Existence and continuity of global attractors for ternary mixtures of solids

Mirelson M. Freitas; Anderson J. A. Ramos; Baowei Feng; Mauro L. Santos; Helen C. M. Rodrigues

doi:10.3934/dcdsb.2021196

Article Contents

2022, Volume 27, Issue 7: 3563-3583. Doi: 10.3934/dcdsb.2021196

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Existence and continuity of global attractors for ternary mixtures of solids

1.
Faculty of Mathematics, Federal University of Pará, Raimundo Santana Street, S/N, 68721-000, Salinópolis, PA, Brazil
2.
Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China
3.
Institute of Exact and Natural Sciences, Federal University of Pará, Augusto corrêa Street, Number 01, 66075-110, Belém PA, Brazil

^* Corresponding author: Mirelson M. Freitas
^* Corresponding author: Mirelson M. Freitas

Received: December 2020

Revised: June 2021

Published: July 2022

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Abstract

In this paper, we study the long-time dynamics of a system modelinga mixture of three interacting continua with nonlinear damping, sources terms and subjected to small perturbations of autonomousexternal forces with a parameter $ \epsilon $, inspired by the modelstudied by Dell' Oro and Rivera [12]. We establish astabilizability estimate for the associated gradient dynamicalsystem, which as a consequence, implies the existence of a compactglobal attractor with finite fractal dimension andexponential attractors. This estimate is establishedindependent of the parameter $ \epsilon\in[0,1] $. We also prove thesmoothness of global attractors independent of the parameter$ \epsilon\in[0,1] $. Moreover, we show that the family of globalattractors is continuous with respect to the parameter $ \epsilon $ ona residual dense set $ I_*\subset[0,1] $ in the same sense proposed inHoang et al. [15].

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Mathematics Subject Classification: Primary: 35B40, 35L53; Secondary: 35B41, 37L30.

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