Decay of solutions to generalized plate type equations with memory

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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

Abstract
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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 and 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, 15 pp.
[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 $\mathbb{R}^N2$ with periodic coefficients, Electron. J. Differential Equations, 2008, 23 pp.
[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.
[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]	Y. Enomoto, On a thermoelastic plate equation in an exterior domain, Math. Meth. Appl. Sci., 25 (2002), 443-472. doi: 10.1002/mma.290.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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]	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.

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, 15 pp.
[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 $\mathbb{R}^N2$ with periodic coefficients, Electron. J. Differential Equations, 2008, 23 pp.
[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.
[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]	Y. Enomoto, On a thermoelastic plate equation in an exterior domain, Math. Meth. Appl. Sci., 25 (2002), 443-472. doi: 10.1002/mma.290.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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]	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.

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