Asymptotically autonomous dynamics for parabolic equations

Given an asymptotically autonomous system, it is shown that the components of the pullback attractor converge to the global attractor if and only if the pullback attractor is forward compact, this result reduces two uniformness conditions in the theoretical result given by Kloeden and Simsen (J. Math. Anal. Appl., 2015, 2017). Some constructions from the forward limit-set of a pullback attractor are established. Some applications are given for two classes of non-autonomous quasi-linear parabolic equations: one has a parabolic operator with spatially variable exponents and another has a weakly dissipative nonlinearity. For two equations, it is shown that the pullback attractor is forward compact and upper semi-continuous to the global attractor. Moreover, there is an inclusion between the global attractor and the forward limit-set of pullback attractors.