ABSTRACT We consider a transmission problem for a string composed by two components: one of them is a viscoelastic material (with viscoelasticity of memory type), and the other is an elastic material (without dissipation effective over this component). Additionally, we consider that in one end is attached a tip load. The main result is that the model is exponentially stable if and only if the memory effect is effective over the string. When there is no memory effect, then there is a lack of exponential stability, but the tip load produces a polynomial rate of decay. That is, the tip load is not strong enough to stabilize exponentially the system, but produces a polynomial rate of decay.

RESUMO Neste trabalho estudamos o comportamento assintótico das placas de Kirchhoff com amortecimento intermediário. O amortecimento considerado contempla as dissipações de tipo friccional e de Kelvin-Voigt. Mostramos que o semigrupo dessas equações decai polinomialmente no tempo com a taxa t −1 /(2−2θ ), onde θ é um parâmetro no intervalo [0, 1[. Além disso, provamos que a taxa de decaimento encontrada é ótima.

ABSTRACT In this paper we study the asymptotic behavior of Kirchhoff plates with intermediate damping. The damping considered contemplates the frictional and the Kelvin-Voigt type dampings. We show that the semigroup those equations decays polynomially in time at least with the rate t −1 /(2−2θ ), where θ is a parameter in the interval [0, 1[. Moreover, we prove that this decay rate is optimal.