Forward attracting sets of reaction-diffusion equations on variable domains

Peter E. Kloeden; Meihua Yang

doi:10.3934/dcdsb.2019015

Article Contents

2019, Volume 24, Issue 3: 1259-1271. Doi: 10.3934/dcdsb.2019015

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Forward attracting sets of reaction-diffusion equations on variable domains

Peter E. Kloeden and
Meihua Yang^,

School of Mathematics and Statistics, and Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

¹Corresponding author
¹Corresponding author

Dedicated to the memory of V. S. Mel’nik.

Received: September 30, 2017

Revised: January 31, 2018

Early access: January 2019

Published: January 2019

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Abstract

Reaction-diffusion equations on time-variable domains are instrinsically nonautonomous even if the coefficients in the equation do not depend explicitly on time. Thus the appropriate asymptotic concepts, such as attractors, are nonautonomous. Forward attracting sets based on omega-limit sets are considered in this paper. These are related to the Vishik uniform attractor but are not as restrictive since they depend only on the dynamics in the distant future. They are usually not invariant. Here it is shown that they are asymptotically positively invariant, in general, and, if the future dynamics is appropriately uniform, also asymptotically negatively invariant as well as upper semi continuous dependence in a parameter will be established. These results also apply to reaction-diffusion equations on a fixed domain.

Keywords:

Nonautonomous dynamical system,
nonautonomous pullback attractor,
forward attracting set,
parabolic partial differential equation,
time-varying domains.

Mathematics Subject Classification: Primary: 37B55; Secondary: 37L30, 35K57.

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References

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