Exponential attractors for semigroups in Banach spaces

Let {S(t)}t⩾0{S(t)}t⩾0 be a semigroup on a Banach space XX, and AA be the global attractor for {S(t)}t⩾0{S(t)}t⩾0.We assume that there exists a T∗T∗ such that S≜S(T∗)S≜S(T∗) is of class C1C1 on a bounded absorbing set Bϵ0(A)Bϵ0(A) and S:Bϵ0(A)→Bϵ0(A)S:Bϵ0(A)→Bϵ0(A), and furthermore, the linearized operator LL at each point of Bϵ0(A)Bϵ0(A) can be decomposed as L=K+CL=K+C with KK compact and ‖C‖<λ<1‖C‖<λ<1; then we prove the existence of an exponential attractor for the discrete semigroup {Sn}n=1∞ in the Banach space XX. And then we apply the standard approach of Eden et al. (1994) [9] to obtain the continuous case. Here Bϵ0(A)Bϵ0(A) denotes the ϵ0ϵ0-neighborhood of AA in Banach space XX, and ‖C‖‖C‖ denotes the norm of the operator CC.We prove, as a simple application, the existence of an exponential attractor for some nonlinear reaction–diffusion equations with polynomial growth nonlinearity of arbitrary order.