Sufficient conditions for decay estimates of the local energy and a behavior of the total energy of dissipative wave equations in exterior domains

Abstract

Decaying properties of the local energy for the dissipative wave equations with the Dirichlet boundary conditions in exterior domains are discussed. For the dissipation coefficient, natural conditions ensuring that waves trapped by obstacles may lose their energy are considered. Under this setting, two sufficient conditions for getting the decay estimates for the energy in bounded regions (i.e. the local energy) are given. These conditions bring some relaxation on classes of the dissipation coefficient which uniformly decaying estimates for the local energy hold. Further, decaying properties of the total energy are also discussed.

Uniform energy decay for wave equations with unbounded damping coefficients

“…Concerning the decay or non-decay property of the total or local energy to problem (1.1)-(1.2) with x-dependent variable damping coefficients, many research manuscripts are already published by Alloui-Ibrahim-Khenissi [1], Bouclet-Royer [2], Daoulatli [5], Ikehata [7], Ikehata-Todorova-Yordanov [10], Joly-Royer [11], Kawashita [13], Khader [12], Matsumura [16], Mochizuki [18], Mochizuki-Nakazawa [19], Nakao [21], Nishiyama [23], Nishihara [22], Radu-Todorova-Yordanov [24], Sobajima-Wakasugi [26], Todorova-Yordanov [28], Uesaka [29] and Wakasugi [30], Zhang [32] and the references therein. However, we should emphasize that those cases are quite restricted to the bounded damping coefficient case, i.e., a ∈ L ∞ (R n ).…”