American Institute of Mathematical Sciences

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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

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 & 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.  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

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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

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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

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A. D. Drozdov and V. B. Kolmanovskii, "Stability in Viscoelasticity,", North-Holland Publishing Co., Amsterdam, 1994., 38 ().   Google Scholar

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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.  Google Scholar

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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

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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

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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. Google Scholar

[11]

Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermo-viscoelasticity, Quart. Appl. Math., 54 (1996), 21-31.  Google Scholar

[12]

Z. Liu and S. Zheng, "Semi-Groups Associated with Dissipative Systems," Chapman $&$ Hall/CRC, London, 1999.  Google Scholar

[13]

J. E. Muñoz Rivera, Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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 $R^N$ with periodic coefficients, Electronic J. Differential Equations, 46 (2008), 23 pp. 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]

A. D. Drozdov and V. B. Kolmanovskii, "Stability in Viscoelasticity,", North-Holland Publishing Co., Amsterdam, 1994., 38 ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[11]

Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermo-viscoelasticity, Quart. Appl. Math., 54 (1996), 21-31.  Google Scholar

[12]

Z. Liu and S. Zheng, "Semi-Groups Associated with Dissipative Systems," Chapman $&$ Hall/CRC, London, 1999.  Google Scholar

[13]

J. E. Muñoz Rivera, Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math., 52 (1994), 628-648.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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