Exponential attractors for a class of reaction-diffusion problems with time delays

We consider a reaction-diffusion system subject to homogeneous Neumann boundary conditions on a given bounded domain. The reaction term depends on the population densities as well as on their past histories in a very general way. This class of systems is widely used in population dynamics modelling. Due to its generality, the longtime behavior of the solutions can display a certain complexity. Here we prove a qualitative result which can be considered as a common denominator of a large family of specific models. More precisely, we demonstrate the existence of an exponential attractor, provided that a bounded invariant region exists and the past history decays exponentially fast. This result will be achieved by means of a suitable adaptation of the l-trajectory method coming back to the seminal paper of Malek and Necas.