Uniform Exponential Attractors for Non-Autonomous Dynamical Systems and an Application to the damped hyperbolic equation with sub-quintic growth
In this paper, we develop an abstract framework for the theory of uniform exponential attractors in non-autonomous dynamical systems. Our construction employs time-dependent absorbing sets while ensuring exponential attraction among the constructed sets. Moreover, we show that the resulting exponential attractors are uniform-that is, they remain stable under perturbations. As an application of this abstract theory, we study the weakly damped hyperbolic equation with sub-quintic nonlinearity. Using the proposed approach, we construct exponential attractors for the process generated by the Shatah-Struwe solution, which satisfies both energy and Strichartz estimates. Furthermore, we establish the Holder continuity of these exponential attractors with respect to the external forcing term.