Robustness of attractors for a class of damped wave equations

In this paper we prove the continuity, that is, upper and lower semicontinuity, of the global attractors for a class of three-dimensional damped wave equations, when perturbed with respect to a parameter $ p\geq 0 $. Upper semicontinuity is obtained at each $ p\geq 0 $. Lower semicontinuity (and, in turn, continuity) is achieved at $ p = 0 $ and, dropping the anti–damping term, it is also obtained at each $ p>0 $. Furthermore, we prove the permanence of the gradient structure for sufficiently small $ p $, which is also referred to in the literature as the topological structural stability at $ p = 0 $.