American Institute of Mathematical Sciences

Advanced
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.  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.  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.  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.  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.  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.  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.   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.  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.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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.   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.  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.  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.  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.  doi: 10.1142/S0219891610002207.  Google Scholar

[1]

Priyanjana M. N. Dharmawardane. Decay property of regularity-loss type for quasi-linear hyperbolic systems of viscoelasticity. Conference Publications, 2013, 2013 (special) : 197-206. doi: 10.3934/proc.2013.2013.197
[2]

Yongqin Liu, Shuichi Kawashima. Decay property for a plate equation with memory-type dissipation. Kinetic & Related Models, 2011, 4 (2) : 531-547. doi: 10.3934/krm.2011.4.531
[3]

Hideo Kubo. On the pointwise decay estimate for the wave equation with compactly supported forcing term. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1469-1480. doi: 10.3934/cpaa.2015.14.1469
[4]

Baowei Feng, Abdelaziz Soufyane. New general decay results for a von Karman plate equation with memory-type boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1757-1774. doi: 10.3934/dcds.2020092
[5]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095
[6]

Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure & Applied Analysis, 2017, 16 (2) : 533-556. doi: 10.3934/cpaa.2017027
[7]

Ralf Kirsch, Sergej Rjasanow. The uniformly heated inelastic Boltzmann equation in Fourier space. Kinetic & Related Models, 2010, 3 (3) : 445-456. doi: 10.3934/krm.2010.3.445
[8]

Shikuan Mao, Yongqin Liu. Decay property for solutions to plate type equations with variable coefficients. Kinetic & Related Models, 2017, 10 (3) : 785-797. doi: 10.3934/krm.2017031
[9]

Mohammad M. Al-Gharabli, Aissa Guesmia, Salim A. Messaoudi. Existence and a general decay results for a viscoelastic plate equation with a logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 159-180. doi: 10.3934/cpaa.2019009
[10]

Li-Ming Yeh. Pointwise estimate for elliptic equations in periodic perforated domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1961-1986. doi: 10.3934/cpaa.2015.14.1961
[11]

Hizia Bounadja, Belkacem Said Houari. Decay rates for the moore-gibson-thompson equation with memory. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020074
[12]

Anushaya Mohapatra, William Ott. Memory loss for nonequilibrium open dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3747-3759. doi: 10.3934/dcds.2014.34.3747
[13]

Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28 (1) : 263-289. doi: 10.3934/era.2020016
[14]

Miao Liu, Weike Wang. Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1203-1222. doi: 10.3934/cpaa.2014.13.1203
[15]

Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179
[16]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064
[17]

Wenjun Liu, Zhijing Chen, Zhiyu Tu. New general decay result for a fourth-order Moore-Gibson-Thompson equation with memory. Electronic Research Archive, 2020, 28 (1) : 433-457. doi: 10.3934/era.2020025
[18]

Barbara Brandolini, Francesco Chiacchio, Jeffrey J. Langford. Estimates for sums of eigenvalues of the free plate via the fourier transform. Communications on Pure & Applied Analysis, 2020, 19 (1) : 113-122. doi: 10.3934/cpaa.2020007
[19]

Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. Conference Publications, 2005, 2005 (Special) : 605-610. doi: 10.3934/proc.2005.2005.605
[20]

Yuto Miyatake, Tai Nakagawa, Tomohiro Sogabe, Shao-Liang Zhang. A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation. Journal of Computational Dynamics, 2019, 6 (2) : 361-383. doi: 10.3934/jcd.2019018

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences