Time-dependent attractor for the Oscillon equation

Time-dependent asymptotic behavior of the solution for plate equations with linear memory

“…In the case when ε(t) is a positive decreasing function which vanish at infinity, the problem (1.1) becomes more complex and interesting; the reason is that the corresponding dynamical system is still understood under the non-autonomous framework even the forcing term in the equation is independent of time t. In order to deal with these problems, in [5], Conti, Pata and Temam presented a notion of time-dependent attractor exploiting the minimality with respect to the pullback attraction property, and gave a sufficient condition proving the existence of time-dependent attractor based on the theory established by Plinio, Duane and Temam ( [8]). Besides, they applied the new methods into the following weak damped wave equations with time-dependent speed of propagation ε(t)u tt + αu t − ∆u + f (u) = g(x).…”

Time-Dependent Global Attractor for a Class of Nonclassical Parabolic Equations

“…In this paper, we borrow some ideas from the following previous contributors: Conti et al in [17] who introduced the theory of time-dependent global attractors and apply the theory to the wave equations; Caraballo et al who introduced a one-parameter family of Banach spaces in the context of cocycles for nonautonomous and random dynamical systems in [18] as well as time-dependent spaces [19] in the context of stochastic partial differential equations; and Flandoli and Schmalfuss in [20] who introduced a family of metric spaces depending on a parameter and applied it to the stochastic form of Navier-Stokes equations. In this paper, based on the recent theory of time-dependent global attractors of Conti et al [17] and di Plinio et al [21], we prove the existence of time-dependent global attractors as well as the regularity of the time-dependent global attractor for a class of nonclassical parabolic equations as described by (1).…”

Asymptotic structure of the attractor for processes on time-dependent spaces

“…We begin by recalling some basic definitions and results from [1,2,8] about processes on time-dependent spaces. For t ∈ R, let Z t be a family of normed spaces.…”

“…The model above, which can be seen as a nonlinear damped wave equation with time-dependent speed of propagation 1/ε(t), has been studied in detail in [1]. In that work, problem (1.1) is shown to generate a process U(t, τ ), acting on a family of time-dependent spaces {Z t }, possessing the time-dependent global attractor in the sense of [2]. Loosely speaking, this is the smallest family {A t } (where each A t lies in its own t-indexed space Z t ) which attracts bounded subsets in a pullback way.…”

“…The first part of this work is devoted to the general theory of attractors for processes in time-dependent spaces, specifically developed in [1,2,8] to handle evolution problems where the coefficients of the differential operators depend explicitly on time. Due to the presence of time-dependent terms at a functional level, the standard theory of attractors fails to capture the dissipation mechanism involved in the dynamical system, and thus it is not suitable to study its longterm behavior.…”