# American Institute of Mathematical Sciences

June  2017, 37(6): 3487-3502. doi: 10.3934/dcds.2017148

## Global exponential κ-dissipative semigroups and exponential attraction

 1 Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China 2 School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan 430074, China 3 Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Jin Zhang

Received  July 2015 Revised  January 2017 Published  February 2017

Globally exponential $κ-$dissipativity, a new concept of dissipativity for semigroups, is introduced. It provides a more general criterion for the exponential attraction of some evolutionary systems. Assuming that a semigroup $\{S(t)\}_{t≥q 0}$ has a bounded absorbing set, then $\{S(t)\}_{t≥q 0}$ is globally exponentially $κ-$dissipative if and only if there exists a compact set $\mathcal{A}^*$ that is positive invariant and attracts any bounded subset exponentially. The set $\mathcal{A}^*$ need not be finite dimensional. This result is illustrated with an application to a damped semilinear wave equation on a bounded domain.

Citation: Jin Zhang, Peter E. Kloeden, Meihua Yang, Chengkui Zhong. Global exponential κ-dissipative semigroups and exponential attraction. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3487-3502. doi: 10.3934/dcds.2017148
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