Exponential decay of the energy of a nonlinear system of Klein–Gordon equations with localized dampings in bounded and unbounded domains

Abstract

We study the exponential decay of the energy of the nonlinear system of Klein–Gordon equations u_tt−Δu+m₁u+f(u,v)+a(x)u_t=0, v_tt−Δv+m₂v+g(u,v)+b(x)v_t=0,(x,t)∈O×(0,∞),u=v=0 on Γ₀×(0,∞), where O is abounded or unbounded domain in R^N with smooth boundary ro Γ₀:=∂O;a,b∈L^∞₊O,a,b≥constant>0, a.e. in some appropriated open subset of O, and f, g satisfy some suitable conditions. The exponential decay of the energy is established by adapting to the system multiplier techniques of J.L. Lions, some techniques developed by E. Zuazua for a single equation and a unique continuation principle of A. Ruiz.