Evolutionary system, global attractor, trajectory attractor and applications to the nonautonomous reaction-diffusion systems

TL;DR

This paper advances the theory of evolutionary systems by establishing the existence of strongly compact trajectory attractors for nonautonomous reaction-diffusion systems, providing new insights into their long-term behavior without restrictive nonlinear conditions.

Contribution

It introduces a framework for strongly compact trajectory attractors in asymptotically compact systems and applies it to reaction-diffusion equations without extra nonlinear assumptions.

Findings

Existence of strongly compact strong trajectory attractors.

Strong equicontinuity of complete trajectories on the global attractor.

Finite strong uniform tracking property for the system.

Abstract

In [Adv. Math., 267(2014), 277-306], Cheskidov and Lu develop a new framework of the evolutionary system that deals directly with the notion of a uniform global attractor due to Haraux, and by which a trajectory attractor is able to be defined for the original system under consideration. The notion of a trajectory attractor was previously established for a system without uniqueness by considering a family of auxiliary systems including the original one. In this paper, we further prove the existence of a notion of a strongly compact strong trajectory attractor if the system is asymptotically compact. As a consequence, we obtain the strong equicontinuity of all complete trajectories on global attractor and the finite strong uniform tracking property. Then we apply the theory to a general nonautonomous reaction-diffusion systems. In particular, we obtain the structure of uniform global attractors without any additional condition on nonlinearity other than those guarantee the existence of a uniform absorbing set. Finally, we construct some interesting examples of such nonlinearities. It is not known whether they can be handled by previous frameworks.