RATE OF CONVERGENCE OF GLOBAL ATTRACTORS OF SOME PERTURBED REACTION-DIFFUSION PROBLEMS

Author(s)

Arrieta, Jose M. ^[1] ; Bezerra, Flank D. M. ^[2] ; Carvalho, Alexandre N. ^[3]

Affiliation

^[1] Univ Complutense Madrid, Fac Matemat, Dept Matemat Aplicada, E-28040 Madrid - Spain

^[2] Univ Fed Paraiba, Ctr Ciencias Exatas & Nat, BR-58051900 Joao Pessoa, Paraiba - Brazil

^[3] Univ Sao Paulo, Inst Ciencias Matemat & Comp, BR-13560970 Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS; v. 41, n. 2, p. 229-253, JUN 2013.
• Web of Science Citations: 4

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) (.) of a reaction-diffusion equation in a bounded domain Omega is perturbed to a(epsilon)(.). We show that the equilibria and the local unstable manifolds of the perturbed problem are at a distance given by the order of parallel to a(epsilon) - a(0)parallel to(infinity). Moreover, the perturbed nonlinear semigroups are at a distance parallel to a(epsilon) - a(0)parallel to(theta)(infinity) with theta < 1 but arbitrarily close to 1. Nevertheless, we can only prove that the distance of attractors is of order parallel to a(epsilon) - a(0)parallel to(theta)(infinity) 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. (AU)