American Institute of Mathematical Sciences

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March  2014, 7(1): 121-131. doi: 10.3934/krm.2014.7.121

Decay of solutions to generalized plate type equations with memory

Shikuan Mao 1, and  Yongqin Liu 1,

1. 

School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China, China

Received  July 2013 Revised  October 2013 Published  December 2013

In this paper we focus on the initial value problem of an inertial model for a generalized plate equation with memory in $\mathbb{R}^n\ (n\geq1)$. We study the decay and the regularity-loss property for this type of equations in the spirit of [10,13]. The novelty of this paper is that we extend the order of derivatives from integer to fraction and refine the results of the even-dimensional case in the related literature [10,13].
Keywords: Plate equation, regularity-loss type, memory., pointwise estimate in the Fourier space, decay.
Mathematics Subject Classification: 35L30, 35B4.
Citation: Shikuan Mao, Yongqin Liu. Decay of solutions to generalized plate type equations with memory. Kinetic & Related Models, 2014, 7 (1) : 121-131. doi: 10.3934/krm.2014.7.121
References:
[1]

M. E. Bradley and S. Lenhart, Bilinear spatial control of the velocity term in a Kirchhoff plate equation,, Electron. J. Differential Equations, 2001 ().   Google Scholar

[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.  Google Scholar

[3]

R. C. Charão, E. Bisognin, V. Bisognin and A. F. Pazoto, Asymptotic behavior for a dissipative plate equation in $\mathbbR^N$ with periodic coefficients,, Electron. J. Differential Equations, 2008 ().   Google Scholar

[4]

C. R. da Luz and R. C. Charão, Asymptotic properties for a semilinear plate equation in unbounded domains, J. Hyperbolic Differ. Equ., 6 (2009), 269-294. doi: 10.1142/S0219891609001824.  Google Scholar

[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.  Google Scholar

[6]

Y. Enomoto, On a thermoelastic plate equation in an exterior domain, Math. Meth. Appl. Sci., 25 (2002), 443-472. doi: 10.1002/mma.290.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

H. J. Lee, Uniform decay for solution of the plate equation with a boundary condition of memory type, Trends in Math., 9 (2006), 51-55. Google Scholar

[10]

Y. Liu, Decay of solutions to an inertial model for a semilinear plate equation with memory, J. Math. Anal. Appl., 394 (2012), 616-632. doi: 10.1016/j.jmaa.2012.04.003.  Google Scholar

[11]

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.  Google Scholar

[12]

Y. Liu and S. Kawashima, Global existence and decay of solutions for a quasi-linear dissipative plate equation, J. Hyperbolic Diff. Equ., 8 (2011), 591-614. doi: 10.1142/S0219891611002500.  Google Scholar

[13]

Y. Liu and S. Kawashima, Decay property for a plate equation with memory-type dissipation, Kinet. Relat. Mod., 4 (2011), 531-547. doi: 10.3934/krm.2011.4.531.  Google Scholar

[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.  Google Scholar

[15]

Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation, J. Hyperbolic Differ. Equ., 7 (2010), 471-501. doi: 10.1142/S0219891610002207.  Google Scholar

show all references

References:
[1]

M. E. Bradley and S. Lenhart, Bilinear spatial control of the velocity term in a Kirchhoff plate equation,, Electron. J. Differential Equations, 2001 ().   Google Scholar

[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.  Google Scholar

[3]

R. C. Charão, E. Bisognin, V. Bisognin and A. F. Pazoto, Asymptotic behavior for a dissipative plate equation in $\mathbbR^N$ with periodic coefficients,, Electron. J. Differential Equations, 2008 ().   Google Scholar

[4]

C. R. da Luz and R. C. Charão, Asymptotic properties for a semilinear plate equation in unbounded domains, J. Hyperbolic Differ. Equ., 6 (2009), 269-294. doi: 10.1142/S0219891609001824.  Google Scholar

[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.  Google Scholar

[6]

Y. Enomoto, On a thermoelastic plate equation in an exterior domain, Math. Meth. Appl. Sci., 25 (2002), 443-472. doi: 10.1002/mma.290.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

H. J. Lee, Uniform decay for solution of the plate equation with a boundary condition of memory type, Trends in Math., 9 (2006), 51-55. Google Scholar

[10]

Y. Liu, Decay of solutions to an inertial model for a semilinear plate equation with memory, J. Math. Anal. Appl., 394 (2012), 616-632. doi: 10.1016/j.jmaa.2012.04.003.  Google Scholar

[11]

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.  Google Scholar

[12]

Y. Liu and S. Kawashima, Global existence and decay of solutions for a quasi-linear dissipative plate equation, J. Hyperbolic Diff. Equ., 8 (2011), 591-614. doi: 10.1142/S0219891611002500.  Google Scholar

[13]

Y. Liu and S. Kawashima, Decay property for a plate equation with memory-type dissipation, Kinet. Relat. Mod., 4 (2011), 531-547. doi: 10.3934/krm.2011.4.531.  Google Scholar

[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.  Google Scholar

[15]

Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semi-linear dissipative plate equation, J. Hyperbolic Differ. Equ., 7 (2010), 471-501. doi: 10.1142/S0219891610002207.  Google Scholar

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