Robustness of uniform attractors under perturbations on the symbol space

In this work, we investigated the continuity properties of uniform attractors for a class of non-autonomous evolution processes with perturbations on the symbol space. Within a general abstract framework, we established sufficient conditions for the upper semicontinuity, lower semicontinuity, and residual continuity of the family of uniform attractors AΣℓ in perturbation parameter ℓ. As illustrative applications, we considered in detail a non-autonomous semilinear damped wave equation with Dirichlet boundary conditions, where all the coefficients in the equation are time-dependent. We proved the global well-posedness, the existence of a uniform attractor, and then the continuity properties of the uniform attractor under perturbations on these time-dependent coefficients. Particularly for a sine-Gordon equation, we proved that the uniform attractor varies continuously under perturbations and the optimal convergence rate was given. Finally, a two-dimensional Navier-Stokes system was briefly discussed as well, indicating that our theoretical results apply to a broad number of dynamical systems in mathematical physics.