This paper is concerned with the lower semicontinuity of attractors for semilinear
non-autonomous differential equations in Banach spaces. We require the unperturbed
attractor to be given as the union of unstable manifolds of time-dependent hyperbolic
solutions, generalizing previous results valid only for gradient-like systems in which
the hyperbolic solutions are equilibria. The tools employed are a study of the continuity
of the local unstable manifolds of the hyperbolic solutions and results on the continuity of
the exponential dichotomy of the linearization around each of these solutions
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