American Institute of Mathematical Sciences

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doi: 10.3934/dcdsb.2021015

The global attractor for the wave equation with nonlocal strong damping

Biyue Chen 1, , Chunxiang Zhao 1,2, and  Chengkui Zhong 1,,

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  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.

Keywords: Wave equation, nonlocal strong damping, global attractor, subcritical and critical growth exponent.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

F. Chen, B. 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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

S. Gatti, V. 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.  Google Scholar

[16]

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

[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.   Google Scholar

[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.  Google Scholar

[19]

M. A. Jorge Silva, V. 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.  Google Scholar

[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.  Google Scholar

[21]

T. Kobayashi, H. 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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

F. J. Meng, M. 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.  Google Scholar

[25]

Q. F. Ma, S. 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.  Google Scholar

[26]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

Z. J. Yang, P. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

C. X. Zhao, C. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

F. Chen, B. 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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

S. Gatti, V. 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.  Google Scholar

[16]

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

[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.   Google Scholar

[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.  Google Scholar

[19]

M. A. Jorge Silva, V. 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.  Google Scholar

[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.  Google Scholar

[21]

T. Kobayashi, H. 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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

F. J. Meng, M. 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.  Google Scholar

[25]

Q. F. Ma, S. 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.  Google Scholar

[26]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

Z. J. Yang, P. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

C. X. Zhao, C. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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