A note on the polynomial stability of a weakly damped elastic abstract system

Abstract

In this paper, we analyze the following abstract system $$\left\{\begin{array}{ll} u_{tt} +Au+ Bu_t =0,\\ u(0) =u_0,\,\,u_t(0) = u_1,\end{array}\right.$$ where A is a self-adjoint, positive definite operator on a Hilbert space H, B (the dissipation operator) is another positive operator satisfying \({cA^{\alpha}u \leq Bu \leq CA^{\alpha}u}\) for some constants 0 < c < C. The case of \({0 \leq \alpha \leq 1}\) has been well investigated in the literature. Our contribution is to prove that the associated semigroup is polynomially stable when \({\alpha < 0}\). Moreover, we obtain the optimal order of polynomial stability.