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

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 timedependent 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.

Asymptotic Regularity and Existence of Time-Dependent Attractors for Second-Order Undamped Evolution Equations with Memory

“…In addition, Meng et al [20] gave some necessary and sufficient conditions to guarantee the existence of a time-dependent attractor. Liu et al [23,24] considered the longtime dynamical behavior and achieved the existence of time-dependent attractor for the plate equation on a bounded domain or unbounded domain via an operator decomposition or a contractive function method, respectively. Furthermore, the time-dependent asymptotic behavior of the nonclassical reaction-diffusion equation was studied in [25,26].…”

Time-dependent attractor of wave equations with nonlinear damping and linear memory

“…Liu and Ma ([ ]) studied the existence of the pullback attractors for the plate equation with time-dependent forcing term on the strong time-dependent Hilbert space. Successively, exploiting the methods and framework of [22,24], Liu and Ma obtained the existence and regularity of the time-dependent attractor for the plate equation with critical growth nonlinearity, as well as the asymptotic structure in [28].…”

Asymptotic behavior for stochastic plate equations with memory and additive noise on unbounded domains

“…For the stochastic plate equations, if µ = 0 and the forcing term g(x, t) = g(x), then the existence of a random attractor of (1.1)-(1.2) on bounded domain have been proved in [15,16,12,14]; if µ = 0, the existence of random attractors for plate equations with memory and additive white noise on bounded domain were considered in [19,20]. Recently, on the unbounded domain, the authors investigated the asymptotic behavior for stochastic plate equation with additive noise and multiplicative noise (see [33,32,30,31] for details).…”