Necessary and sufficient conditions for the existence of exponential attractors for semigroups, and applications

Exponential Attractors for Parabolic Equations with Dynamic Boundary Conditions

“…Theorem 2 (see [14]). Assume that is a bounded absorbing set for discrete dynamical system ( ) in ; then the following are equivalent.…”

“…In [14], the authors established some necessary and sufficient conditions for the existence of exponential attractors for continuous and norm-to-weak continuous semigroup and provided a new method for proving the existence of exponential attractors by combining with the flattering property. Motivated by some ideas in [14][15][16], we combine asymptotic a prior estimate with the enhanced flattening property and show sufficient and necessary existence of exponential attractors in uniformly convex Banach spaces. As an application, we prove the existence of exponential attractors for the reaction-diffusion equation with dynamic boundary condition.…”

Pullback Exponential Attractors for Nonautonomous Reaction–Diffusion Equations

“…We refer to [Li et al, 2012;Pražák, 2003] for the necessary and sufficient condition needed to prove the existence of the exponential attractor. The authors in [Pražák, 2003] presented an abstract result on the existence of the exponential attractor under the necessary and sufficient condition.…”

“…They used the squeezing condition to verify the result. Li et al [2012] employed the measure of noncompactness to prove the existence of the exponential attractor. They also utilized the flattening condition to verify the exponential decay of the measure of noncompactness.…”

Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness

“…The intention of this article is to show the existence of a compact positively invariant set A * which exponentially attracts every bounded set for multi-valued semidynamical systems. It is worth mentioning that here we do not consider the finite dimensionality of the set A * , since it is difficult to show that multi-valued systems possess some kind of smoothing property, which was used in the construction of the exponential attractors for the single-valued case, see, e.g., [2,12,13,14,15,24,34], and in fact, many single-valued semigroups have infinite dimensional global attractors, see, e.g., [33,36].…”