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June  2022, 15(6): 1455-1467. doi: 10.3934/dcdss.2022031

Sharper and finer energy decay rate for an elastic string with localized Kelvin-Voigt damping

Zhong-Jie Han 1, , Zhuangyi Liu 2,, and  Jing Wang 3,

1. 

School of Mathematics, Tianjin University, Tianjin 300354, China
2. 

Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812-3000, USA
3. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 102488, China

*Corresponding author: Zhuangyi Liu

Received  March 2021 Revised  November 2021 Published  June 2022 Early access  February 2022

Fund Project: The first author is supported by the Natural Science Foundation of China grant NSFC-62073236

This paper is on the asymptotic behavior of the elastic string equation with localized Kelvin-Voigt damping
$ u_{tt}(x, t)-[u_{x}(x, t)+b(x)u_{x, t}(x, t)]_{x} = 0, \; x\in(-1, 1), \; t>0, $
where
$ b(x) = 0 $
 on
$ x\in (-1, 0] $
, and
$ b(x) = a(x)>0 $
 on
$ x\in (0, 1) $
. It is known that the Geometric Optics Condition for exponential stability does not apply to Kelvin-Voigt damping. Under the assumption that
$ a'(x) $
 has a singularity at
$ x = 0 $
, we investigate the decay rate of the solution which depends on the order of the singularity.
When
$ a(x) $
 behaves like
$ x^{\alpha}(-\log x)^{-\beta} $
 near
$ x = 0 $
 for
$ 0\le{\alpha}<1, \;0\le\beta $
 or
$ 0<{\alpha}<1, \;\beta<0 $
, we show that the system can achieve a mixed polynomial-logarithmic decay rate.
As a byproduct, when
$ \beta = 0 $
, we obtain the decay rate
$ t^{-\frac{ 3-\alpha-\varepsilon}{2(1-{\alpha})}} $
 of solution for arbitrarily small
$ \varepsilon>0 $
, which improves the rate
$ t^{-\frac{1}{1-{\alpha}}} $
 obtained in [14]. The new rate is again consistent with the exponential decay rate in the limit case
$ \alpha\to 1^- $
. This is a step toward the goal of obtaining the optimal decay rate eventually.
Keywords: Polynomial-logarithmic decay, frequency domain analysis, localized Kelvin-Voigt damping, wave equation, $ C_0 $ semigroup.
Mathematics Subject Classification: Primary: 35B35, 35B40; Secondary: 93D20.
Citation: Zhong-Jie Han, Zhuangyi Liu, Jing Wang. Sharper and finer energy decay rate for an elastic string with localized Kelvin-Voigt damping. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1455-1467. doi: 10.3934/dcdss.2022031
References:
[1]

M. Alves, J. M. Rivera, M. Sepúlveda, O. V. Villagrán and M. Z. Garay, The asymptotic behavior of the linear transmission problem in viscoelasticity, Math. Nachr., 287 (2014), 483-497.  doi: 10.1002/mana.201200319.

[2]

C. J. K. Batty, R. Chill and Y. Tomilov, Fine scales of decay of operator semigroups, J. Eur. Math. Soc., 18 (2016), 853-929.  doi: 10.4171/JEMS/605.

[3]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1.

[4]

E. N. Batuev and V. D. Stepanov, Weighted inequalities of Hardy type, Siberian Math. J., 30 (1989), 8-16.  doi: 10.1007/BF01054210.

[5]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.

[6]

S. Chen, K. Liu and Z. Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping, SIAM J. Appl. Math., 59 (1999), 651-668.  doi: 10.1137/S0036139996292015.

[7]

B. Z. Guo, J. M. Wang and G. D. Zhang, Spectral analysis of a wave equation with Kelvin-Voigt damping, Z. Angew. Math. Mech., 90 (2010), 323-342.  doi: 10.1002/zamm.200900275.

[8]

J. Hao and Z. Liu, Stability of an abstract system of coupled hyperbolic and parabolic equations, Z. Angew. Math. Phys., 64 (2013), 1145-1159.  doi: 10.1007/s00033-012-0274-0.

[9]

F. Huang, On the mathematical model for linear elastic systems with analytic damping, SIAM J. Control Optim., 26 (1988), 714-724.  doi: 10.1137/0326041.

[10]

K. Liu and Z. Liu, Exponential decay of the energy of the Euler Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098.  doi: 10.1137/S0363012996310703.

