Stability of regular pullback attractors for non-autonomous dynamical systems: theoretical results and applications

We study the stability of regular pullback attractors for non-autonomous dynamical systems generated by partial differential equations with non-autonomous forcing term. We first introduce the concept of the backward compact regular pullback attractor. We then establish the existence theorem of the backward compact regular pullback attractor. Finally, we discuss the stability (backward controllability and asymptotic autonomy) of the backward compact regular pullback attractor in the regular space. As an application of theoretical results, we consider the double time-delayed Newton-Boussinesq equations with non-autonomous forcing term. Since the solution has no high regularity, we use the spectrum decomposition technique to prove the asymptotic compactness of the solution operator in the regular space.