We consider the semilinear wave equation posed in an inhomogeneous medium Ω with smooth boundary ∂Ω subject to a nonlinear damping distributed around a neighborhood ω of the boundary according to the geometric control condition. We show that the energy of the wave equation goes uniformly to zero for all initial data of finite energy phase-space.

We consider the semilinear wave equation posed in an inhomogeneous medium Ω with smooth boundary ∂Ω subject to a local viscoelastic damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase‐space. As far as we know, this is the first stabilization result for a semilinear wave equation with localized Kelvin–Voigt damping.

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We study the asymptotic behavior of a linear plate equation with effects of rotational inertia and a fractional damping in the memory term: utt−γΔutt+βnormalΔ2u−∫0∞gfalse(sfalse)normalΔ2θufalse(t−sfalse)ds=0,where θ≤1 and the kernel g is exponentially decreasing. The main result of this work is the polynomial decay of their solutions when θ<1. We prove that the solutions decay with the rate t−1/false(4−4θfalse) and also that the decay rate is optimal. Furthermore, when θ=1, we obtain the exponential decay of the solutions.