Pullback Exponential Attractor for Second Order Nonautonomous Lattice System

Abstract

We first present some sufficient conditions for the existence of a pullback exponential attractor for continuous process on the product space of the weighted spaces of infinite sequences. Then we prove the existence and continuity of a pullback exponential attractor for second order lattice system with time-dependent coupled coefficients in the weighted space of infinite sequences. Moreover, we obtain the upper bound of fractal dimension and attracting rate for the attractor.

The Existence of Exponential Attractor for Discrete Ginzburg-Landau Equation

“…Using ( ) to take inner product (⋅, ⋅) with both sides of (15) and then taking the real part, we obtain…”

“…Also there are some references investigating the exponential attractors for lattice dynamical systems (LDSs). We can see [11][12][13] for the exponential attractors for firstorder LDSs; see [14,15] for the pullback exponential attractors for first-and second-order LDSs; see [16,17] for second-order nonautonomous LDSs and discrete Zakharov equations for the uniform exponential attractors.…”

“…There are many references studying the asymptotic behavior of general LDSs. For instance, we can refer to [23][24][25] for the existence of global attractor, to [26][27][28] for the uniform attractor, to [11,14,15] for the exponential and pullback exponential attractor, and to [29,30] for the random attractor. Also, there are some concrete applications of the above theory to the discrete PDEs.…”

Pullback-Forward Dynamics for Damped Schrödinger Equations with Time-Dependent Forcing

“…This paper deals with dynamics for the nonautonomous Schrödinger equation; that is, the force is time-dependent. To the best of our knowledge, there is no literature treating nonautonomous dynamics (including random dynamics) for the Schrödinger equation, even in the simple case for the existence of a pullback attractor, although the theory and application of pullback attractors had been widely developed for many other PDEs (see [12][13][14][15][16]), and for pullback random attractors, see, for example, [17][18][19][20].…”