[11]

K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phys., 53 (2002), 265-280.  doi: 10.1007/s00033-002-8155-6.

[12]

K. Liu, Z. Liu and Q. Zhang, Eventual differentiability of a string with local Kelvin-Voigt damping, ESAIM Control Optim. Calc. Var., 23 (2017), 443-454.  doi: 10.1051/cocv/2015055.

[13]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equations, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4.

[14]

Z. Liu and Q. Zhang, Stability of a string with Local Kelvin–Voigt damping and nonsmooth coefficient at interface, SIAM J. Control and Optim., 54 (2016), 1859-1871.  doi: 10.1137/15M1049385.

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[16]

M. Renardy, On localized Kelvin-Voigt damping, Z. Angew. Math. Mech., 84 (2004), 280-283.  doi: 10.1002/zamm.200310100.

[17]

J. Rozendaal, D. Seifert and R. Stahn, Optimal rates of decay for operators semigroups on Hilbert spaces, Adv. Math., 346 (2019), 359-388.  doi: 10.1016/j.aim.2019.02.007.

[18]

G. Q. Xu and N. E. Mastorakis, Spectrum of an operator arising elastic system with local K-V damping, Z. Angew. Math. Mech., 88 (2008), 483-496.  doi: 10.1002/zamm.200700109.

show all references

References:
[1]

M. Alves, J. M. Rivera, M. Sepúlveda, O. V. Villagrán and M. Z. Garay, The asymptotic behavior of the linear transmission problem in viscoelasticity, Math. Nachr., 287 (2014), 483-497.  doi: 10.1002/mana.201200319.

[2]

C. J. K. Batty, R. Chill and Y. Tomilov, Fine scales of decay of operator semigroups, J. Eur. Math. Soc., 18 (2016), 853-929.  doi: 10.4171/JEMS/605.

[3]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1.

[4]

E. N. Batuev and V. D. Stepanov, Weighted inequalities of Hardy type, Siberian Math. J., 30 (1989), 8-16.  doi: 10.1007/BF01054210.

[5]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.

[6]

S. Chen, K. Liu and Z. Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping, SIAM J. Appl. Math., 59 (1999), 651-668.  doi: 10.1137/S0036139996292015.

[7]

B. Z. Guo, J. M. Wang and G. D. Zhang, Spectral analysis of a wave equation with Kelvin-Voigt damping, Z. Angew. Math. Mech., 90 (2010), 323-342.  doi: 10.1002/zamm.200900275.

[8]

J. Hao and Z. Liu, Stability of an abstract system of coupled hyperbolic and parabolic equations, Z. Angew. Math. Phys., 64 (2013), 1145-1159.  doi: 10.1007/s00033-012-0274-0.

[9]

F. Huang, On the mathematical model for linear elastic systems with analytic damping, SIAM J. Control Optim., 26 (1988), 714-724.  doi: 10.1137/0326041.

[10]

K. Liu and Z. Liu, Exponential decay of the energy of the Euler Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098.  doi: 10.1137/S0363012996310703.

[11]

K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phys., 53 (2002), 265-280.  doi: 10.1007/s00033-002-8155-6.

[12]

K. Liu, Z. Liu and Q. Zhang, Eventual differentiability of a string with local Kelvin-Voigt damping, ESAIM Control Optim. Calc. Var., 23 (2017), 443-454.  doi: 10.1051/cocv/2015055.

[13]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equations, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4.

[14]

Z. Liu and Q. Zhang, Stability of a string with Local Kelvin–Voigt damping and nonsmooth coefficient at interface, SIAM J. Control and Optim., 54 (2016), 1859-1871.  doi: 10.1137/15M1049385.

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[16]

M. Renardy, On localized Kelvin-Voigt damping, Z. Angew. Math. Mech., 84 (2004), 280-283.  doi: 10.1002/zamm.200310100.

[17]

J. Rozendaal, D. Seifert and R. Stahn, Optimal rates of decay for operators semigroups on Hilbert spaces, Adv. Math., 346 (2019), 359-388.  doi: 10.1016/j.aim.2019.02.007.

[18]

G. Q. Xu and N. E. Mastorakis, Spectrum of an operator arising elastic system with local K-V damping, Z. Angew. Math. Mech., 88 (2008), 483-496.  doi: 10.1002/zamm.200700109.

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