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In odd dimension space, under a microlocal geometric condition, we give the rate of decay of the local energy for solutions of the wave equation on exterior domain, with localized nonlinear damping.
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Moez Daoulatli, Local energy decay for the nonlinear dissipative wave equation in an exterior domain. Port. Math. 64 (2007), no. 1, pp. 39–65DOI 10.4171/PM/1775