Decay property for a plate equation with memory-type dissipation

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June 2011, 4(2): 531-547. doi: 10.3934/krm.2011.4.531

Decay property for a plate equation with memory-type dissipation

Yongqin Liu ^1, and Shuichi Kawashima ^2,

1.	Department of Mathematics, North China Electric Power University, Beijing 102208, China
2.	Faculty of Mathematics, Kyushu University, Fukuoka 819-0395

Received September 2010 Revised December 2010 Published April 2011

Abstract
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In this paper we focus on the initial value problem of the semi-linear plate equation with memory in multi-dimensions $(n\geq1)$, the decay structure of which is of regularity-loss property. By using Fourier transform and Laplace transform, we obtain the fundamental solutions and thus the solution to the corresponding linear problem. Appealing to the point-wise estimate in the Fourier space of solutions to the linear problem, we get estimates and properties of solution operators, by exploiting which decay estimates of solutions to the linear problem are obtained. Also by introducing a set of time-weighted Sobolev spaces and using the contraction mapping theorem, we obtain the global in-time existence and the optimal decay estimates of solutions to the semi-linear problem under smallness assumption on the initial data.

Keywords: regularity-loss type., point-wise estimate in the Fourier space, Semi-linear plate equation with memory, decay estimates.

Mathematics Subject Classification: 35G25, 35L30, 35B4.

Citation: 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

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 $R^N$ with periodic coefficients, Electronic J. Differential Equations, 46 (2008), 23 pp.
[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]	A. D. Drozdov and V. B. Kolmanovskii, "Stability in Viscoelasticity,", North-Holland Publishing Co., Amsterdam, 1994., 38 ().
[7]	M. Fabrizio and B. Lazzari, On the existence and the asymptotic stability of solutions for linear viscoelastic solids, Arch. Rational Mech. Anal., 116 (1991), 139-152. doi: 10.1007/BF00375589.
[8]	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.
[9]	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.
[10]	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.
[11]	Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermo-viscoelasticity, Quart. Appl. Math., 54 (1996), 21-31.
[12]	Z. Liu and S. Zheng, "Semi-Groups Associated with Dissipative Systems," Chapman $&$ Hall/CRC, London, 1999.
[13]	J. E. Muñoz Rivera, Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648.
[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), 692-704. doi: 10.1016/S0022-247X(03)00511-0.
[15]	J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst., 9 (2003), 1625-1639. doi: 10.3934/dcds.2003.9.1625.
[16]	A. F. Pazoto, J. C. Vila Bravo and J. E. Muñoz Rivera, Asymptotic stability of semi-groups associated to linear weak dissipative systems, Math. Computer Modeling, 40 (2004), 387-392. doi: 10.1016/j.mcm.2003.10.048.
[17]	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.
[18]	X. Zhang and E. Zuazua, On the optimality of the observability inequalities for Kirchhoff plate systems with potentials in unbounded domains,, Springer, 2008, 233--243., (). doi: 10.1007/978-3-540-75712-2_19.

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 $R^N$ with periodic coefficients, Electronic J. Differential Equations, 46 (2008), 23 pp.
[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]	A. D. Drozdov and V. B. Kolmanovskii, "Stability in Viscoelasticity,", North-Holland Publishing Co., Amsterdam, 1994., 38 ().
[7]	M. Fabrizio and B. Lazzari, On the existence and the asymptotic stability of solutions for linear viscoelastic solids, Arch. Rational Mech. Anal., 116 (1991), 139-152. doi: 10.1007/BF00375589.
[8]	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.
[9]	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.
[10]	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.
[11]	Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermo-viscoelasticity, Quart. Appl. Math., 54 (1996), 21-31.
[12]	Z. Liu and S. Zheng, "Semi-Groups Associated with Dissipative Systems," Chapman $&$ Hall/CRC, London, 1999.
[13]	J. E. Muñoz Rivera, Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648.
[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), 692-704. doi: 10.1016/S0022-247X(03)00511-0.
[15]	J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst., 9 (2003), 1625-1639. doi: 10.3934/dcds.2003.9.1625.
[16]	A. F. Pazoto, J. C. Vila Bravo and J. E. Muñoz Rivera, Asymptotic stability of semi-groups associated to linear weak dissipative systems, Math. Computer Modeling, 40 (2004), 387-392. doi: 10.1016/j.mcm.2003.10.048.
[17]	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.
[18]	X. Zhang and E. Zuazua, On the optimality of the observability inequalities for Kirchhoff plate systems with potentials in unbounded domains,, Springer, 2008, 233--243., (). doi: 10.1007/978-3-540-75712-2_19.

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