Abstract
In this article, we consider the long-time behavior of solutions for the plate equation with linear memory. Within the theory of process on time-dependent spaces, we investigate the existence of the time-dependent attractor by using the operator decomposition technique and compactness of translation theorem and more detailed estimates. Furthermore, the asymptotic structure of time-dependent attractor, which converges to the attractor of fourth order parabolic equation with memory, is proved. Besides, we obtain a further regular result.
Highlights
Let Ω be an open bounded set of Rn(n ≥ 5) with smooth boundary ∂Ω
As in ([4, 18]), we introduce a new variable η as follows η = ηt(x, s) = u(x, t) − u(x, t − s), (x, s) ∈ Ω × R+ = [0, +∞), t ≥ τ, (2.1)
Let Xt be a family of normed spaces, we introduce the R−ball of Xt
Summary
Let Ω be an open bounded set of Rn(n ≥ 5) with smooth boundary ∂Ω. We consider the following equations ε(t)utt + αut + ∆2u + ∞ 0 μ(s)∆2(u(t) − u(t s))ds + f (u) =g(x), in Ω × (τ, ∞), u(x, t) = ∂u(x,t) ∂n= 0, x ∈ ∂Ω, t ∈ R, u(x, t) = u0(x, t), ut(x, t) = ∂tu0(x, t), x ∈ Ω, t ≤ τ, (1.1). Linear memory, time-dependent attractor, regularity, compactness transitivity theorem, asymptotic structure. A time-dependent absorbing set for the process U (t, τ ) is a uniformly bounded family B = {Bt}t∈R with the following property: for every R > 0 there exists a t0 such that τ ≤ t − t0 ⇒ U (t, τ )Bτ (R) ⊂ Bt. Definition 2.4.
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