-
Previous Article
On a linear runs and tumbles equation
- KRM Home
- This Issue
-
Next Article
Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces
Decay property for solutions to plate type equations with variable coefficients
School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China |
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 [
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. Google Scholar |
[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. Google Scholar |
[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 & 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 & 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 & 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 & Continuous Dynamical Systems - A, 2020, 40 (3) : 1757-1774. doi: 10.3934/dcds.2020092 |
[6] |
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 |
[7] |
Roberto Triggiani, Jing Zhang. Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay. Evolution Equations & Control Theory, 2018, 7 (1) : 153-182. doi: 10.3934/eect.2018008 |
[8] |
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 |
[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] |
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 |
[12] |
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 |
[13] |
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 |
[14] |
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 |
[15] |
Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, Sandro Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793 |
[16] |
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 |
[17] |
John Sylvester. An estimate for the free Helmholtz equation that scales. Inverse Problems & Imaging, 2009, 3 (2) : 333-351. doi: 10.3934/ipi.2009.3.333 |
[18] |
Zongming Guo, Juncheng Wei. Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2561-2580. doi: 10.3934/dcds.2014.34.2561 |
[19] |
Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2261-2281. doi: 10.3934/dcds.2014.34.2261 |
[20] |
Robert M. Strain. Optimal time decay of the non cut-off Boltzmann equation in the whole space. Kinetic & Related Models, 2012, 5 (3) : 583-613. doi: 10.3934/krm.2012.5.583 |
2018 Impact Factor: 1.38
Tools
Metrics
Other articles
by authors
[Back to Top]