Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE

Yangrong Li; Lianbing She; Jinyan Yin

doi:10.3934/dcdsb.2018058

Article Contents

2018, Volume 23, Issue 4: 1535-1557. Doi: 10.3934/dcdsb.2018058

This issue Previous Article Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise Next Article Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity

Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Yangrong Li, Email: liyr@swu.edu.cn
* Corresponding author: Yangrong Li, Email: liyr@swu.edu.cn

Received: March 31, 2017

Revised: July 31, 2017

Early access: February 2018

Published: April 2018

This work is supported by National Natural Science Foundation of China grant 11571283 and Natural Science Foundation of Guizhou Province (KY[2016]103).

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Abstract

This paper is concerned with the robustness of a pullback attractor as the time tends to infinity. A pullback attractor is called forward (resp. backward) compact if the union over the future (resp. the past) is pre-compact. We prove that the forward (resp. backward) compactness is a necessary and sufficient condition such that a pullback attractor is upper semi-continuous to a compact set at positive (resp. negative) infinity, and also obtain the minimal limit-set. We further prove the lower semi-continuity of the pullback attractor and get the maximal limit-set at infinity. Some criteria for such robustness are established when the evolution process is forward or backward omega-limit compact. Those theoretical criteria are applied to prove semi-uniform compactness and robustness at infinity in pullback dynamics for a Ginzburg-Landau equation with variable coefficients and a forward or backward tempered nonlinearity.

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Mathematics Subject Classification: Primary: 37L55, 37L30; Secondary: 35B41.

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