# American Institute of Mathematical Sciences

November  2021, 20(11): 3911-3936. doi: 10.3934/cpaa.2021137

## Dynamics of solutions to a semilinear plate equation with memory

 1 Department of Mathematics, Yibin University, Yibin, Sichuan, 644000, China 2 School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

* Corresponding author

Received  March 2021 Revised  July 2021 Published  November 2021 Early access  August 2021

Fund Project: The work of Jinxing Liu is supported by the Talent project of Yibin University (No. 2018RC17), and the work of Jun Zhou is supported by the Basic and Advanced Research Project of CQC-STC grant cstc2016jcyjA0018, NSFC 11201380

In this paper we consider an initial-boundary value problem of a semilinear regularity-loss-type plate equation with memory in a bounded domain of $\mathbb{R}^n$ ($n = 1,2,\cdots$). By using the Faedo-Galërkin method and some theories of ordinary differential equations, we obtain the local existence and uniqueness of weak solutions. Then, we study the dynamics of the weak solutions, such as global existence and finite time blow-up, by energy estimation and some ordinary differential inequalities. Moreover, the upper bound of blow-up time for the blow-up solutions is also considered.

Citation: Jinxing Liu, Xiongrui Wang, Jun Zhou, Xu Liu. Dynamics of solutions to a semilinear plate equation with memory. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3911-3936. doi: 10.3934/cpaa.2021137
##### References:
 [1] Nouri Boumaza and Billel Gheraibia, General decay and blowup of solutions for a degenerate viscoelastic equation of Kirchhoff type with source term, J. Math. Anal. Appl., 489 (2020), 124185. doi: 10.1016/j.jmaa.2020.124185. [2] Wenhui Chen and Tuan Anh Dao, On the Cauchy problem for semilinear regularity-loss-type $\sigma$-evolution models with memory term, Nonlinear Anal. Real World Appl., 59 (2021), 103265. doi: 10.1016/j.nonrwa.2020.103265. [3] Marcelo Moreira Cavalcanti and Higidio Portillo Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.  doi: 10.1137/s0363012902408010. [4] Cleverson Roberto da Luz and Ruy Coimbra Charão, Asymptotic properties for a semilinear plate equation in unbounded domains, J. Hyperbolic Differ. Equ., 6 (2009), 269-294.  doi: 10.1142/S0219891609001824. [5] Priyanjana M. N. Dharmawardane, Tohru Nakamura and Shuichi Kawashima, Decay estimates of solutions for quasi-linear hyperbolic systems of viscoelasticity, SIAM J. Math. Anal., 44 (2012), 1976-2001.  doi: 10.1137/11083900x. [6] A. Guesmia and S. A. Messaoudi, A new approach to the stability of an abstract system in the presence of infinite history, J. Math. Anal. Appl., 416 (2014), 212-228.  doi: 10.1016/j.jmaa.2014.02.030. [7] A. Guesmia, S. A. Messaoudi and C. M. Webler, Well-posedness and optimal decay rates for the viscoelastic Kirchhoff equation, Bol. Soc. Parana. Mat., 35 (2017), 203-224.  doi: 10.5269/bspm.v35i3.31395. [8] Aissa Guesmia and Salim A. Messaoudi, A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Anal. Real World Appl., 13 (2012), 476-485.  doi: 10.1016/j.nonrwa.2011.08.004. [9] Jamilu Hashim Hassan and Salim A. Messaoudi, General decay estimate for a class of weakly dissipative second-order system with memory, Math. Meth. Appl. Sci., 42, (2019), 2842–2853. doi: 10.1002/mma.5554. [10] Jamilu Hashim Hassan and Salim A. Messaoudi, A note on the polynomial decay of a weakly dissipative viscoelastic system, Math. Nachr., 293 (2020), 1961-1967.  doi: 10.1002/mana.201900182. [11] Jum-Ran Kang, General stability of solutions for a viscoelastic wave equations of Kirchhoff type with acoustic boundary conditions, Math. Methods Appl. Sci., 39 (2016), 2953-2964.  