Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations

Yangrong Li; Renhai Wang; Lianbing She

doi:10.3934/eect.2018030

Article Contents

2018, Volume 7, Issue 4: 617-637. Doi: 10.3934/eect.2018030

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Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations

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

* Corresponding author: Yangrong Li
* Corresponding author: Yangrong Li

Received: August 2017

Revised: May 2018

Published: September 2018

Y. Li and R. Wang are supported by National Natural Science Foundation of China grant 11571283 and L. She is supported by the Natural Science Foundation of Guizhou Province:KY[2016]103.

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Abstract

This paper analyzes the time-dependence and backward controllability of pullback attractors for the trajectory space generated by a non-autonomous evolution equation without uniqueness. A pullback trajectory attractor is called backward controllable if the norm of its union over the past is controlled by a continuous function, and backward compact if it is backward controllable and pre-compact in the past on the underlying space. We then establish two existence theorems for such a backward compact trajectory attractor, which leads to the existence of a pullback attractor with the backward compactness and backward boundedness in two original phase spaces respectively. An essential criterion is the existence of an increasing, compact and absorbing brochette. Applying to the non-autonomous Jeffreys-Oldroyd equations with a backward controllable force, we obtain a backward compact trajectory attractor, and also a pullback attractor with backward compactness in the negative-exponent Sobolev space and backward boundedness in the Lebesgue space.

Keywords:

Trajectory attractor,
pullback attractor,
backward controllability,
backward compactness,
nonautonomous Jeffreys-Oldroyd equations.

Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 37B55.

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