We study a decay property of solutions for the wave equation with a localized dissipation and a boundary dissipation in an exterior domain Ω with the boundary ∂Ω, = Γ0 ∪ Γ1, Γ0 ∩ Γ1 = Ø. We impose the homogeneous Dirichlet condition on Γ0 and a dissipative Neumann condition on Γ1. Further, we assume that a localized dissipation a(x)ut is effective near infinity and in a neighborhood of a certain part of the boundary Γ0. Under these assumptions we derive an energy decay like E(t) ≤ C(1 + t)-1 and some related estimates.
ASJC Scopus subject areas
- 数学 (全般)