Continuity of attractors for parabolic problems with localized large diffusion

Continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions

“…Also, for the case n = 1 the continuity has been proved in [9], where the authors prove that the limiting problem is generically Morse-Smale and, using the fact that the spectra of the associate linear operators posses large gaps, C 1 -convergence of the invariant manifolds leading topological equivalence of the family of attractors for small values of . For the case of homogeneous Dirichlet boundary conditions, the upper and lower semicontinuity of attractors as tends to zero is studied in [8]. Here we extend these results to the case of nonlinear Neumann boundary conditions.…”

“…For instance, concerning to singular perturbations of the domains, we can cite [4] for Dumbbell type domains and [18], where this agenda is applied for examples on thin domains. We also refer [2] for a highly oscillating boundary problem, [8] for localized large diffusion problems with homogeneous Dirichlet boundary conditions and [11] for a general scheme developed to treat the continuity of the attractors of semilinear parabolic problems.…”

Lower semicontinuity of attractors for non-autonomous dynamical systems

“…Such results, in general, rely on reproducing the structures present in the limiting attractor inside the perturbed attractors. Because of this, a gradient-like structure for the attractors of the limiting problem has always been taken as the starting point (for autonomous problems by Hale and Raugel [19], Stuart and Humphries [29] and various other authors [1,2,5,8,9]; for non-autonomous perturbations of autonomous systems by Langa et al [23] and Carvalho et al [11]). The following result, however, requires only that the limiting attractor be the closure of the union of a (possibly infinite) number of unstable manifolds of hyperbolic global solutions.…”

“…Our results suggest a new set of problems for further study. It should be possible to establish lower semicontinuity under singular perturbations of dynamical systems, although results on the robustness of exponential dichotomies would need to be proved first (see the papers [1,2,5,8,9] for examples of lower semicontinuity results in singularly perturbed gradient systems). It would also be very interesting to describe the geometrical structure of the perturbed attractors; note that the results of Langa et al [23] or Carvalho et al [11] are not applicable here.…”

“…Their proof of this result relies on A. N. Carvalho et al the continuity of the equilibria and lower semicontinuity of the local unstable manifolds under perturbation (see also Stuart and Humphries [29], and Kostin [22] and Elliot and Kostin [16], who use a similar argument based on the continuity of unstable manifolds but show that this remains applicable in certain systems with non-hyperbolic equilibria when the unperturbed problem possesses a Lyapunov functional). Several applications have been considered in [1,2,5,8,9] (including more complicated situations where even the continuity of the equilibria is not straightforward), which maintain the hypothesis of a gradient structure for the limit problem. In all of these papers (with the exception of [16] and [22]), a key step was to show the uniformity (with respect to the underlying parameter) of the exponential dichotomy of the linearization around each hyperbolic equilibrium.…”