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AIMS Mathematics

2017, Volume 2, Issue 1: 1-15. doi: 10.3934/Math.2017.1.1
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Research article

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

  • 1.
    Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba-ken 278-8510, Japan
  • 2.
    Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602 Japan
  • 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.


    加载中
    [1] M. Ikawa, Mixed problems for hyperbolic equations of second order, J. Math. Soc. Japan, 20 (1968), 580-608.
    [2] M. Ikawa, M. Ikawa, Hyperbolic partial differential equations and wave phenomena, American Mathematical Society, Providence, RI, 2000.
    [3] R. Ikehata, Some remarks on the wave equation with potential type damping coefficients, Int. J. Pure Appl. Math., 21 (2005), 19-24.
    [4] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976), 169-189.
    [5] A. Matsumura, Energy decay of solutions of dissipative wave equations, Proc. Japan Acad., Ser. A, 53 (1977), 232-236.
    [6] K. Mochizuki, Scattering theory for wave equations with dissipative terms, Publ. Res. Inst. Math. Sci., 12 (1976), 383-390.
    [7] K. Nishihara, Lp-Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z., 244 (2003), 631-649.
    [8] P. Radu, G. Todorova, and B. Yordanov, Higher order energy decay rates for damped wave equations with variable coefficients, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 609-629.
    [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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