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On the robustness of pullback attractors for a nonlocal reaction-diffusion equation under perturbation

Rubén Caballero, Pedro Marín-Rubio, José Valero

Abstract

A parametric family of reaction-diffusion equations with nonlocal viscosity is considered. Existence of solutions and actually of pullback attractors is known from previous works. In this paper we obtain a robustness result of the attractors toward the corresponding minimal pullback attractor of the limiting problem. This result extends the ones obtained in \cite{5}. Actually here all terms (reactions, external forces and nonlocal viscosity functions) may vary with the parameter. The upper semicontinuous convergence of attractors is obtained under rather general assumptions and in a fully non-autonomous context using the framework of tempered universes.

On the robustness of pullback attractors for a nonlocal reaction-diffusion equation under perturbation

Abstract

A parametric family of reaction-diffusion equations with nonlocal viscosity is considered. Existence of solutions and actually of pullback attractors is known from previous works. In this paper we obtain a robustness result of the attractors toward the corresponding minimal pullback attractor of the limiting problem. This result extends the ones obtained in \cite{5}. Actually here all terms (reactions, external forces and nonlocal viscosity functions) may vary with the parameter. The upper semicontinuous convergence of attractors is obtained under rather general assumptions and in a fully non-autonomous context using the framework of tempered universes.
Paper Structure (3 sections, 14 theorems, 57 equations)

This paper contains 3 sections, 14 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Dynamical systems and attractors
  3. Robustness result in $L^2$

Key Result

Theorem 4

Assume that (A1) holds. Then for any $u_\tau\in L^2(\Omega )$ there exists at least one weak solution to $(P_\eta).$ The set of weak solutions to $(P_\eta)$ with initial datum $u_\tau$ at time $\tau$ will be denoted by $\Phi_\eta (\tau,u_\tau).$

Theorems & Definitions (37)

  • Remark 1
  • Definition 2
  • Remark 3
  • Theorem 4
  • Definition 5
  • Proposition 6
  • Definition 7
  • Definition 8
  • Theorem 9
  • Remark 10
  • ...and 27 more