# American Institute of Mathematical Sciences

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March  2019, 24(3): 1259-1271. doi: 10.3934/dcdsb.2019015

## Forward attracting sets of reaction-diffusion equations on variable domains

 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

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

Received  October 2017 Revised  February 2018 Published  March 2019 Early access  January 2019

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.

Citation: Peter E. Kloeden, Meihua Yang. Forward attracting sets of reaction-diffusion equations on variable domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1259-1271. doi: 10.3934/dcdsb.2019015
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