# American Institute of Mathematical Sciences

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

## Decay property for solutions to plate type equations with variable coefficients

 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.

Citation: 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
##### 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. 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 $\mathbb{R}^N$ with periodic coefficients, Electronic J. Differential Equations, 2008 (2008), 1-23. Google Scholar [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. 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] 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. Google Scholar [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. Google Scholar [8] L. Hörmander, Analysis of Linear Partial Differential Operators, Vol. Ⅲ, Springer-Verlag, 2007. doi: 10.1007/978-3-540-49938-1. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] N. Mori and S. Kawashima, Decay property of the Timoshenko-Cattaneo system, Anal. Appl., 14 (2016), 393-413. doi: 10.1142/S0219530515500062. Google Scholar [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. Google Scholar [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. Google Scholar [21] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. Ⅰ, Academic Press, New York, 1980. Google Scholar [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. Google Scholar [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. Google Scholar

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##### 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. 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 $\mathbb{R}^N$ with periodic coefficients, Electronic J. Differential Equations, 2008 (2008), 1-23. Google Scholar [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. 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] 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. Google Scholar [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. Google Scholar [8] L. Hörmander, Analysis of Linear Partial Differential Operators, Vol. Ⅲ, Springer-Verlag, 2007. doi: 10.1007/978-3-540-49938-1. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] N. Mori and S. Kawashima, Decay property of the Timoshenko-Cattaneo system, Anal. Appl., 14 (2016), 393-413. doi: 10.1142/S0219530515500062. Google Scholar [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. Google Scholar [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. Google Scholar [21] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. Ⅰ, Academic Press, New York, 1980. Google Scholar [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. Google Scholar [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. Google Scholar
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