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 u tt − κ 0 ∆u + t 0 div[a(x)g(t − s)∇u(s)] ds + f (u) + b(x)h(u t) = 0 in Ω × R + , (0.1) where a, b are nonnegative functions, a ∈ C 1 (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.

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Abstract.In this paper we study a transmission problem for thermoelastic plates. We prove that the problem is well-posed in the sense that there exists only one solution which is as regular as the initial data. Moreover, we prove that the local thermal effect is strong enough to produce uniform rate of decay of the solution. More precisely, there exist positive constants C and 7 such that the total energy E(t) satisfies E(t) < CE{0)e"7t.

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In this paper, we consider a coupled system of two wave equations. One of these equations is conservative, and the other has damping and delay terms. If the damping acts with more force than the delay term, we show polynomial stability for strong solutions of the system. Explicit decay rates are found, and the optimality of those rates is discussed. On the other hand, if the damping acts with the same or less force than the delay term, then we obtain a result of instability by constructing a sequence of time delays and initial data such that the solutions are not asymptotically stable.