March  2017, 16(2): 533-556. doi: 10.3934/cpaa.2017027

Asymptotic behavior of solutions to a nonlinear plate equation with memory

School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

Received  June 2016 Revised  October 2016 Published  January 2016

Fund Project: The author is supported by the National Natural Science Foundation of China (Grant Nos. 11201144, 11201142), by the Project-sponsored by SRF for ROCS, SEM (Grant No. 2013B010) and by the Fundamental Research Funds for the Central Universities (Grant Nos. 2014MS57, 2014MS63, 2014ZZD10).

In this paper we consider the initial value problem of a nonlinear plate equation with memory-type dissipation in multi-dimensional space. Due to the memory effect, more technique is needed to deal with the global existence and decay property of solutions compared with the frictional dissipation. The model we study is an inertial one and the rotational inertia plays an important role in the part of energy estimates. By exploiting the time-weighted energy method we prove the global existence and asymptotic decay of solutions under smallness and suitable regularity assumptions on the initial data.

Citation: 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
References:
[1]

M. E. Bradley and S. Lenhart, Bilinear spatial control of the velocity term in a Kirchhoff plate equation, Electron. J. Differ. Equations, 2001 (2001), 1-15.  Google Scholar

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C. Buriol, Energy decay rates for the Timoshenko system of thermoelastic plates, Nonlinear Analysis-TMA, 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 semilinear plate equation in unbounded domains, J. Hyperbol. Differ. Eq., 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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R. Denk, R. Racke and Y. Shibata, Lp theory for the linear thermoelastic plate equations in bounded and exterior domains, Adv. Differential Equations, 14 (2009), 685-715.  Google Scholar

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Y. Enomoto, On a thermoelastic plate equation in an exterior domain, Math. Meth. Appl. Sci., 25 (2002), 443-472. doi: 10.1002/mma.290.abs.  Google Scholar

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I. Lasiecka and J. Ong, Global solvability and uniform decays of solutions to quasilinear equation with nonlinear boundary dissipation, Comm. Partial Differential Equations, 24 (1999), 2069-2107. doi: 10.1080/03605309908821495.  Google Scholar

[8]

I. Lasiecka, S. Maad and A. Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system, Nonlinear Differ. Equ. Appl., 15 (2008), 689-715. doi: 10.1007/s00030-008-0011-8.  Google Scholar

[9]

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

[10]

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

[11]

Y. Liu and S. Kawashima, Global existence and decay of solutions for a quasi-linear dissipative plate equation, J. Hyperbol. Differ. Eq., 8(2011), 591-614. doi: 10.1142/S0219891611002500.  Google Scholar

[12]

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

[13]

Y. Liu and S. Kawashima, Decay property for the Timoshenko system with memory-type dissipation, Math. Models Meth. Appl. Sci., 22 (2012), 1-19. doi: 10.1142/S0218202511500126.  Google Scholar

[14]

Y. Liu and S. Kawashima, Global existence and asymptotic decay of solutions to the nonlinear Timoshenko system with memory, Nonlinear Analysis-TMA, 84 (2013), 1-17. doi: 10.1016/j.na.2013.02.005.  Google Scholar

[15]

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

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

[18]

G. Perla Menzala and E. Zuazua, Timoshenko's plate equations as a singular limit of the dinamical von Kármán system, J. Math. Pures et Appli., 79 (2000), 73-94. doi: 10.1016/S0021-7824(00)00149-5.  Google Scholar

[19]

Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semilinear dissipative plate equation, J. Hyperbol. Differ. Eq., 7 (2010), 471-501. doi: 10.1142/S0219891610002207.  Google Scholar

[20]

R. Teman, Navier-Stokes Equations, Studies in Mathematics and Its Applications, Vol. 2, Revised Edition, North-Holland, Amsterdam, New York, Oxford, 1979.  Google Scholar

[21]

X. Zhang and E. Zuazua, On the optimality of the observability inequalities for Kirchhoff plate systems with potentials in unbounded domains, in Hyperbolic Problems: Theory, Numerics and Applications, 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, Electron. J. Differ. Equations, 2001 (2001), 1-15.  Google Scholar

[2]

C. Buriol, Energy decay rates for the Timoshenko system of thermoelastic plates, Nonlinear Analysis-TMA, 64 (2006), 92-108. doi: 10.1016/j.na.2005.06.010.  Google Scholar

[3]

C. R. da Luz and R. C. Charão, Asymptotic properties for a semilinear plate equation in unbounded domains, J. Hyperbol. Differ. Eq., 6 (2009), 269-294. doi: 10.1142/S0219891609001824.  Google Scholar

[4]

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

[5]

R. Denk, R. Racke and Y. Shibata, Lp theory for the linear thermoelastic plate equations in bounded and exterior domains, Adv. Differential Equations, 14 (2009), 685-715.  Google Scholar

[6]

Y. Enomoto, On a thermoelastic plate equation in an exterior domain, Math. Meth. Appl. Sci., 25 (2002), 443-472. doi: 10.1002/mma.290.abs.  Google Scholar

[7]

I. Lasiecka and J. Ong, Global solvability and uniform decays of solutions to quasilinear equation with nonlinear boundary dissipation, Comm. Partial Differential Equations, 24 (1999), 2069-2107. doi: 10.1080/03605309908821495.  Google Scholar

[8]

I. Lasiecka, S. Maad and A. Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system, Nonlinear Differ. Equ. Appl., 15 (2008), 689-715. doi: 10.1007/s00030-008-0011-8.  Google Scholar

[9]

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

[10]

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

[11]

Y. Liu and S. Kawashima, Global existence and decay of solutions for a quasi-linear dissipative plate equation, J. Hyperbol. Differ. Eq., 8(2011), 591-614. doi: 10.1142/S0219891611002500.  Google Scholar

[12]

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

[13]

Y. Liu and S. Kawashima, Decay property for the Timoshenko system with memory-type dissipation, Math. Models Meth. Appl. Sci., 22 (2012), 1-19. doi: 10.1142/S0218202511500126.  Google Scholar

[14]

Y. Liu and S. Kawashima, Global existence and asymptotic decay of solutions to the nonlinear Timoshenko system with memory, Nonlinear Analysis-TMA, 84 (2013), 1-17. doi: 10.1016/j.na.2013.02.005.  Google Scholar

[15]

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

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

[18]

G. Perla Menzala and E. Zuazua, Timoshenko's plate equations as a singular limit of the dinamical von Kármán system, J. Math. Pures et Appli., 79 (2000), 73-94. doi: 10.1016/S0021-7824(00)00149-5.  Google Scholar

[19]

Y. Sugitani and S. Kawashima, Decay estimates of solutions to a semilinear dissipative plate equation, J. Hyperbol. Differ. Eq., 7 (2010), 471-501. doi: 10.1142/S0219891610002207.  Google Scholar

[20]

R. Teman, Navier-Stokes Equations, Studies in Mathematics and Its Applications, Vol. 2, Revised Edition, North-Holland, Amsterdam, New York, Oxford, 1979.  Google Scholar

[21]

X. Zhang and E. Zuazua, On the optimality of the observability inequalities for Kirchhoff plate systems with potentials in unbounded domains, in Hyperbolic Problems: Theory, Numerics and Applications, Springer, (2008), 233-243. doi: 10.1007/978-3-540-75712-2_19.  Google Scholar

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