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Zeitschrift für angewandte Mathematik und Physik

, Volume 66, Issue 4, pp 1799–1804 | Cite as

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

  • Zhuangyi LiuEmail author
  • Qiong Zhang
Article

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 <  cC. 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.

Mathematics Subject Classification

35B40 47D03 93D05 

Keywords

Dissipation Polynomial stability Semigroup 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of MinnesotaDuluthUSA
  2. 2.School of MathematicsBeijing Institute of TechnologyBeijingChina

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