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Sharper and finer energy decay rate for an elastic string with localized Kelvin-Voigt damping
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 |
$ u_{tt}(x, t)-[u_{x}(x, t)+b(x)u_{x, t}(x, t)]_{x} = 0, \; x\in(-1, 1), \; t>0, $ |
$ b(x) = 0 $ |
$ x\in (-1, 0] $ |
$ b(x) = a(x)>0 $ |
$ x\in (0, 1) $ |
$ a'(x) $ |
$ x = 0 $ |
$ a(x) $ |
$ x^{\alpha}(-\log x)^{-\beta} $ |
$ x = 0 $ |
$ 0\le{\alpha}<1, \;0\le\beta $ |
$ 0<{\alpha}<1, \;\beta<0 $ |
$ \beta = 0 $ |
$ t^{-\frac{ 3-\alpha-\varepsilon}{2(1-{\alpha})}} $ |
$ \varepsilon>0 $ |
$ t^{-\frac{1}{1-{\alpha}}} $ |
$ \alpha\to 1^- $ |
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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