Some results on the asymptotic behavior for hyperbolic problems in cylindrical domains becoming unbounded

We study here the asymptotic behavior of the solution of a hyperbolic problem defined on a cylindrical domain [0,T]×p(−ℓ,ℓ)×ω⊂[0,T]×Rn when ℓ→∞. We show that, under very general assumptions, the solution of this problem converges faster than any power of towards the solution of another hyperbolic problem, defined on [0,T]×ω, in any bounded subdomain. We give both necessary and sufficient conditions for this convergence to occur.