September  2013, 12(5): 2229-2266. doi: 10.3934/cpaa.2013.12.2229

Energy decay for Maxwell's equations with Ohm's law in partially cubic domains

1. 

Université d'Orléans, Laboratoire MAPMO, CNRS UMR 7349, Fédération Denis Poisson, FR CNRS 2964, Bâtiment de Mathématiques, B.P. 6759, 45067 Orléans Cedex 2, France

Received  January 2012 Revised  June 2012 Published  January 2013

We prove a polynomial energy decay for the Maxwell's equations with Ohm's law in partially cubic domains with trapped rays. We extend the results of polynomial decay for the scalar damped wave equation in partially rectangular or cubic domain. Our approach have some similitude with the construction of reflected gaussian beams.
Citation: Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229
References:
[1]

Math. Methods Appl. Sci., 21 (1998), 823-864. Google Scholar

[2]

SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055.  Google Scholar

[3]

Ph.D thesis, Université de Rennes 1, 2005. Google Scholar

[4]

Math. Res. Lett., 14 (2007), 35-47.  Google Scholar

[5]

World Scientific, Singapore, 1996.  Google Scholar

[6]

Masson, Paris, 1988.  Google Scholar

[7]

Dunod, Paris, 1972.  Google Scholar

[8]

J. Math. Anal. Appl., 329 (2007), 1375-1396. doi: 10.1016/j.jmaa.2006.06.101.  Google Scholar

[9]

Masson, Paris, 1988.  Google Scholar

[10]

Math. Res. Lett., 16 (2009), 881-894.  Google Scholar

[11]

ESAIM Control Optim. Calc. Var., 5 (2000), 87-137. doi: 10.1051/cocv:2000103.  Google Scholar

[12]

J. Diff. Eq., 240 (2007), 92-124. doi: 10.1016/j.jde.2007.05.016.  Google Scholar

[13]

in "Studies in Partial Differential Equations" (eds. W. Littman), MAA studies in Mathematics, 23 (1982), 206-248.  Google Scholar

[14]

Nonlinear Analysis, 75 (2012), 2024-2036. doi: 10.1016/j.na.2011.10.003.  Google Scholar

[15]

J. Math. Phys., 26 (1985), 861-863. doi: 10.1063/1.526579.  Google Scholar

show all references

References:
[1]

Math. Methods Appl. Sci., 21 (1998), 823-864. Google Scholar

[2]

SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055.  Google Scholar

[3]

Ph.D thesis, Université de Rennes 1, 2005. Google Scholar

[4]

Math. Res. Lett., 14 (2007), 35-47.  Google Scholar

[5]

World Scientific, Singapore, 1996.  Google Scholar

[6]

Masson, Paris, 1988.  Google Scholar

[7]

Dunod, Paris, 1972.  Google Scholar

[8]

J. Math. Anal. Appl., 329 (2007), 1375-1396. doi: 10.1016/j.jmaa.2006.06.101.  Google Scholar

[9]

Masson, Paris, 1988.  Google Scholar

[10]

Math. Res. Lett., 16 (2009), 881-894.  Google Scholar

[11]

ESAIM Control Optim. Calc. Var., 5 (2000), 87-137. doi: 10.1051/cocv:2000103.  Google Scholar

[12]

J. Diff. Eq., 240 (2007), 92-124. doi: 10.1016/j.jde.2007.05.016.  Google Scholar

[13]

in "Studies in Partial Differential Equations" (eds. W. Littman), MAA studies in Mathematics, 23 (1982), 206-248.  Google Scholar

[14]

Nonlinear Analysis, 75 (2012), 2024-2036. doi: 10.1016/j.na.2011.10.003.  Google Scholar

[15]

J. Math. Phys., 26 (1985), 861-863. doi: 10.1063/1.526579.  Google Scholar

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