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Applied Mathematics & Optimization

, Volume 79, Issue 1, pp 131–144 | Cite as

Asymptotic Behavior for a Class of Logarithmic Wave Equations with Linear Damping

  • Qingying Hu
  • Hongwei ZhangEmail author
  • Gongwei Liu
Article

Abstract

We consider the initial boundary value problem for a class of logarithmic wave equations with linear damping. By constructing a potential well and using the logarithmic Sobolev inequality, we prove that, if the solution lies in the unstable set or the initial energy is negative, the solution will grow as an exponential function in the \(H^1_0(\Omega )\) norm as time goes to infinity. If the solution lies in a smaller set compared with the stable set, we can estimate the decay rate of the energy. These results are extensions of earlier results.

Keywords

Logarithmic wave equation Initial boundary value problem Exponential growth Energy decay 

Mathematics Subject Classification

35L20 35L70 35B40 35Q40 

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (nos. 11520677, 11601122) and the Basic Research Foundation of Henan University of Technology (171164).

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsHenan University of TechnologyZhengzhouPeople’s Republic of China

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