Energy decay and periodic solution for the wave equation in an exterior domain with half-linear and nonlinear boundary dissipations

Abstract

We first consider the wave equation in an exterior domain Ω in R N with two separated boundary parts Γ 0 , Γ 1 . On Γ 0 , the Dirichlet condition u | Γ 0 = 0 is imposed, while on Γ 1 , Neumann type nonlinear boundary dissipation ∂ u / ∂ ν = − g ( u t ) is assumed. Further, a ‘half-linear’ localized dissipation is attached on Ω . For such a situation we derive a precise rate of decay of the energy E ( t ) for solutions of the initial boundary value problem. We impose no geometrical condition on the shape of the boundary ∂ Ω = Γ 0 ∪ Γ 1 . Secondly, when a T periodic forcing term works we prove the existence of a T periodic solution on R under an additional growth assumption on ρ ( x , v ) and g ( v ) .