# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2021110
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## Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term

 College of Mathematical Sciences, Harbin Engineering University, No. 145 Nantong Street, Harbin 150001, China

* Corresponding author: Yanbing Yang

Received  July 2021 Revised  August 2021 Early access October 2021

Fund Project: The work was supported by the Heilongjiang Postdoctoral Research Start-up Funding Project (No. LBH-Q20013 and No. LBH-Q20086), the National Natural Science Foundation of China (No. 11801114 and No. 11871017) and the Research Funds for the Central Universities

We mainly focus on the asymptotic behavior analysis for certain fourth-order nonlinear wave equations with strain term, nonlinear weak damping term and source term. We establish two theorems on the asymptotic behavior of the solution depending on some conditions related to the relationship among the forced strain term, the nonlinear weak damping term and source terms.

Citation: Chao Yang, Yanbing Yang. Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021110
##### References:
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Phys (Beijing)., 35 (2001), 647-650.  doi: 10.1088/0253-6102/35/6/647.  Google Scholar [29] X.-G. Yang, M. J. D Nascimento and M. L. Pelicer, Uniform attractors for non-autonomous plate equations with $p$-Laplacian perturbation and critical nonlinearities, Discrete Contin. Dyn. Syst., 40 (2020), 1937-1961.  doi: 10.3934/dcds.2020100.  Google Scholar [30] Z.-J. Yang, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations, 187 (2003), 520-540.   Google Scholar

