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.

[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.

[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.

[4]

P. M. N. DharmawardaneJ. 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.

[5]

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

[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.

[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.

[8]

I. LasieckaS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[17]

J. E. Muñoz RiveraM. 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.

[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.

[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.

[20]

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

[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.

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.

[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.

[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.

[4]

P. M. N. DharmawardaneJ. 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.

[5]

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

[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.

[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.

[8]

I. LasieckaS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[17]

J. E. Muñoz RiveraM. 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.

[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.

[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.

[20]

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

[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.

[1]

Tingting Liu, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4595-4616. doi: 10.3934/dcdsb.2018178

[2]

Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383

[3]

Mykhailo Potomkin. Asymptotic behavior of thermoviscoelastic Berger plate. Communications on Pure & Applied Analysis, 2010, 9 (1) : 161-192. doi: 10.3934/cpaa.2010.9.161

[4]

Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009

[5]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021

[6]

Yongqin Liu, Shuichi Kawashima. Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1113-1139. doi: 10.3934/dcds.2011.29.1113

[7]

Yun Li, Fuke Wu, George Yin. Asymptotic behavior of gene expression with complete memory and two-time scales based on the chemical Langevin equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-22. doi: 10.3934/dcdsb.2019125

[8]

Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465

[9]

Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609

[10]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[11]

Weijiu Liu. Asymptotic behavior of solutions of time-delayed Burgers' equation. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 47-56. doi: 10.3934/dcdsb.2002.2.47

[12]

Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861

[13]

Moncef Aouadi, Alain Miranville. Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory. Evolution Equations & Control Theory, 2015, 4 (3) : 241-263. doi: 10.3934/eect.2015.4.241

[14]

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

[15]

Hiroshi Takeda. Large time behavior of solutions for a nonlinear damped wave equation. Communications on Pure & Applied Analysis, 2016, 15 (1) : 41-55. doi: 10.3934/cpaa.2016.15.41

[16]

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93

[17]

Per Christian Moan, Jitse Niesen. On an asymptotic method for computing the modified energy for symplectic methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1105-1120. doi: 10.3934/dcds.2014.34.1105

[18]

Ryo Ikehata, Marina Soga. Asymptotic profiles for a strongly damped plate equation with lower order perturbation. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1759-1780. doi: 10.3934/cpaa.2015.14.1759

[19]

Jáuber Cavalcante Oliveira, Jardel Morais Pereira, Gustavo Perla Menzala. Long time dynamics of a multidimensional nonlinear lattice with memory. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2715-2732. doi: 10.3934/dcdsb.2015.20.2715

[20]

Guofu Lu. Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1567-1586. doi: 10.3934/dcdsb.2016011

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (10)
  • HTML views (1)
  • Cited by (0)

Other articles
by authors

[Back to Top]