doi: 10.1002/mma.3742. [12] Howard A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt} = -Au+{F}(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.1090/S0002-9947-1974-0344697-2. [13] Howard A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015. [14] J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites non Linéaires, Gauthier-Villars, Paris, 1969. [15] Yongqin 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. [16] Yongqin Liu, Asymptotic behavior of solutions to a nonlinear plate equation with memory, Commun. Pure Appl. Anal., 16 (2017), 533-556.  doi: 10.3934/cpaa.2017027. [17] Yongqin Liu and Shuichi Kawashima, Decay property for a plate equation with memory-type dissipation, Kinet. Relat. Models, 4 (2011), 531-547.  doi: 10.1007/978-3-540-75712-2_19. [18] Yongqin Liu and Shuichi Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation, Discrete Contin. Dyn. Syst., 29 (2011), 1113-1139.  doi: 10.1007/978-3-540-75712-2_19. [19] Yongqin Liu and Shuichi Kawashima, Global existence and decay of solutions for a quasi-linear dissipative plate equation, J. Hyperbolic Differ. Equ., 8 (2011), 591-614.  doi: 10.1142/S0219891611002500. [20] Yongqin Liu and Shuichi Kawashima, Decay property for the Timoshenko system with memory-type dissipation, Math. Models Methods Appl. Sci., 22 (2012), 1150012. doi: 10.1142/S0218202511500126. [21] Yongqin Liu and Yoshihiro Ueda, Decay estimate and asymptotic profile for a plate equation with memory, J. Differ. Equ., 268 (2020), 2435-2463.  doi: 10.1016/j.jde.2019.09.007. [22] Salim A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal., 69 (2008), 2589-2598.  doi: 10.1016/j.na.2007.08.035. [23] Salim A. Messaoudi and Muhammad M. Al-Gharabli, A general stability result for a nonlinear wave equation with infinite memory, Appl. Math. Lett., 26 (2013), 1082-1086.  doi: 10.1016/j.aml.2013.06.002. [24] Sun-Hye Park, Mi Jin Lee and Jum-Ran Kang, Blow-up results for viscoelastic wave equations with weak damping, Appl. Math. Lett., 80 (2018), 20-26.  doi: 10.1016/j.aml.2018.01.002. [25] M. Pellicer and B. Said-Houari, Wellposedness and decay rates for the Cauchy problem of the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Appl. Math. Optim., 80 (2019), 447-478.  doi: 10.1007/s00245-017-9471-8. [26] Reinhard Racke and Yoshihiro Ueda, Dissipative structures for thermoelastic plate equations in ${\mathbb{R}^n}$, Adv. Differ. Equ., 21 (2016), 601–630. [27] J. E. Munoz 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. [28] J. E. Munoz Rivera and M. G. Naso, Asymptotic stability of semigroups associated to weak dissipative systems with memory, J. Math. Anal. Appl., 326 (2007), 691-707.  doi: 10.1016/j.jmaa.2006.03.022. [29] Haitao Song and Chengkui Zhong, Blow-up of solutions of a nonlinear viscoelastic wave equation, Nonlinear Anal. Real World Appl., 11 (2010), 3877-3883.  doi: 10.1016/j.nonrwa.2010.02.015. [30] Yousuke Sugitani and Shuichi Kawashima, Decay estimates of solutions to a semilinear dissipative plate equation, J. Hyperbolic Differ. Equ., 7 (2010), 471-501.  doi: 10.1142/S0219891610002207. [31] Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, in Applied Mathematical Sciences, Springer-Verlag, New York, 1997. doi: https://link.springer.com/book/10.1007/978-1-4684-0313-8. [32] Yoshihiro Ueda, Optimal decay estimates of a regularity-loss type system with constraint condition, J. Differ. Equ., 264 (2018), 679-701.  doi: 10.1016/j.jde.2017.09.020. [33] Yoshihiro Ueda, Renjun Duan and Shuichi Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal., 205 (2012), 239-266.  doi: 10.1007/s00205-012-0508-5. [34] Shun-Tang Wu, Exponential energy decay of solutions for an integro-differential equation with strong damping, J. Math. Anal. Appl., 364 (2010), 609-617.  doi: 10.1016/j.jmaa.2009.11.046. [35] Zhifeng Yang and Zhaogang Gong, Blow-up solutions for viscoelastic equations of Kirchhoff type with arbitrary positive initial energy, Electron. J. Differ. Equ., 2016 (2016), 1–12. [36] Songmu Zheng, Nonlinear Evolution Equations, in Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: https://www.routledge.com/Nonlinear-Evolution-Equations/Zheng/p/book/9781584884521.

