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Decay property for a plate equation with memory-type dissipation
1. | Department of Mathematics, North China Electric Power University, Beijing 102208, China |
2. | Faculty of Mathematics, Kyushu University, Fukuoka 819-0395 |
References:
[1] |
M. E. Bradley and S. Lenhart, Bilinear spatial control of the velocity term in a Kirchhoff plate equation, Electronic J. Differential Equations, 2001 (2001), 1-15. |
[2] |
C. Buriol, Energy decay rates for the Timoshenko system of thermoelastic plates, Nonlinear Analysis, 64 (2006), 92-108.
doi: 10.1016/j.na.2005.06.010. |
[3] |
R. C. Charão, E. Bisognin, V. Bisognin and A. F. Pazoto, Asymptotic behavior for a dissipative plate equation in $R^N$ with periodic coefficients, Electronic J. Differential Equations, 46 (2008), 23 pp. |
[4] |
C. R. da Luz and R. C. Charão, Asymptotic properties for a semi-linear plate equation in unbounded domains, J. Hyperbolic Differential Equations, 6 (2009), 269-294.
doi: 10.1142/S0219891609001824. |
[5] |
P. M. N. Dharmawardane, J. E. Muñoz Rivera and S. Kawashima, Decay property for second order hyperbolic systems of viscoelastic materials, J. Math. Anal. Appl., 366 (2010), 621-635.
doi: 10.1016/j.jmaa.2009.12.019. |
[6] |
A. D. Drozdov and V. B. Kolmanovskii, "Stability in Viscoelasticity,", North-Holland Publishing Co., Amsterdam, 1994., 38 ().
|
[7] |
M. Fabrizio and B. Lazzari, On the existence and the asymptotic stability of solutions for linear viscoelastic solids, Arch. Rational Mech. Anal., 116 (1991), 139-152.
doi: 10.1007/BF00375589. |
[8] |
T. Hosono and S. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Meth. Appl. Sci., 16 (2006), 1839-1859.
doi: 10.1142/S021820250600173X. |
[9] |
K. Ide and S. Kawashima, Decay property of regularity-loss type and nonlinear effects for dissipative Timoshenko system, Math. Models Meth. Appl. Sci., 18 (2008), 1001-1025.
doi: 10.1142/S0218202508002930. |
[10] |
Y. Liu and S. Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation, Discrete Contin. Dyn. Syst., 29 (2011), 1113-1139. |
[11] |
Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermo-viscoelasticity, Quart. Appl. Math., 54 (1996), 21-31. |
[12] |
Z. Liu and S. Zheng, "Semi-Groups Associated with Dissipative Systems," Chapman $&$ Hall/CRC, London, 1999. |
[13] |
J. E. Muñoz Rivera, Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648. |
[14] |
J. E. Muñoz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl., 286 (2003), 692-704.
doi: 10.1016/S0022-247X(03)00511-0. |
[15] |
J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst., 9 (2003), 1625-1639.
doi: 10.3934/dcds.2003.9.1625. |
[16] |
A. F. Pazoto, J. C. Vila Bravo and J. E. Muñoz Rivera, Asymptotic stability of semi-groups associated to linear weak dissipative systems, Math. Computer Modeling, 40 (2004), 387-392.
doi: 10.1016/j.mcm.2003.10.048. |
[17] |
Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation, J. Hyperbolic Differential Equations, 7 (2010), 471-501.
doi: 10.1142/S0219891610002207. |
[18] |
X. Zhang and E. Zuazua, On the optimality of the observability inequalities for Kirchhoff plate systems with potentials in unbounded domains,, Springer, 2008, 233--243., ().
doi: 10.1007/978-3-540-75712-2_19. |
show all references
References:
[1] |
M. E. Bradley and S. Lenhart, Bilinear spatial control of the velocity term in a Kirchhoff plate equation, Electronic J. Differential Equations, 2001 (2001), 1-15. |
[2] |
C. Buriol, Energy decay rates for the Timoshenko system of thermoelastic plates, Nonlinear Analysis, 64 (2006), 92-108.
doi: 10.1016/j.na.2005.06.010. |
[3] |
R. C. Charão, E. Bisognin, V. Bisognin and A. F. Pazoto, Asymptotic behavior for a dissipative plate equation in $R^N$ with periodic coefficients, Electronic J. Differential Equations, 46 (2008), 23 pp. |
[4] |
C. R. da Luz and R. C. Charão, Asymptotic properties for a semi-linear plate equation in unbounded domains, J. Hyperbolic Differential Equations, 6 (2009), 269-294.
doi: 10.1142/S0219891609001824. |
[5] |
P. M. N. Dharmawardane, J. E. Muñoz Rivera and S. Kawashima, Decay property for second order hyperbolic systems of viscoelastic materials, J. Math. Anal. Appl., 366 (2010), 621-635.
doi: 10.1016/j.jmaa.2009.12.019. |
[6] |
A. D. Drozdov and V. B. Kolmanovskii, "Stability in Viscoelasticity,", North-Holland Publishing Co., Amsterdam, 1994., 38 ().
|
[7] |
M. Fabrizio and B. Lazzari, On the existence and the asymptotic stability of solutions for linear viscoelastic solids, Arch. Rational Mech. Anal., 116 (1991), 139-152.
doi: 10.1007/BF00375589. |
[8] |
T. Hosono and S. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Meth. Appl. Sci., 16 (2006), 1839-1859.
doi: 10.1142/S021820250600173X. |
[9] |
K. Ide and S. Kawashima, Decay property of regularity-loss type and nonlinear effects for dissipative Timoshenko system, Math. Models Meth. Appl. Sci., 18 (2008), 1001-1025.
doi: 10.1142/S0218202508002930. |
[10] |
Y. Liu and S. Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation, Discrete Contin. Dyn. Syst., 29 (2011), 1113-1139. |
[11] |
Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermo-viscoelasticity, Quart. Appl. Math., 54 (1996), 21-31. |
[12] |
Z. Liu and S. Zheng, "Semi-Groups Associated with Dissipative Systems," Chapman $&$ Hall/CRC, London, 1999. |
[13] |
J. E. Muñoz Rivera, Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648. |
[14] |
J. E. Muñoz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl., 286 (2003), 692-704.
doi: 10.1016/S0022-247X(03)00511-0. |
[15] |
J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst., 9 (2003), 1625-1639.
doi: 10.3934/dcds.2003.9.1625. |
[16] |
A. F. Pazoto, J. C. Vila Bravo and J. E. Muñoz Rivera, Asymptotic stability of semi-groups associated to linear weak dissipative systems, Math. Computer Modeling, 40 (2004), 387-392.
doi: 10.1016/j.mcm.2003.10.048. |
[17] |
Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation, J. Hyperbolic Differential Equations, 7 (2010), 471-501.
doi: 10.1142/S0219891610002207. |
[18] |
X. Zhang and E. Zuazua, On the optimality of the observability inequalities for Kirchhoff plate systems with potentials in unbounded domains,, Springer, 2008, 233--243., ().
doi: 10.1007/978-3-540-75712-2_19. |
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