doi: 10.3934/eect.2022025
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Asymptotic behavior of the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity

1. 

School of Mathematics and Big Data, Anhui University of Science and Technology

2. 

Department of Mathematics, Nanjing University, China

* Corresponding author: Chengkui Zhong

Received  August 2021 Revised  April 2022 Early access May 2022

Fund Project: The work is supported by National Natural Science Foundation of China (No.11731005; No.11801071)

In this paper, we prove the existence of the global attractor for the wave equation with nonlocal weak damping, nonlocal anti-damping and critical nonlinearity.

Citation: Chunyan Zhao, Chengkui Zhong, Zhijun Tang. Asymptotic behavior of the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity. Evolution Equations and Control Theory, doi: 10.3934/eect.2022025
References:
[1]

J. ArrietaA. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations, 17 (1992), 841-866.  doi: 10.1080/03605309208820866.

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992.

[3]

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.

[4]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 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 $\mathbb{R}^3$, Discrete Contin. Dynam. Systems, 7 (2001), 719-735.  doi: 10.3934/dcds.2001.7.719.

[6]

A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310.  doi: 10.2140/pjm.2002.207.287.

[7]

A. N. CarvalhoJ. W. Cholewa and T. Dlotko, Damped wave equations with fast growing dissipative nonlinearities, Discrete Contin. Dyn. Syst., 24 (2009), 1147-1165.  doi: 10.3934/dcds.2009.24.1147.

[8]

M. M. CavalcantiV. N. Domingos CavalcantiM. A. J. Silva and C. M. Webler, Exponential stability for the wave equation with degenerate nonlocal weak damping, Israel J. Math., 219 (2017), 189-213.  doi: 10.1007/s11856-017-1478-y.

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, R.I., 2002.

[10]

I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106. 

[11]

I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262.  doi: 10.1016/j.jde.2011.08.022.

[12]

I. Chueshov, Dynamics of Quasi-stable Dissipative Systems, Springer, Switzerland, 2015. doi: 10.1007/978-3-319-22903-4.

[13]

I. ChueshovM. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132.

[14]

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

[15]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-Posedness and Long Time Dynamics, Springer Science & Business Media, 2010. doi: 10.1007/978-0-387-87712-9.

[16]

M. A. J. da Silva and V. Narciso, Attractors and their properties for a class of nonlocal extensible beams, Discrete Contin. Dyn. Syst., 35 (2015), 985-1008.  doi: 10.3934/dcds.2015.35.985.

[17]

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

[18]

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin 1985. doi: 10.1007/978-3-662-00547-7.

[19]

P. DingZ. Yang and Y. Li, Global attractor of the Kirchhoff wave models with strong nonlinear damping, Appl. Math. Lett., 76 (2018), 40-45.  doi: 10.1016/j.aml.2017.07.008.

[20]

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

[21]

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

[22]

E. Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent, J. Differential Equations, 116 (1995), 431-447.  doi: 10.1006/jdeq.1995.1042.

[23]

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

[24]

J.-M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (1987), 273-319. 

[25]

M. Grasselli and V. Pata, On the damped semilinear wave equation with critical exponent, Discrete Contin. Dyn. Syst., (2003), 351–358.

[26]

J. K. Hale, Functional Differential Equations, Springer-Verlag, New York, 1971.

[27]

J. K. Hale, Asymptotic behaviour and dynamics in infinite dimensions, in Nonlinear Differential Equations, Pitman, Boston, MA, 1985, 1–42.

[28]

J. K. Hale and G. Raugel, Attractors for dissipative evolutionary equations, International Conference on Differential Equations, World Sci. Publ., River Edge, NJ, 1 (1993), 3–22.

[29]

A. Haraux, Two remarks on hyperbolic dissipative problems, Nonlinear Partial Differential Equations and their Applications, 122 (1985), 161-179. 

[30]

A. Haraux, Semi-linear hyperbolic problems in bounded domains, Math. Rep., 3 (1987), 1-281. 

[31]

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

[32]

A. K. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.  doi: 10.1016/j.jmaa.2005.05.031.

[33]

A. K. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations, 230 (2006), 702-719.  doi: 10.1016/j.jde.2006.06.001.

[34] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, 1991.  doi: 10.1017/CBO9780511569418.
[35]

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

[36]

P. D. Lax, Functional Analysis, Wiley-Interscience [John Wiley & Sons], New York, 2002.

[37]

P. P. D. Lazo, Quasi-linear Wave Equation with Damping and Source Terms, Ph.D thesis, Federal University of Rio de Janeiro, Brazil, 1997.

