Frictional versus Viscoelastic Damping in a Semilinear Wave Equation

In this article we show exponential and polynomial decay rates for the partially viscoelastic nonlinear wave equation subject to a nonlinear and localized frictional damping. The equation that model this problem is given by utt − κ0∆u + ∫ t 0 div[a(x)g(t− s)∇u(s)] ds + f(u) + b(x)h(ut) = 0 in Ω× R, (0.1) where a, b are nonnegative functions, a ∈ C(Ω), b ∈ L∞(Ω), satisfying the assumption a(x) + b(x) ≥ δ > 0 ∀x ∈ Ω, (0.2) and f and h are power like functions. We observe that the assumption (0.2) gives us a wide assortment of possibilities to choose the functions a(x) and b(x) and the most interesting case occurs when one has simultaneous and complementary damping mechanisms. Taking this point of view into account, a distinctive feature of our paper is exactly to consider different and localized damping mechanisms acting in the domain but not necessarily ‘strategically localized dissipations’ as considered in the prior literature.