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Applied Mathematics & Optimization

, Volume 78, Issue 1, pp 61–101 | Cite as

Energy Decay Rate of the Wave Equations on Riemannian Manifolds with Critical Potential

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Abstract

Decay of the energy for the Cauchy problem of the wave equation on Riemannian manifolds with a variable damping term \(V(x)u_{t}\) is considered, where \(V(x) \ge V_{0}(1 + \rho ^{2})^{-\frac{1}{2}}\)(\(\rho \) being a distance function under the Riemannian metric). Some relations among the decay rates of energy, the size of the coefficients \(V_{0}\), and the radial curvatures of the Riemannian metric are presented.

Keywords

Damped wave equation Critical potential Riemannian metric Distance function of a metric 

Notes

Acknowledgements

This work is supported by the National Science Foundation of China, Grants Nos. 61473126 and 61573342, and Key Research Program of Frontier Sciences, CAS, No. QYZDJ-SSW-SYS011.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China

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