December  2021, 14(12): 4643-4658. doi: 10.3934/dcdss.2021110

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 Published  December 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 and Continuous Dynamical Systems - S, 2021, 14 (12) : 4643-4658. doi: 10.3934/dcdss.2021110
References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
[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.

[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.

[4]

V. BarrosC. 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.

[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.

[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. 

[7]

H. DiY. 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.

[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.

[9]

L. H. FatoriM. A. Jorge SilvaT. 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.

[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.

[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.

[12]

J. HanR. 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.

[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.

[14]

V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Research in Applied Mathematics, Masson, Paris, France, 1994.

[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.

[16]

W. LianV. D. RădulescuR. XuY. 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.

[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.

[18]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.

[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.

[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.

[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.

[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.

[23]

A. MohammedV. 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.

[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.

[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.

[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.

[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.

[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.

[29]

X.-G. YangM. 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.

[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. 

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
[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.

[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.

[4]

V. BarrosC. 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.

[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.

[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. 

[7]

H. DiY. 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.

[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.

[9]

L. H. FatoriM. A. Jorge SilvaT. 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.

[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.

[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.

[12]

J. HanR. 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.

[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.

[14]

V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Research in Applied Mathematics, Masson, Paris, France, 1994.

[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.

[16]

W. LianV. D. RădulescuR. XuY. 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.

[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.

[18]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.

[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.

[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.

[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.

[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.

[23]

A. MohammedV. 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.

[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.

[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.

[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.

[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.

[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.

[29]

X.-G. YangM. 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.

[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. 

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