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

Zhuangyi Liu; Qiong Zhang

doi:10.1007/s00033-015-0517-y

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

Zhuangyi Liu^*, Qiong Zhang

^*Corresponding author for this work

School of Mathematics and Statistics

University of Minnesota Duluth

Research output: Contribution to journal › Article › peer-review

24 Citations (Scopus)

Abstract

In this paper, we analyze the following abstract system (Formula Presented.) where A is a self-adjoint, positive definite operator on a Hilbert space H, B (the dissipation operator) is another positive operator satisfying c^Aαu≤Bu≤C^Aαu for some constants 0 < c < C. The case of 0≤α≤1 has been well investigated in the literature. Our contribution is to prove that the associated semigroup is polynomially stable when α<0. Moreover, we obtain the optimal order of polynomial stability.

Original language	English
Pages (from-to)	1799-1804
Number of pages	6
Journal	Zeitschrift fur Angewandte Mathematik und Physik
Volume	66
Issue number	4
DOIs	https://doi.org/10.1007/s00033-015-0517-y
Publication status	Published - 10 Aug 2015

Keywords

35B40
47D03
93D05

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10.1007/s00033-015-0517-y

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abstract = "In this paper, we analyze the following abstract system (Formula Presented.) where A is a self-adjoint, positive definite operator on a Hilbert space H, B (the dissipation operator) is another positive operator satisfying cAαu≤Bu≤CAαu for some constants 0 < c < C. The case of 0≤α≤1 has been well investigated in the literature. Our contribution is to prove that the associated semigroup is polynomially stable when α<0. Moreover, we obtain the optimal order of polynomial stability.",

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AU - Zhang, Qiong

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N2 - In this paper, we analyze the following abstract system (Formula Presented.) where A is a self-adjoint, positive definite operator on a Hilbert space H, B (the dissipation operator) is another positive operator satisfying cAαu≤Bu≤CAαu for some constants 0 < c < C. The case of 0≤α≤1 has been well investigated in the literature. Our contribution is to prove that the associated semigroup is polynomially stable when α<0. Moreover, we obtain the optimal order of polynomial stability.

AB - In this paper, we analyze the following abstract system (Formula Presented.) where A is a self-adjoint, positive definite operator on a Hilbert space H, B (the dissipation operator) is another positive operator satisfying cAαu≤Bu≤CAαu for some constants 0 < c < C. The case of 0≤α≤1 has been well investigated in the literature. Our contribution is to prove that the associated semigroup is polynomially stable when α<0. Moreover, we obtain the optimal order of polynomial stability.

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KW - 47D03

KW - 93D05

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JF - Zeitschrift fur Angewandte Mathematik und Physik

IS - 4

ER -

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

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Keywords

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