show all references

##### References:
 [1] Nouri Boumaza and Billel Gheraibia, General decay and blowup of solutions for a degenerate viscoelastic equation of Kirchhoff type with source term, J. Math. Anal. Appl., 489 (2020), 124185. doi: 10.1016/j.jmaa.2020.124185. [2] Wenhui Chen and Tuan Anh Dao, On the Cauchy problem for semilinear regularity-loss-type $\sigma$-evolution models with memory term, Nonlinear Anal. Real World Appl., 59 (2021), 103265. doi: 10.1016/j.nonrwa.2020.103265. [3] Marcelo Moreira Cavalcanti and Higidio Portillo Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.  doi: 10.1137/s0363012902408010. [4] Cleverson Roberto da Luz and Ruy Coimbra Charão, Asymptotic properties for a semilinear plate equation in unbounded domains, J. Hyperbolic Differ. Equ., 6 (2009), 269-294.  doi: 10.1142/S0219891609001824. [5] Priyanjana M. N. Dharmawardane, Tohru Nakamura and Shuichi Kawashima, Decay estimates of solutions for quasi-linear hyperbolic systems of viscoelasticity, SIAM J. Math. Anal., 44 (2012), 1976-2001.  doi: 10.1137/11083900x. [6] A. Guesmia and S. A. Messaoudi, A new approach to the stability of an abstract system in the presence of infinite history, J. Math. Anal. Appl., 416 (2014), 212-228.  doi: 10.1016/j.jmaa.2014.02.030. [7] A. Guesmia, S. A. Messaoudi and C. M. Webler, Well-posedness and optimal decay rates for the viscoelastic Kirchhoff equation, Bol. Soc. Parana. Mat., 35 (2017), 203-224.  doi: 10.5269/bspm.v35i3.31395. [8] Aissa Guesmia and Salim A. Messaoudi, A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Anal. Real World Appl., 13 (2012), 476-485.  doi: 10.1016/j.nonrwa.2011.08.004. [9] Jamilu Hashim Hassan and Salim A. Messaoudi, General decay estimate for a class of weakly dissipative second-order system with memory, Math. Meth. Appl. Sci., 42, (2019), 2842–2853. doi: 10.1002/mma.5554. [10] Jamilu Hashim Hassan and Salim A. Messaoudi, A note on the polynomial decay of a weakly dissipative viscoelastic system, Math. Nachr., 293 (2020), 1961-1967.  doi: 10.1002/mana.201900182. [11] Jum-Ran Kang, General stability of solutions for a viscoelastic wave equations of Kirchhoff type with acoustic boundary conditions, Math. Methods Appl. Sci., 39 (2016), 2953-2964.  doi: 10.1002/mma.3742. [12] Howard A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt} = -Au+{F}(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.1090/S0002-9947-1974-0344697-2. [13] Howard A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015. [14] J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites non Linéaires, Gauthier-Villars, Paris, 1969. [15] Yongqin 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. [16] Yongqin Liu, Asymptotic behavior of solutions to a nonlinear plate equation with memory, Commun. Pure Appl. Anal., 16 (2017), 533-556.  doi: 10.3934/cpaa.2017027. [17] Yongqin Liu and Shuichi Kawashima, Decay property for a plate equation with memory-type dissipation, Kinet. Relat. Models, 4 (2011), 531-547.  doi: 10.1007/978-3-540-75712-2_19. [18] Yongqin Liu and Shuichi Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation, Discrete Contin. Dyn. Syst., 29 (2011), 1113-1139.  