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

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).