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

TL;DRAbstract

We study the exponential decay of the energy of the nonlinear system of Klein–Gordon equations u tt −Δu+m 1 u+f(u,v)+a(x)u t =0, v tt −Δv+m 2 v+g(u,v)+b(x)v t =0,(x,t)∈O×(0,∞),u=v=0 on Γ 0 ×(0,∞), where O is abounded or unbounded domain in R N with smooth boundary ro Γ 0 :=∂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.

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