We derive an energy decay estimate for solutions to the initial-boundary value problem of a semilinear wave equation in exterior domains with a nonlinear localized dissipation. Our equation includes an absorbing term like |u|αu, α ≥ 0, and can be regarded as a generalized Klein-Gordon equation at least if α is closed to 0. This observation plays an essential role in our argument.
Mitsuhiro NAKAO. "Energy decay for a nonlinear generalized Klein-Gordon equation in exterior domains with a nonlinear localized dissipative term." J. Math. Soc. Japan 64 (3) 851 - 883, July, 2012. https://doi.org/10.2969/jmsj/06430851