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

We first consider the wave equation in an exterior domain ΩΩ in RNRN with two separated boundary parts Γ0Γ0, Γ1Γ1. On Γ0Γ0, the Dirichlet condition u|Γ0=0u|Γ0=0 is imposed, while on Γ1Γ1, Neumann type nonlinear boundary dissipation ∂u/∂ν=−g(ut)∂u/∂ν=−g(ut) is assumed. Further, a ‘half-linear’ localized dissipation is attached on ΩΩ. For such a situation we derive a precise rate of decay of the energy E(t)E(t) for solutions of the initial boundary value problem. We impose no geometrical condition on the shape of the boundary ∂Ω=Γ0∪Γ1∂Ω=Γ0∪Γ1. Secondly, when a TT periodic forcing term works we prove the existence of a TT periodic solution on RR under an additional growth assumption on ρ(x,v)ρ(x,v) and g(v)g(v).