[38]

P. P. D. Lazo, Global solutions for a nonlinear wave equation, Appl. Math. Comput., 200 (2008), 596-601.  doi: 10.1016/j.amc.2007.11.056.

[39]

Y. Li and Z. Yang, Optimal attractors of the Kirchhoff wave model with structural nonlinear damping, J. Differential Equations, 268 (2020), 7741-7773.  doi: 10.1016/j.jde.2019.11.084.

[40]

Q. MaS. Wang and C. Zhong, Necessary and sufficient 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.

[41]

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.

[42]

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

[43]

I. Perai, Multiplicity of Solutions for the p-Laplacian, 1997.

[44]

G. Raugel, Une équation des ondes avec amortissement non linéaire dans le cas critique en dimension trois, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 177-182. 

[45]

G. Raugel, Global attractors in partial differential equations, Handbook of Dynamical Systems, Vol. 2, North-Holland, 2002,885–982. doi: 10.1016/S1874-575X(02)80038-8.

[46]

R. E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, AMS, Providence, 1997. doi: 10.1090/surv/049.

[47]

J. Simon, Régularité de la solution d'une équation non linéaire dans RN, Journées d'Analyse Non Linéaire, Lecture Notes in Math., Springer, Berlin, 665 (1978), 205–227.

[48]

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

[49] C. D. Sogge, Lectures on Non-linear Wave Equations, 2$^{nd}$ edition, International Press, Boston, MA, 2008. 
[50]

C. SunM. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping, J. Differential Equations, 227 (2006), 427-443.  doi: 10.1016/j.jde.2005.09.010.

[51]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[52]

Z. YangP. 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.

[53]

Z. Yang, Z. Liu and 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.

[54]

C. ZhaoC. Zhao and C. Zhong, The global attractor for a class of extensible beams with nonlocal weak damping, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 935-955.  doi: 10.3934/dcdsb.2019197.

[55]

C. Zhao, C. Zhao and C. Zhong, Asymptotic behaviour of the wave equation with nonlocal weak damping and anti-damping, J. Math. Anal. Appl., 490 (2020), 124186, 16 pp. doi: 10.1016/j.jmaa.2020.124186.

show all references

References:
[1]

J. ArrietaA. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations, 17 (1992), 841-866.  doi: 10.1080/03605309208820866.

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992.

[3]

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.

[4]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 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 $\mathbb{R}^3$, Discrete Contin. Dynam. Systems, 7 (2001), 719-735.  doi: 10.3934/dcds.2001.7.719.

[6]

A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310.  doi: 10.2140/pjm.2002.207.287.

[7]

A. N. CarvalhoJ. W. Cholewa and T. Dlotko, Damped wave equations with fast growing dissipative nonlinearities, Discrete Contin. Dyn. Syst., 24 (2009), 1147-1165.  doi: 10.3934/dcds.2009.24.1147.

[8]

M. M. CavalcantiV. N. Domingos CavalcantiM. A. J. Silva and C. M. Webler, Exponential stability for the wave equation with degenerate nonlocal weak damping, Israel J. Math., 219 (2017), 189-213.  doi: 10.1007/s11856-017-1478-y.

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, R.I., 2002.

[10]

I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106. 

[11]

I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262.  doi: 10.1016/j.jde.2011.08.022.

[12]

I. Chueshov, Dynamics of Quasi-stable Dissipative Systems, Springer, Switzerland, 2015. doi: 10.1007/978-3-319-22903-4.

[13]

I. ChueshovM. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132.

[14]

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

[15]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-Posedness and Long Time Dynamics, Springer Science & Business Media, 2010. doi: 10.1007/978-0-387-87712-9.

[16]

M. A. J. da Silva and V. Narciso, Attractors and their properties for a class of nonlocal extensible beams, Discrete Contin. Dyn. Syst., 35 (2015), 985-1008.  doi: 10.3934/dcds.2015.35.985.

