American Institute of Mathematical Sciences

Advanced
September  2017, 10(3): 785-797. doi: 10.3934/krm.2017031

Decay property for solutions to plate type equations with variable coefficients

Shikuan Mao and  Yongqin Liu ,

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

* Corresponding author: Yongqin Liu

Received  June 2016 Revised  August 2016 Published  December 2016

In this paper we consider the initial value problem of a rotational inertial model for plate type equations with variable coefficients and memory in $\mathbb{R}^n\ (n≥q1)$. We study the decay and the regularity-loss property for this equation in the spirit of [12,15], and characterize the decay and regularity property by a function in the spectral space.

Keywords: Plate equation, pointwise estimate in the spectral space, decay, regularity-loss type, memory.
Mathematics Subject Classification: 35L30, 35B40.
Citation: Shikuan Mao, Yongqin Liu. Decay property for solutions to plate type equations with variable coefficients. Kinetic and Related Models, 2017, 10 (3) : 785-797. doi: 10.3934/krm.2017031
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 $\mathbb{R}^N$ with periodic coefficients, Electronic J. Differential Equations, 2008 (2008), 1-23. 

[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]

M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit, London Mathematical Society Lecture Note Series, 268 Cambridge University Press, 1999. doi: 10.1017/CBO9780511662195.

[7]

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

[8]

L. Hörmander, Analysis of Linear Partial Differential Operators, Vol. Ⅲ, Springer-Verlag, 2007. doi: 10.1007/978-3-540-49938-1.

[9]

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.

[10]

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.

[11]

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. 

[12]

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.

[13]

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.  doi: 10.3934/dcds.2011.29.1113.

[14]

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

[15]

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.

[16]

S. Mao and Y. Liu, Decay of solutions to generalized plate type equations with memory, Kinet. Relat. Mod., 7 (2014), 121-131.  doi: 10.3934/krm.2014.7.121.

[17]

N. Mori and S. Kawashima, Decay property for the Timoshenko system with Fourier's type heat conduction, J. Hyperbolic Differential Equations, 11 (2014), 135-157.  doi: 10.1142/S0219891614500039.

[18]

N. Mori and S. Kawashima, Decay property of the Timoshenko-Cattaneo system, Anal. Appl., 14 (2016), 393-413.  doi: 10.1142/S0219530515500062.

[19]

N. Mori, J. Xu and S. Kawashima, Global existence and optimal decay rates for the Timoshenko system: The case of equal wave speeds, J. Differ. Equations., 258 (2015), 1494-1518.  doi: 10.1016/j.jde.2014.11.003.

[20]

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.

[21] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. Ⅰ, Academic Press, New York, 1980. 
[22]

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.

[23]

J. Xu, N. Mori and S. Kawashima, Global existence and minimal decay regularity for the Timoshenko system: The case of non-equal wave speeds, J. Differ. Equations., 259 (2015), 5533-5553.  doi: 10.1016/j.jde.2015.06.041.

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 $\mathbb{R}^N$ with periodic coefficients, Electronic J. Differential Equations, 2008 (2008), 1-23. 

[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]

M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit, London Mathematical Society Lecture Note Series, 268 Cambridge University Press, 1999. doi: 10.1017/CBO9780511662195.

[7]

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

[8]

L. Hörmander, Analysis of Linear Partial Differential Operators, Vol. Ⅲ, Springer-Verlag, 2007. doi: 10.1007/978-3-540-49938-1.

[9]

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.

[10]

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.

[11]

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. 

[12]

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.

[13]

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.  doi: 10.3934/dcds.2011.29.1113.

[14]

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

[15]

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.

[16]

S. Mao and Y. Liu, Decay of solutions to generalized plate type equations with memory, Kinet. Relat. Mod., 7 (2014), 121-131.  doi: 10.3934/krm.2014.7.121.

[17]

N. Mori and S. Kawashima, Decay property for the Timoshenko system with Fourier's type heat conduction, J. Hyperbolic Differential Equations, 11 (2014), 135-157.  doi: 10.1142/S0219891614500039.

[18]

N. Mori and S. Kawashima, Decay property of the Timoshenko-Cattaneo system, Anal. Appl., 14 (2016), 393-413.  doi: 10.1142/S0219530515500062.

[19]

N. Mori, J. Xu and S. Kawashima, Global existence and optimal decay rates for the Timoshenko system: The case of equal wave speeds, J. Differ. Equations., 258 (2015), 1494-1518.  doi: 10.1016/j.jde.2014.11.003.

[20]

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.

[21] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. Ⅰ, Academic Press, New York, 1980. 
[22]

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.

[23]

J. Xu, N. Mori and S. Kawashima, Global existence and minimal decay regularity for the Timoshenko system: The case of non-equal wave speeds, J. Differ. Equations., 259 (2015), 5533-5553.  doi: 10.1016/j.jde.2015.06.041.

[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 and Related Models, 2011, 4 (2) : 531-547. doi: 10.3934/krm.2011.4.531
[3]

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
[4]

Hideo Kubo. On the pointwise decay estimate for the wave equation with compactly supported forcing term. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1469-1480. doi: 10.3934/cpaa.2015.14.1469
[5]

Baowei Feng, Abdelaziz Soufyane. New general decay results for a von Karman plate equation with memory-type boundary conditions. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1757-1774. doi: 10.3934/dcds.2020092
[6]

Tingting Liu, Qiaozhen Ma, Ling Xu. Attractor of the Kirchhoff type plate equation with memory and nonlinear damping on the whole time-dependent space. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022046
[7]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095
[8]

Ruy Coimbra Charão, Alessandra Piske, Ryo Ikehata. A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2215-2255. doi: 10.3934/dcds.2021189
[9]

Salim A. Messaoudi, Ilyes Lacheheb. A general decay result for the Cauchy problem of plate equations with memory. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022026
[10]

Roberto Triggiani, Jing Zhang. Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay. Evolution Equations and Control Theory, 2018, 7 (1) : 153-182. doi: 10.3934/eect.2018008
[11]

Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure and Applied Analysis, 2017, 16 (2) : 533-556. doi: 10.3934/cpaa.2017027
[12]

Jinxing Liu, Xiongrui Wang, Jun Zhou, Xu Liu. Dynamics of solutions to a semilinear plate equation with memory. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3911-3936. doi: 10.3934/cpaa.2021137
[13]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Mostafa Zahri. Theoretical and computational decay results for a memory type wave equation with variable-exponent nonlinearity. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022010
[14]

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 and Applied Analysis, 2019, 18 (1) : 159-180. doi: 10.3934/cpaa.2019009
[15]

Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012
[16]

Li-Ming Yeh. Pointwise estimate for elliptic equations in periodic perforated domains. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1961-1986. doi: 10.3934/cpaa.2015.14.1961
[17]

Hizia Bounadja, Belkacem Said Houari. Decay rates for the Moore-Gibson-Thompson equation with memory. Evolution Equations and Control Theory, 2021, 10 (3) : 431-460. doi: 10.3934/eect.2020074
[18]

Anushaya Mohapatra, William Ott. Memory loss for nonequilibrium open dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3747-3759. doi: 10.3934/dcds.2014.34.3747
[19]

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
[20]

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

2021 Impact Factor: 1.398

Tools

Metrics

  • PDF downloads (96)
  • HTML views (50)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy