Exponential attractors for semigroups in Banach spaces

Abstract

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 ≜ S ( T ∗ ) is of class C 1 on a bounded absorbing set B ϵ 0 ( A ) and S : B ϵ 0 ( A ) → B ϵ 0 ( A ) , and furthermore, the linearized operator L at each point of B ϵ 0 ( A ) can be decomposed as L = K + C with K compact and ‖ C ‖ < λ < 1 ; then we prove the existence of an exponential attractor for the discrete semigroup { S n } n = 1 ∞ 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 ϵ 0 ( A ) denotes the ϵ 0 -neighborhood of A in Banach space X , and ‖ C ‖ 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.