Figure 1.
Star-shaped viscoelastic network and the prescribed conditions
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June
2022, 15(6): 1421-1438.
doi: 10.3934/dcdss.2022073
Polynomial stability in viscoelastic network of strings
| 1. | UR Analysis and Control of PDEs, UR13ES64, ISCAE, University of Manouba, Tunisia |
| 2. | UR Analysis and Control of PDEs, UR13ES64, Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunisia |
In this paper we consider star-shaped viscoelastic networks, and study the large-time behaviour of these networks by proving polynomial decay rates. The energy decay rate depends on the irrationality properties of the lengths of the rods.
Keywords:
Network of strings,
local viscoelasticity,
frequency approach,
energy decay,
polynomial stability.
Mathematics Subject Classification:
35Q74, 35B30, 35B35.
Citation:
Karim El Mufti, Rania Yahia. Polynomial stability in viscoelastic network of strings. Discrete and Continuous Dynamical Systems - S,
2022, 15
(6)
: 1421-1438.
doi: 10.3934/dcdss.2022073
![]()
References:
| [1] |
M. Alves, J. Muñoz Rivera, M. Sepùlveda, O. Vera 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] |
K. Ammari, Z. Liu and F. Shel,
Stability of the wave equations on a tree with local Kelvin-Voigt damping, Semigroup Forum, 100 (2020), 364-382.
doi: 10.1007/s00233-019-10064-7. |
| [3] |
A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt,
Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.
doi: 10.1002/mana.200410429. |
| [4] |
C. J. K. Batty, R. Chill and Y. Tomilov,
Fine scales of decay of operator semigroups, J. Eur. Math. Soc. (JEMS), 18 (2016), 853-929.
doi: 10.4171/JEMS/605. |
| [5] |
C. J. K. Batty and T. Duyckaerts,
Non-uniform stability for bounded semigroups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.
doi: 10.1007/s00028-008-0424-1. |
| [6] |
A. Bchatnia, K. El Mufti and R. Yahia,
Stability of an infinite star-shaped network of strings by a Kelvin-Voigt damping, Math. Methods Appl. Sci., 45 (2022), 2024-2041.
doi: 10.1002/mma.7903. |
| [7] |
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. |
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H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
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R. Dàger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures, Mathématiques & Applications (Berlin), 50. Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-37726-3. |
| [10] |
Z.-J. Han and E. Zuazua,
Decay rates for elastic-thermoelastic star-shaped networks, Netw. Heterog. Media, 12 (2017), 461-488.
doi: 10.3934/nhm.2017020. |
| [11] |
F. Hassine,
Stability of a star-shaped network with local Kelvin-Voigt damping and non-smooth coefficient at interface, J. Differential Equations, 297 (2021), 1-24.
doi: 10.1016/j.jde.2021.06.017. |
| [12] |
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. |
| [13] |
K. Liu and Z. Liu,
Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control. Optim., 36 (1998), 1086-1098.
doi: 10.1137/S0363012996310703. |
| [14] |
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. |
| [15] |
Z. Liu and R. Quintanilla,
Energy decay rate of a mixed type Ⅱ and type Ⅲ thermolastic system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1433-1444.
doi: 10.3934/dcdsb.2010.14.1433. |
| [16] |
Z. Liu and B. Rao,
Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew.Math. Phys., 56 (2005), 630-644.
doi: 10.1007/s00033-004-3073-4. |
| [17] |
Z. Liu and Q. Zhang,
Stability of a string with local Kelvin-Voigt damping and nonsmooth coefficient at interface, SIAM J. Control Optim., 54 (2016), 1859-1871.
doi: 10.1137/15M1049385. |
| [18] |
Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics, 398. Chapman & Hall/CRC, Boca Raton, FL, 1999. |
| [19] |
B. Muckenhoupt,
Hardy's inequality with weights, Stud. Math., 44 (1972), 31-38.
doi: 10.4064/sm-44-1-31-38. |
| [20] |
M. Renardy,
On localized Kelvin-Voigt damping, Z. Angew. Math. Mech., 84 (2004), 280-283.