[17]

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

[18]

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin 1985. doi: 10.1007/978-3-662-00547-7.

[19]

P. DingZ. Yang and Y. Li, Global attractor of the Kirchhoff wave models with strong nonlinear damping, Appl. Math. Lett., 76 (2018), 40-45.  doi: 10.1016/j.aml.2017.07.008.

[20]

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

[21]

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

[22]

E. Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent, J. Differential Equations, 116 (1995), 431-447.  doi: 10.1006/jdeq.1995.1042.

[23]

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

[24]

J.-M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (1987), 273-319. 

[25]

M. Grasselli and V. Pata, On the damped semilinear wave equation with critical exponent, Discrete Contin. Dyn. Syst., (2003), 351–358.

[26]

J. K. Hale, Functional Differential Equations, Springer-Verlag, New York, 1971.

[27]

J. K. Hale, Asymptotic behaviour and dynamics in infinite dimensions, in Nonlinear Differential Equations, Pitman, Boston, MA, 1985, 1–42.

[28]

J. K. Hale and G. Raugel, Attractors for dissipative evolutionary equations, International Conference on Differential Equations, World Sci. Publ., River Edge, NJ, 1 (1993), 3–22.

[29]

A. Haraux, Two remarks on hyperbolic dissipative problems, Nonlinear Partial Differential Equations and their Applications, 122 (1985), 161-179. 

[30]

A. Haraux, Semi-linear hyperbolic problems in bounded domains, Math. Rep., 3 (1987), 1-281. 

[31]

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

[32]

A. K. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.  doi: 10.1016/j.jmaa.2005.05.031.

[33]

A. K. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations, 230 (2006), 702-719.  doi: 10.1016/j.jde.2006.06.001.

[34] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, 1991.  doi: 10.1017/CBO9780511569418.
[35]

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

[36]

P. D. Lax, Functional Analysis, Wiley-Interscience [John Wiley & Sons], New York, 2002.

[37]

P. P. D. Lazo, Quasi-linear Wave Equation with Damping and Source Terms, Ph.D thesis, Federal University of Rio de Janeiro, Brazil, 1997.

[38]

P. P. D. Lazo, Global solutions for a nonlinear wave equation, Appl. Math. Comput., 200 (2008), 596-601.  doi: 10.1016/j.amc.2007.11.056.

[39]

Y. Li and Z. Yang, Optimal attractors of the Kirchhoff wave model with structural nonlinear damping, J. Differential Equations, 268 (2020), 7741-7773.  doi: 10.1016/j.jde.2019.11.084.

[40]

Q. MaS. Wang and C. Zhong, Necessary and sufficient 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.

[41]

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.

[42]

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

[43]

I. Perai, Multiplicity of Solutions for the p-Laplacian, 1997.

[44]

G. Raugel, Une équation des ondes avec amortissement non linéaire dans le cas critique en dimension trois, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 177-182. 

[45]

G. Raugel, Global attractors in partial differential equations, Handbook of Dynamical Systems, Vol. 2, North-Holland, 2002,885–982. doi: 10.1016/S1874-575X(02)80038-8.

[46]

R. E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, AMS, Providence, 1997. doi: 10.1090/surv/049.

[47]

J. Simon, Régularité de la solution d'une équation non linéaire dans RN, Journées d'Analyse Non Linéaire, Lecture Notes in Math., Springer, Berlin, 665 (1978), 205–227.

[48]

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

[49] C. D. Sogge, Lectures on Non-linear Wave Equations, 2$^{nd}$ edition, International Press, Boston, MA, 2008. 
[50]

C. SunM. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping, J. Differential Equations, 227 (2006), 427-443.  doi: 10.1016/j.jde.2005.09.010.

[51]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[52]

Z. YangP. 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.

[53]

Z. Yang, Z. Liu and 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.

[54]

C. ZhaoC. Zhao and C. Zhong, The global attractor for a class of extensible beams with nonlocal weak damping, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 935-955.  doi: 10.3934/dcdsb.2019197.

[55]

C. Zhao, C. Zhao and C. Zhong, Asymptotic behaviour of the wave equation with nonlocal weak damping and anti-damping, J. Math. Anal. Appl., 490 (2020), 124186, 16 pp. doi: 10.1016/j.jmaa.2020.124186.

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