Research article

Remarks on an elliptic problem arising in weighted energy estimates for wave equations with space-dependent damping term in an exterior domain

  • Received: 17 August 2016 Accepted: 01 November 2016 Published: 22 November 2016
  • This paper is concerned with weighted energy estimates and diffusion phenomena for the initial-boundary problem of the wave equation with space-dependent damping term in an exterior domain. In this analysis, an elliptic problem was introduced by Todorova and Yordanov. This attempt was quite useful when the coefficient of the damping term is radially symmetric. In this paper, by modifying their elliptic problem, we establish weighted energy estimates and diffusion phenomena even when the coefficient of the damping term is not radially symmetric.

    Citation: Sobajima Motohiro, Wakasugi Yuta. Remarks on an elliptic problem arising in weighted energy estimates for wave equations with space-dependent damping term in an exterior domain[J]. AIMS Mathematics, 2017, 2(1): 1-15. doi: 10.3934/Math.2017.1.1

    Related Papers:

  • This paper is concerned with weighted energy estimates and diffusion phenomena for the initial-boundary problem of the wave equation with space-dependent damping term in an exterior domain. In this analysis, an elliptic problem was introduced by Todorova and Yordanov. This attempt was quite useful when the coefficient of the damping term is radially symmetric. In this paper, by modifying their elliptic problem, we establish weighted energy estimates and diffusion phenomena even when the coefficient of the damping term is not radially symmetric.


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    [9] P. Radu, G. Todorova, and B. Yordanov, Decay estimates for wave equations with variable coefficients, Trans. Amer. Math. Soc., 362 (2010), 2279-2299.
    [10] M. Sobajima and Y. Wakasugi, Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain, J. Differential Equations, 261 (2016), 5690-5718.
    [11] G. Todorova, and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489.
    [12] G. Todorova, and B. Yordanov, Weighted L2-estimates for dissipative wave equations with variable coefficients, J. Differential Equations, 246 (2009), 4497-4518.
    [13] Y.Wakasugi, On diffusion phenomena for the linear wave equation with space-dependent damping, J. Hyp. Diff. Eq., 11 (2014), 795-819.
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  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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