Article Contents
Article Contents

# Asymptotic behavior of solutions to a nonlinear plate equation with memory

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.

Mathematics Subject Classification: Primary: 35G25, 35L30; Secondary: 35B40.

 Citation:

•  [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. 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. [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. [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. 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. [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 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. [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.