Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms— an intrinsic approach

Let Ω be an open bounded connected domain in Rn, with local Lipschitz boundary Γ. Define QT ≡ [0, T ]×Ω, ΣT ≡ [0, T ]× Γ, and let ‖ · ‖ stand for L2(Ω) norm.

Consider the following model of the wave equation with localized damping χg(ut) and source term f(u) :{

utt −∆u+ χg(ut) = f(u) in QT u(0) = u0, ut(0) = u1

(1.1)

The functions g (resp. f) represent Nemytski operators associated with scalar, continuous, real-valued functions g(s) (resp. f(s)). Map g, assumed monotone increasing, models dissipation. Instead, function f corresponds to a source. The dissipation acts on small subportion Ωχ of Ω, and χ is the characteristic function of this subset. We postpone the description of Ωχ, for now it suffices to say that Ωχ covers a thin layer (a collar) near a portion of the boundary.