doi: 10.1002/zamm.200310100. |
| [21] |
W. M. Schmidt,
Simultaneous approximation to algebraic numbers by rationals, Acta Math., 125 (1970), 189-201.
doi: 10.1007/BF02392334. |
| [22] |
V. D. Stepanov,
Weighted Hardy inequality, Siberian Math. J, 28 (1987), 515-517.
|
| [23] |
Q. Zhang,
Exponential stability of an elastic string with local Kelvin-Voigt damping, Z. Angew. Math. Phys., 61 (2010), 1009-1015.
doi: 10.1007/s00033-010-0064-5. |
show all references
References:
| [1] |
M. Alves, J. Muñoz Rivera, M. Sepùlveda, O. Vera 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] |
K. Ammari, Z. Liu and F. Shel,
Stability of the wave equations on a tree with local Kelvin-Voigt damping, Semigroup Forum, 100 (2020), 364-382.
doi: 10.1007/s00233-019-10064-7. |
| [3] |
A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt,
Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.
doi: 10.1002/mana.200410429. |
| [4] |
C. J. K. Batty, R. Chill and Y. Tomilov,
Fine scales of decay of operator semigroups, J. Eur. Math. Soc. (JEMS), 18 (2016), 853-929.
doi: 10.4171/JEMS/605. |
| [5] |
C. J. K. Batty and T. Duyckaerts,
Non-uniform stability for bounded semigroups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.
doi: 10.1007/s00028-008-0424-1. |
| [6] |
A. Bchatnia, K. El Mufti and R. Yahia,
Stability of an infinite star-shaped network of strings by a Kelvin-Voigt damping, Math. Methods Appl. Sci., 45 (2022), 2024-2041.
doi: 10.1002/mma.7903. |
| [7] |
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. |
| [8] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
| [9] |
R. Dàger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures, Mathématiques & Applications (Berlin), 50. Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-37726-3. |
| [10] |
Z.-J. Han and E. Zuazua,
Decay rates for elastic-thermoelastic star-shaped networks, Netw. Heterog. Media, 12 (2017), 461-488.
doi: 10.3934/nhm.2017020. |
| [11] |
F. Hassine,
Stability of a star-shaped network with local Kelvin-Voigt damping and non-smooth coefficient at interface, J. Differential Equations, 297 (2021), 1-24.
doi: 10.1016/j.jde.2021.06.017. |
| [12] |
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. |
| [13] |
K. Liu and Z. Liu,
Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control. Optim., 36 (1998), 1086-1098.
doi: 10.1137/S0363012996310703. |
| [14] |
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. |
| [15] |
Z. Liu and R. Quintanilla,
Energy decay rate of a mixed type Ⅱ and type Ⅲ thermolastic system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1433-1444.
doi: 10.3934/dcdsb.2010.14.1433. |
| [16] |
Z. Liu and B. Rao,
Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew.Math. Phys., 56 (2005), 630-644.
doi: 10.1007/s00033-004-3073-4. |
| [17] |
Z. Liu and Q. Zhang,
Stability of a string with local Kelvin-Voigt damping and nonsmooth coefficient at interface, SIAM J. Control Optim., 54 (2016), 1859-1871.
doi: 10.1137/15M1049385. |
| [18] |
Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics, 398. Chapman & Hall/CRC, Boca Raton, FL, 1999. |
| [19] |
B. Muckenhoupt,
Hardy's inequality with weights, Stud. Math., 44 (1972), 31-38.
doi: 10.4064/sm-44-1-31-38. |
| [20] |
M. Renardy,
On localized Kelvin-Voigt damping, Z. Angew. Math. Mech., 84 (2004), 280-283.
doi: 10.1002/zamm.200310100. |
| [21] |
W. M. Schmidt,
Simultaneous approximation to algebraic numbers by rationals, Acta Math., 125 (1970), 189-201.
doi: 10.1007/BF02392334. |
| [22] |
V. D. Stepanov,
Weighted Hardy inequality, Siberian Math. J, 28 (1987), 515-517.
|
| [23] |
Q. Zhang,
Exponential stability of an elastic string with local Kelvin-Voigt damping, Z. Angew. Math. Phys., 61 (2010), 1009-1015.
doi: 10.1007/s00033-010-0064-5. |
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