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##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.   Google Scholar [2] L. J. An and A. Peirce, A weakly nonlinear analysis of elasto-plastic-microstructure models, SIAM J. Appl. Math., 55 (1995), 136-155.  doi: 10.1137/S0036139993255327.  Google Scholar [3] L. J. An and A. Peirce, The effect of microstructure on elastic-plastic models, SIAM J. Appl. Math., 54 (1994), 708-730.  doi: 10.1137/S0036139992238498.  Google Scholar [4] V. Barros, C. Nonato and C. Raposo, Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights, Electron. Res. Arch., 28 (2020), 205-220.  doi: 10.3934/era.2020014.  Google Scholar [5] T. Cazenave and Z. Han, Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation, Discrete Contin. Dyn. Syst., 40 (2020), 4801-4819.  doi: 10.3934/dcds.2020202.  Google Scholar [6] G. W. Chen and Z. J. Yang, Existence and non-existence of global solutions for a class of nonlinear wave equations, Math. Meth. Appl. Sci., 23 (2000), 615-631.   Google Scholar [7] H. Di, Y. Shang and J. Yu, Existence and uniform decay estimates for the fourth order wave eqation with nonlinnear boundary damping and interior source, Electron. Res. Arch., 28 (2020), 221-261.  doi: 10.3934/era.2020015.  Google Scholar [8] J. A. Esquivel-Avila, Dynamics around the ground state of a nonlinear evolution equation, Nonlinear Anal., 63 (2005), 331-343.  doi: 10.1016/j.na.2005.02.108.  Google Scholar [9] L. H. Fatori, M. A. Jorge Silva, T. F. Ma and Z. Yang, Long-time behavior of a class of thermoelastic plates with nonlinear strain, J. Differential Equations, 259 (2015), 4831-4862.  doi: 10.1016/j.jde.2015.06.026.  Google Scholar [10] V. Georgiev and G. Todorova, Existence of solutions of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051.  Google Scholar [11] E. H. Gomes Tavares, M. A. Jorge Silva and T. F. Ma, Sharp decay rates for a class of nonlinear viscoelastic plate models, Commun. Contemp. Math., 20 (2018), 1750010, 21 pp. doi: 10.1142/S0219199717500109.  Google Scholar [12] J. Han, R. Xu and Y. Yang, Asymptotic behavior and finite time blow up for damped fourth order nonlinear evolution equation, Asymptotic. Anal., 122 (2021), 349-369.  doi: 10.3233/ASY-201621.  Google Scholar [13] M. A. Jorge Silva and T. F. Ma, On a viscoelastic plate equation with history setting and perturbation of $p$-Laplacian type, IMA J. Appl. Math., 78 (2013), 1130-1146.  doi: 10.1093/imamat/hxs011.  Google Scholar [14] V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Research in Applied Mathematics, Masson, Paris, France, 1994.  Google Scholar [15] J. Kreulich, Asymptotic behavior of evolution systems in arbitrary Banach spaces using general almost periodic splittings, Adv. Nonlinear Anal., 8 (2019), 1-28.  doi: 10.1515/anona-2016-0075.  Google Scholar [16] W. Lian, V. D. Rădulescu, R. Xu, Y. Yang and N. Zhao, Global well-posedness for a class of fourth order nonlinear strongly damped wave equations, Adv. Calc. Var., 14 (2021), 589-611.  doi: 10.1515/acv-2019-0039.  Google Scholar [17] W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar [18] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.  Google Scholar [19] Y. Liu, Long-time behavior of a class of viscoelastic plate equations, Electron. Res. Arch., 28 (2020), 311-326.  doi: 10.3934/era.2020018.  Google Scholar [20] Y. Liu and R. Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differential Equations., 244 (2008), 200-228.  doi: 10.1016/j.jde.2007.10.015.  Google Scholar [21] Y. Liu and R. Xu, Fourth order wave equations with nonlinear strain and source terms, J. Math. Anal. Appl., 331 (2007), 585-607.  doi: 10.1016/j.jmaa.2006.09.010.  Google Scholar [22] T. F. Ma and M. L. Pelicer, Attractors for weakly damped beam equations with $p$-Laplacian, Discrete Contin. Dyn. Syst., 2013 (2013), 525-534.  doi: 10.3934/proc.2013.2013.525.  Google Scholar [23] A. Mohammed, V. D. Rădulescu and A. Vitolo, Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness, Adv. Nonlinear Anal., 9 (2020), 39-64.  doi: 10.1515/anona-2018-0134.  Google Scholar [24] T. T. Nguyen, Asymptotic limit and decay estimates for a class of dissipative linear hyperbolic systems in several dimensions, Discrete Contin. Dyn. Syst., 39 (2019), 1651-1684.  doi: 10.3934/dcds.2019073.  Google Scholar [25] F. Shakeri and M. Dehghan, A hybrid Legendre tau method for the solution of a class of nonlinear wave equations with nonlinear dissipative terms, Numer. Methods Partial Differential Equations, 27 (2011), 1055-1071.  doi: 10.1002/num.20569.  Google Scholar [26] J. Shen, Y. Yang, S. Chen and R. Xu, Finite time blow up of fourth order wave equations with nonlinear strain and source terms at high energy level, Internat. J. Math., 24 (2013), 1350043, 8 pp. doi: 10.1142/S0129167X13500432.  Google Scholar [27] Y. Wang and Y. Wang, On the initial-boundary problem for fourth order wave equations with damping, strain and source terms, J. Math. Anal. Appl., 405 (2013), 116-127.  doi: 10.1016/j.jmaa.2013.03.060.  Google Scholar [28] Z.-Y. Yan, Similarity reduction and integrability for the nonlinear wave equations from EPM model, Commun. Theor. Phys (Beijing)., 35 (2001), 647-650.  doi: 10.1088/0253-6102/35/6/647.  Google Scholar [29] X.-G. Yang, M. J. D Nascimento and M. L. Pelicer, Uniform attractors for non-autonomous plate equations with $p$-Laplacian perturbation and critical nonlinearities, Discrete Contin. Dyn. Syst., 40 (2020), 1937-1961.  doi: 10.3934/dcds.2020100.  Google Scholar [30] Z.-J. Yang, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations, 187 (2003), 520-540.   Google Scholar
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