Regular approach to attractors of singularly perturbed equations

Gradient semi-dynamical systems, which depend on parameter(s) $\lambda$ and possess a finite number of hyperbolic equilibrium points, are considered. Under certain assumptions it is proved that the global attractor $\mathfrak{M}_\lambda$ is Hölder continuous in $\lambda$ in the Hausdorff metric. As an intermediate result it is shown that $\mathfrak{M}_\lambda$ uniformly in $\lambda$ exponentially attracts every bounded set. The results are applied to prove the convergence (in the Hausdorff metric) of the global attractor of an abstract damped hyperbolic equation with a small parameter $\varepsilon$ by the second-order time derivative — to the attractor of a corresponding parabolic equation.