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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 & 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

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.   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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