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

José M. Arrieta, Flank D. M. Bezerra, Alexandre N. Carvalho

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 $a_\eps(\cdot)$. We show that the equilibria and the local unstable manifolds of the perturbed problem are at a distance given by the order of $\|a_\eps-a_0\|_\infty$. Moreover, the perturbed nonlinear semigroups are at a distance $\|a_\eps-a_0\|_\infty^\theta$ with $\theta< 1$ but arbitrarily close to 1.
Nevertheless, we can only prove that the distance of attractors is of order $\|a_\eps-a_0\|_\infty^\beta$
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.

Keywords


Attractors; continuity; uniform exponential attraction; rate of convergence\

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