Rate of convergence of global attractors of some perturbed reaction-diffusion problems

Abstract

In this paper we treat the problem of the rate of convergence of attractors of dynamical systems for some autonomous semilinear parabolic problems. We consider a prototype problem, where the diffusion $a_0(\cdot)$ of a reaction-diffusion equation in a bounded domain $\Omega$ is perturbed to $a_\varepsilon(\cdot)$. We show that the equilibria and the local unstable manifolds of the perturbed problem are at a distance given by the order of $\|a_\varepsilon-a_0\|_\infty$. Moreover, the perturbed nonlinear semigroups are at a distance $\|a_\varepsilon-a_0\|_\infty^\theta$ with $\theta< 1$ but arbitrarily close to $1$. Nevertheless, we can only prove that the distance of attractors is of order $\|a_\varepsilon-a_0\|_\infty^\beta$ for some $\beta< 1$, which depends on some other parameters of the problem and may be significantly smaller than $1$. We also show how this technique can be applied to other more complicated problems.