doi: 10.1007/978-3-540-75712-2_19. [19] Yongqin Liu and Shuichi Kawashima, Global existence and decay of solutions for a quasi-linear dissipative plate equation, J. Hyperbolic Differ. Equ., 8 (2011), 591-614.  doi: 10.1142/S0219891611002500. [20] Yongqin Liu and Shuichi Kawashima, Decay property for the Timoshenko system with memory-type dissipation, Math. Models Methods Appl. Sci., 22 (2012), 1150012. doi: 10.1142/S0218202511500126. [21] Yongqin Liu and Yoshihiro Ueda, Decay estimate and asymptotic profile for a plate equation with memory, J. Differ. Equ., 268 (2020), 2435-2463.  doi: 10.1016/j.jde.2019.09.007. [22] Salim A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal., 69 (2008), 2589-2598.  doi: 10.1016/j.na.2007.08.035. [23] Salim A. Messaoudi and Muhammad M. Al-Gharabli, A general stability result for a nonlinear wave equation with infinite memory, Appl. Math. Lett., 26 (2013), 1082-1086.  doi: 10.1016/j.aml.2013.06.002. [24] Sun-Hye Park, Mi Jin Lee and Jum-Ran Kang, Blow-up results for viscoelastic wave equations with weak damping, Appl. Math. Lett., 80 (2018), 20-26.  doi: 10.1016/j.aml.2018.01.002. [25] M. Pellicer and B. Said-Houari, Wellposedness and decay rates for the Cauchy problem of the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Appl. Math. Optim., 80 (2019), 447-478.  doi: 10.1007/s00245-017-9471-8. [26] Reinhard Racke and Yoshihiro Ueda, Dissipative structures for thermoelastic plate equations in ${\mathbb{R}^n}$, Adv. Differ. Equ., 21 (2016), 601–630. [27] J. E. Munoz 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. [28] J. E. Munoz Rivera and M. G. Naso, Asymptotic stability of semigroups associated to weak dissipative systems with memory, J. Math. Anal. Appl., 326 (2007), 691-707.  doi: 10.1016/j.jmaa.2006.03.022. [29] Haitao Song and Chengkui Zhong, Blow-up of solutions of a nonlinear viscoelastic wave equation, Nonlinear Anal. Real World Appl., 11 (2010), 3877-3883.  doi: 10.1016/j.nonrwa.2010.02.015. [30] Yousuke Sugitani and Shuichi Kawashima, Decay estimates of solutions to a semilinear dissipative plate equation, J. Hyperbolic Differ. Equ., 7 (2010), 471-501.  doi: 10.1142/S0219891610002207. [31] Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, in Applied Mathematical Sciences, Springer-Verlag, New York, 1997. doi: https://link.springer.com/book/10.1007/978-1-4684-0313-8. [32] Yoshihiro Ueda, Optimal decay estimates of a regularity-loss type system with constraint condition, J. Differ. Equ., 264 (2018), 679-701.  doi: 10.1016/j.jde.2017.09.020. [33] Yoshihiro Ueda, Renjun Duan and Shuichi Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal., 205 (2012), 239-266.  doi: 10.1007/s00205-012-0508-5. [34] Shun-Tang Wu, Exponential energy decay of solutions for an integro-differential equation with strong damping, J. Math. Anal. Appl., 364 (2010), 609-617.  doi: 10.1016/j.jmaa.2009.11.046. [35] Zhifeng Yang and Zhaogang Gong, Blow-up solutions for viscoelastic equations of Kirchhoff type with arbitrary positive initial energy, Electron. J. Differ. Equ., 2016 (2016), 1–12. [36] Songmu Zheng, Nonlinear Evolution Equations, in Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: https://www.routledge.com/Nonlinear-Evolution-Equations/Zheng/p/book/9781584884521.
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