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): (equation found) (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.

Publication Title

Free and Moving Boundaries: Analysis, Simulation and Control

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