treat the multivalued dynamical systems associated with time‐dependent subdifferentials, in which the solution is not unique for a given initial state. In particular, we discuss the asymptotic behaviour of our multivalued semiflows from the viewpoint of attractors. In fact, assuming that the time‐dependent subdifferential converges asymptotically to a time‐independent one (in a sense) as time goes to infinity, we construct global attractors for nonautonomous multivalued dynamical systems and its limiting autonomous multivalued dynamical system. Moreover, we discuss the relationship between them.
Highlights
In [8, 12], we considered a nonlinear evolution equation in a real Hilbert space H of the form u (t) + ∂φt u(t) + g t, u(t) f (t) in H, t > s(≥ 0), (1.1)
Where ∂φt is the subdifferential of a time-dependent proper lower semicontinuous (l.s.c.) and convex function φt on H, g(t, ·) is a single-valued perturbation which is small relative to φt, and f is a given forcing term
G(t, ·), and f (t), respectively, converge to a convex function φ∞ on H, a singlevalued operator g∞(·) in H and an element f ∞ in H in appropriate senses as t → +∞, we considered the limiting autonomous system u (t) + ∂φ∞ u(t) + g∞ u(t) f ∞ in H, t ≥ 0
Summary
In [8, 12], we considered a nonlinear evolution equation in a real Hilbert space H of the form u (t) + ∂φt u(t) + g t, u(t) f (t) in H, t > s(≥ 0),. In [8], considering the case when the Cauchy problems (1.1) and (1.2) lose the uniqueness of solutions, we discussed the largetime behaviour of multiple solutions for (1.1) and (1.2) In such a situation, the solution operator E(t, s) (0 ≤ s ≤ t < +∞) for (1.1) is multivalued. In [8] we showed that there exists a global attractor for multivalued evolution operators {E(t, s)} and it is semi-invariant under S(t). In Mel’nik and Valero [9], they constructed the uniform global attractor for (1.4) with φt ≡ φ and f (t) ≡ 0, which implies that the domains of solution operators {E(t, s)} are independent of time t, s ∈ R+. For two sets A and B in H, we define the so-called Hausdorff semi-distance distH (A, B) := sup inf |x − y|H
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