Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents

Some abstract results on the convergence of nonautonomous pullback attractors in asymptotically autonomous problems are established and then applied to quasi-linear parabolic equations with spatially variable exponents in which the parabolic operator is time-dependent. In particular, it is shown that the component subsets of the pullback attractor converge in the Hausdorff semi-distance to the global autonomous attractor.