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December  2021, 26(12): 6207-6228. doi: 10.3934/dcdsb.2021015

The global attractor for the wave equation with nonlocal strong damping

1. 

Department of Mathematics, Nanjing University, Nanjing, 210093, China

2. 

Institute of Applied System Analysis, Jiangsu University, Zhenjiang, 212013, China

* Corresponding author: ckzhong@nju.edu.cn

Received  December 2019 Revised  October 2020 Published  December 2021 Early access  January 2021

Fund Project: The first author is supported by NSFC(11731005)

The paper is devoted to establishing the long-time behavior of solutions for the wave equation with nonlocal strong damping: $ u_{tt}-\Delta u-\|\nabla u_{t}\|^{p}\Delta u_{t}+f(u) = h(x). $ It proves the well-posedness by means of the monotone operator theory and the existence of a global attractor when the growth exponent of the nonlinearity $ f(u) $ is up to the subcritical and critical cases in natural energy space.

Citation: Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6207-6228. doi: 10.3934/dcdsb.2021015
References:
[1]

G. Andrews and J. M. Ball, Asymptotic behavior and changes in phase in one-dimensional nonlinear viscoelasticity, J.Dierential Equations, 44 (1982), 306-341.  doi: 10.1016/0022-0396(82)90019-5.

[2]

D. D. Ang and A. P. N. Dinh, Strong solutions of a quasi-linear wave equation with nonlinear damping term, SIAM J. Math. Anal, 19 (1988), 337-347.  doi: 10.1137/0519024.

[3]

F. Aloui, I. Ben Hassen and A. Haraux, Compactness of trajectories to some nonlinear second order evolution equations and applications, J. Math. Pures Appl., 100 (2013), 295–326. doi: 10.1016/j.matpur.2013.01.002.

[4]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Continuous Dynam. Systems, 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.

[5]

V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $R^{3}$, Discrete Continuous Dynam. Systems, 7 (2001), 719-735.  doi: 10.3934/dcds.2001.7.719.

[6]

A. V. Balakrishnan and L. W. Taylor, Distributed Parameter Nonlinear Damping Models for Flight Structures, Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989.

[7]

F. ChenB. Guo and P. Wang, Long time behavior of strongly damped nonlinear wave eqautions, J. Differential Equations, 147 (1998), 231-241.  doi: 10.1006/jdeq.1998.3447.

[8]

I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, AKTA, Kharkiv, 1999.

[9]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Memoirs of AMS, 2008. doi: 10.1090/memo/0912.

[10]

I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping, Commun. Pure Appl. Anal, 11 (2012), 659-674.  doi: 10.3934/cpaa.2012.11.659.

[11]

J. Clements, On the existence and uniqueness of solutions of the equation $u_tt-\frac{\partial \sigma_{i}(u_xi)}{\partial x_{i}}-\Delta u_{t} = f$, Canad. Math. Bull, 18 (1975), 181-187.  doi: 10.4153/CMB-1975-036-1.

[12]

E. Feireisl, Attractors for wave equations with nonlinear dissipation and critical exponent, C. R. Acad. Sci. Paris Sffer. I Math, 315 (1992), 551-555. 

[13]

E. Feireisl, Finite-dimensional asymptotic behavior of some semilinear damped hyperbolic problems, J. Dynam. Differential Equations, 6 (1994), 23-35.  doi: 10.1007/BF02219186.

[14]

J. M. Ghidaglia and A. Marzocchi, Long-time behaviour of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal., 22 (1991), 879-895.  doi: 10.1137/0522057.

[15]

S. GattiV. Pata and S. Zelik, A Gronwall-type lemma with parameter and dissipative estimates for PDEs, Nonlinear Analysis: Theory, Methods Applications Volume., 70 (2009), 2337-2343.  doi: 10.1016/j.na.2008.03.015.

[16]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. doi: 10.1090/surv/025.

[17]

M. A. Jorge Silva and V. Narciso, Long-time behavior for a plate equation with nonlocal weak damping, Differential Integral Equations, 27 (2014), 931-948. 

[18]

M. A. Jorge Silva and V. Narciso, Long-time dynamics for a class of extensible beams with nonlocal nonlinear dampin, Evol. Equ. Control Theory, 6 (2017), 437-470.  doi: 10.3934/eect.2017023.

[19]

M. A. Jorge SilvaV. Narciso and A. Vicente, On a beam model related to flight structures with nonlocal energy damping, Discrete Continuous Dynam. Systems - B., 24 (2019), 3281-3298.  doi: 10.3934/dcdsb.2018320.

[20]

S. Kawashima and Y. Shibata, Global existence and exponential stability of small solutions to nonlinear viscoelasticity, Comm. Math. Phys, 148 (1992), 189-208.  doi: 10.1007/BF02102372.

[21]

T. KobayashiH. Pecher and Y. Shibata, On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity, Math. Ann., 296 (1993), 215-234.  doi: 10.1007/BF01445103.

[22]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations., 247 (2009), 1120-1155.  doi: 10.1016/j.jde.2009.04.010.

