Exponential attractors for reaction–diffusion equations with arbitrary polynomial growth

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

“…The problem has been studied in [10]; here we will use our method to prove the existence of exponential attractors.…”

“…However, global attractors attract any bounded set in X , but the attraction to it may be arbitrarily slow. The need to overcome this drawback created the notion of the exponential attractors [7][8][9][10][11][12], a compact, positively invariant set of finite fractal dimension which exponentially attract each bounded subset. Though it is no longer uniquely determined, the exponential attractors still contain the global attractor.…”

“…Besides that, in 2002, Málek and Prazák proved the existence of an exponential attractor by the l-trajectory method. These methods have been applied to a variety of concrete equations [7][8][9][10][11][12][13][14].…”

Exponential Attractors for Parabolic Equations with Dynamic Boundary Conditions

“…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.…”

Continuous Dependence on a Parameter of Exponential Attractors for Nonclassical Diffusion Equations

“…erefore, from the compact embedding W 1,2 [0, t; H 1 0 (Ω)] ⊂ L 2 [0, t; L 2 (Ω)], we deduce that n ϵ,t 1 is compact in B. us, assertion (i) is proved. Assertion (ii) can be easily proved by multiplying (70) (with ϵ � 0) with −Δw 0 2 , and this procedure is elementary; here, we omit it (see, e.g., [20] for details).…”