Rate of convergence of attractors for singularly perturbed semilinear problems

Abstract

We exhibit a class of singularly perturbed parabolic problems which the asymptotic behavior can be described by a system of ordinary differential equation. We estimate the convergence of attractors in the Hausdorff metric by rate of convergence of resolvent operators. Application to spatial homogenization and large diffusion except in a neighborhood of a point will be considered.

Large Diffusivity and Rate of Convergence of Attractors in Parabolic Systems

“…This estimate imply with the compact convergence in [3] and [5], that is the operator A −1 ε converges compactly to A −1 ∞ P = P . Note that, if we take ϕ = 1 as a test function in (4.2), we have u ∞ = P u ε , hence (4.1) shows that u ε converge for its average in X ε .…”

“…In order to obtain estimate for the convergence of the attractor A ε of the equation (2.4) to the attractor A ∞ of the (3.4) as d ε → μ following the results of the [5], we assume the nonlinear semigroup T ∞ (•) generated by solutions of the (3.4) is a Morse-Smale semigroup in R n . More precisely,…”

“…This property is known as geometric structural stability and it is the main feature of Morse-Smale problems. In this way we are under the conditions described in [5] where results about rate of convergence of attractor for Morse-Smale problems were obtained. More precisely it is valid the following result.…”

Smoothing effect and asymptotic dynamics of nonautonomous parabolic equations with time-dependent linear operators

“…In papers such as [2,3,4,8,11,12,14] a general routine was conceived and applied in order to guarantee convergence of solutions of Problems (1.2) as ε → 0 + . This routine is based in a detailed study of the behavior of the linear part.…”