[23]

H. Lange and G. P. Menzala, Rates of decay of a nonlocal beam equation, Differential Integral Equations., 10 (1997), 1075-1092. 

[24]

F. J. MengM. H. Yang and C. K. Zhong, Attractors for wave equations with nonlinear damping on timedependent space, Discrete Continuous Dynam. Systems., 21 (2016), 205-225.  doi: 10.3934/dcdsb.2016.21.205.

[25]

Q. F. MaS. H. Wang and C. K. Zhong, Necessary and suffcient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.

[26]

M. Nakao, Energy decay for the quasi-linear wave equation with viscosity, Math. Z., 219 (1995), 289-299.  doi: 10.1007/BF02572366.

[27]

V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533.  doi: 10.1007/s00220-004-1233-1.

[28]

I. Perai, Multiplicity of Solutions for the p-Laplacian, Second School of Nonlinear Functional Analysis and Applications to Differential Equations 21 April-9 May, 1997.

[29]

V. Pata and S. Zelik, Smooth attractor for strongly damped wave equation, Noninearity, 19 (2006), 1495-1506.  doi: 10.1088/0951-7715/19/7/001.

[30]

J. Simon, Compact sets in the space $L_{p}(0, T;B)$, Ann. Mat. Pure Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[31]

R. E. Showalter, Monotone Operator in Banach Spaces and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49. AMS, Providence, RI, 1997. doi: 10.1090/surv/049.

[32]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^nd$ edition, Applied Mathematical Sciences, 68, SpringerVerlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[33]

Z. J. YangP. Y. Ding and L. Li, Longtime dynamics of the Kirchhoff equations with fractional damping and supercritical nonlinearity, J. Math. Anal. Appl., 442 (2016), 485-510.  doi: 10.1016/j.jmaa.2016.04.079.

[34]

Z. J. Yang, Z. M. Liu and P. P. Niu, Exponential attractor for the wave equation with structural damping and supercritical exponent, Commun. Contemp. Math., 18 (2016), 1550055, 13 pp. doi: 10.1142/S0219199715500558.

[35]

Z. J. Yang and Z. M. Liu, Global attractor of the quasi-linear wave equation with strong damping, J. Math. Anal. Appl., 458 (2018), 1292-1306.  doi: 10.1016/j.jmaa.2017.10.021.

[36]

C. X. ZhaoC. Y. Zhao and C. K. Zhong, The global attractor for a class of extensible beams with nonlocal weak damping, Discrete Continuous Dynam. Systems-B, 25 (2020), 935-955.  doi: 10.3934/dcdsb.2019197.

[37]

C. Y. Zhao, C. X. Zhao and C. K. Zhong, Asymptotic behaviour of the wave equation with nonlocal weak damping and anti-damping., J. Math. Anal. Appl., 490 (2020), 124186, 10 pp. doi: 10.1016/j.jmaa.2020.124186.

[38]

C. Zhao, S. Ma and C. Zhong, Long-time behavior for a class of extensible beams with nonlocal weak damping and critical nonlinearity, J. Math. Phys., 61 (2020), 032701, 15 pp. doi: 10.1063/1.5128686.

show all references

References:
[1]

G. Andrews and J. M. Ball, Asymptotic behavior and changes in phase in one-dimensional nonlinear viscoelasticity, J.Dierential Equations, 44 (1982), 306-341.  doi: 10.1016/0022-0396(82)90019-5.

[2]

D. D. Ang and A. P. N. Dinh, Strong solutions of a quasi-linear wave equation with nonlinear damping term, SIAM J. Math. Anal, 19 (1988), 337-347.  doi: 10.1137/0519024.

[3]

F. Aloui, I. Ben Hassen and A. Haraux, Compactness of trajectories to some nonlinear second order evolution equations and applications, J. Math. Pures Appl., 100 (2013), 295–326. doi: 10.1016/j.matpur.2013.01.002.

[4]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Continuous Dynam. Systems, 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.

[5]

V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $R^{3}$, Discrete Continuous Dynam. Systems, 7 (2001), 719-735.  doi: 10.3934/dcds.2001.7.719.

[6]

A. V. Balakrishnan and L. W. Taylor, Distributed Parameter Nonlinear Damping Models for Flight Structures, Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989.

[7]

F. ChenB. Guo and P. Wang, Long time behavior of strongly damped nonlinear wave eqautions, J. Differential Equations, 147 (1998), 231-241.  doi: 10.1006/jdeq.1998.3447.

[8]

I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, AKTA, Kharkiv, 1999.

[9]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Memoirs of AMS, 2008. doi: 10.1090/memo/0912.

[10]

I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping, Commun. Pure Appl. Anal, 11 (2012), 659-674.  doi: 10.3934/cpaa.2012.11.659.

[11]

J. Clements, On the existence and uniqueness of solutions of the equation $u_tt-\frac{\partial \sigma_{i}(u_xi)}{\partial x_{i}}-\Delta u_{t} = f$, Canad. Math. Bull, 18 (1975), 181-187.  doi: 10.4153/CMB-1975-036-1.

[12]

E. Feireisl, Attractors for wave equations with nonlinear dissipation and critical exponent, C. R. Acad. Sci. Paris Sffer. I Math, 315 (1992), 551-555. 

