Exponential attractors for semigroups in Banach spaces

Let {S(t)}(t >= 0) be a semigroup on a Banach space X, and A be the global attractor for {S(t)}(t >= 0). We assume that there exists a T* such that S (sic) S(T*) is of class C-1 on a bounded absorbing set B-epsilon 0 (A) and S : B-epsilon 0 (A) -> B-epsilon 0 (A), and furthermore, the linearized operator L at each point of B-epsilon 0 (A) can be decomposed as L = K + C with K compact and parallel to C parallel to < lambda < 1; then we prove the existence of an exponential attractor for the discrete semigroup {S-n}(n=1)(infinity) in the Banach space X. And then we apply the standard approach of Eden et al. (1994) [9] to obtain the continuous case. Here B-epsilon 0 (A) denotes the epsilon(0)-neighborhood of A in Banach space X, and parallel to C parallel to denotes the norm of the operator C. 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. (C) 2011 Elsevier Ltd. All rights reserved.