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

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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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Minnesota, Duluth, MN, 55812-2496, USA

    Zhuangyi Liu

  2. School of Mathematics, Beijing Institute of Technology, Beijing, 100081, China

    Qiong Zhang

Authors
  1. Zhuangyi Liu
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  2. Qiong Zhang
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Correspondence to Zhuangyi Liu.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 60974033) and Beijing Municipal Natural Science Foundation of China (Grant No. 4132051).

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Liu, Z., Zhang, Q. A note on the polynomial stability of a weakly damped elastic abstract system. Z. Angew. Math. Phys. 66, 1799–1804 (2015). https://doi.org/10.1007/s00033-015-0517-y

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  • DOI: https://doi.org/10.1007/s00033-015-0517-y

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