[13]

E. Feireisl, Finite-dimensional asymptotic behavior of some semilinear damped hyperbolic problems, J. Dynam. Differential Equations, 6 (1994), 23-35.  doi: 10.1007/BF02219186.

[14]

J. M. Ghidaglia and A. Marzocchi, Long-time behaviour of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal., 22 (1991), 879-895.  doi: 10.1137/0522057.

[15]

S. GattiV. Pata and S. Zelik, A Gronwall-type lemma with parameter and dissipative estimates for PDEs, Nonlinear Analysis: Theory, Methods Applications Volume., 70 (2009), 2337-2343.  doi: 10.1016/j.na.2008.03.015.

[16]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. doi: 10.1090/surv/025.

[17]

M. A. Jorge Silva and V. Narciso, Long-time behavior for a plate equation with nonlocal weak damping, Differential Integral Equations, 27 (2014), 931-948. 

[18]

M. A. Jorge Silva and V. Narciso, Long-time dynamics for a class of extensible beams with nonlocal nonlinear dampin, Evol. Equ. Control Theory, 6 (2017), 437-470.  doi: 10.3934/eect.2017023.

[19]

M. A. Jorge SilvaV. Narciso and A. Vicente, On a beam model related to flight structures with nonlocal energy damping, Discrete Continuous Dynam. Systems - B., 24 (2019), 3281-3298.  doi: 10.3934/dcdsb.2018320.

[20]

S. Kawashima and Y. Shibata, Global existence and exponential stability of small solutions to nonlinear viscoelasticity, Comm. Math. Phys, 148 (1992), 189-208.  doi: 10.1007/BF02102372.

[21]

T. KobayashiH. Pecher and Y. Shibata, On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity, Math. Ann., 296 (1993), 215-234.  doi: 10.1007/BF01445103.

[22]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations., 247 (2009), 1120-1155.  doi: 10.1016/j.jde.2009.04.010.

[23]

H. Lange and G. P. Menzala, Rates of decay of a nonlocal beam equation, Differential Integral Equations., 10 (1997), 1075-1092. 

[24]

F. J. MengM. H. Yang and C. K. Zhong, Attractors for wave equations with nonlinear damping on timedependent space, Discrete Continuous Dynam. Systems., 21 (2016), 205-225.  doi: 10.3934/dcdsb.2016.21.205.

[25]

Q. F. MaS. H. Wang and C. K. Zhong, Necessary and suffcient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.

[26]

M. Nakao, Energy decay for the quasi-linear wave equation with viscosity, Math. Z., 219 (1995), 289-299.  doi: 10.1007/BF02572366.

[27]

V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533.  doi: 10.1007/s00220-004-1233-1.

[28]

I. Perai, Multiplicity of Solutions for the p-Laplacian, Second School of Nonlinear Functional Analysis and Applications to Differential Equations 21 April-9 May, 1997.

[29]

V. Pata and S. Zelik, Smooth attractor for strongly damped wave equation, Noninearity, 19 (2006), 1495-1506.  doi: 10.1088/0951-7715/19/7/001.

[30]

J. Simon, Compact sets in the space $L_{p}(0, T;B)$, Ann. Mat. Pure Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[31]

R. E. Showalter, Monotone Operator in Banach Spaces and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49. AMS, Providence, RI, 1997. doi: 10.1090/surv/049.

[32]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^nd$ edition, Applied Mathematical Sciences, 68, SpringerVerlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[33]

Z. J. YangP. Y. Ding and L. Li, Longtime dynamics of the Kirchhoff equations with fractional damping and supercritical nonlinearity, J. Math. Anal. Appl., 442 (2016), 485-510.  doi: 10.1016/j.jmaa.2016.04.079.

[34]

Z. J. Yang, Z. M. Liu and P. P. Niu, Exponential attractor for the wave equation with structural damping and supercritical exponent, Commun. Contemp. Math., 18 (2016), 1550055, 13 pp. doi: 10.1142/S0219199715500558.

[35]

Z. J. Yang and Z. M. Liu, Global attractor of the quasi-linear wave equation with strong damping, J. Math. Anal. Appl., 458 (2018), 1292-1306.  doi: 10.1016/j.jmaa.2017.10.021.

[36]

C. X. ZhaoC. Y. Zhao and C. K. Zhong, The global attractor for a class of extensible beams with nonlocal weak damping, Discrete Continuous Dynam. Systems-B, 25 (2020), 935-955.  doi: 10.3934/dcdsb.2019197.

[37]

C. Y. Zhao, C. X. Zhao and C. K. Zhong, Asymptotic behaviour of the wave equation with nonlocal weak damping and anti-damping., J. Math. Anal. Appl., 490 (2020), 124186, 10 pp. doi: 10.1016/j.jmaa.2020.124186.

[38]

C. Zhao, S. Ma and C. Zhong, Long-time behavior for a class of extensible beams with nonlocal weak damping and critical nonlinearity, J. Math. Phys., 61 (2020), 032701, 15 pp. doi: 10.1063/1.5128686.

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