Global exponential κ-dissipative semigroups and exponential attraction

Jin Zhang; Peter E. Kloeden; Meihua Yang; Chengkui Zhong

doi:10.3934/dcds.2017148

Article Contents

2017, Volume 37, Issue 6: 3487-3502. Doi: 10.3934/dcds.2017148

This issue Previous Article Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations Next Article The Cauchy problem for a generalized Novikov equation

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
* Corresponding author: Jin Zhang

Received: June 30, 2015

Revised: December 31, 2016

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B41; Secondary: 35K57, 37B55.

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