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Longtime behavior of the hyperbolic equations with an arbitrary internal damping

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Abstract

This paper is devoted to a study of the longtime behavior of the hyperbolic equations with an arbitrary internal damping, under sharp regularity assumptions that both the principal part coefficients and the boundary of the space domain (in which the system evolves) are continuously differentiable. For this purpose, we derive a new point-wise inequality for second differential operators with symmetric coefficients. Then, based on a global Carleman estimate, we establish an estimate on the underlying resolvent operator of the equation, via which, we show the logarithmic decay rate for solutions of the hyperbolic equations.

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Authors and Affiliations

  1. School of Mathematics, Sichuan University, Chengdu, 610064, China

    Xiaoyu Fu

  2. Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, 76, India

    Xiaoyu Fu

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  1. Xiaoyu Fu
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Correspondence to Xiaoyu Fu.

Additional information

This work is partially supported by the NSF of China under grant 10901114, the Doctoral Fund for New Teachers of Ministry of Education of China under grant 20090181120084, and the National Basic Research Program of China (973 program) under grant 2011CB808002. The author gratefully acknowledges Professor Xu Zhang for his help.

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Fu, X. Longtime behavior of the hyperbolic equations with an arbitrary internal damping. Z. Angew. Math. Phys. 62, 667–680 (2011). https://doi.org/10.1007/s00033-010-0113-0

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  • DOI: https://doi.org/10.1007/s00033-010-0113-0

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