September  2019, 24(9): 4721-4737. doi: 10.3934/dcdsb.2019027

Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping

1. 

School of Mathematics, Southeast University, Nanjing, 211189, China

2. 

Department of Mathematics, College of Science, Hohai University, Nanjing, 210098, China

3. 

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

* Corresponding author: Chengkui Zhong

Received  August 2017 Published  September 2019 Early access  February 2019

Fund Project: Ma was supported by NSFC Grant (No.11801071), Zhang was supported by NSFC Grant (No.11601117) and Zhong was supported by NSFC Grant (No.11731005).

In this paper, we consider the initial boundary problem for the Kirchhoff type wave equation. We prove that the Kirchhoff wave model is globally well-posed in the sufficiently regular space $ (H^2(\Omega)\cap H^1_0(\Omega))\times H^1_0(\Omega) $, then, we also obtain that the semigroup generated by the equation has a global attractor in the corresponding phase space, in the presence of a quite general nonlinearity of supercritical growth.

Citation: Honglv Ma, Jin Zhang, Chengkui Zhong. Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4721-4737. doi: 10.3934/dcdsb.2019027
References:
[1]

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

[2]

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.

[3]

J. W. Cholewa and T. Doltko, Strongly damped wave equation in uniform spaces, Nonlinear Anal. TMA, 64 (2006), 174-187.  doi: 10.1016/j.na.2005.06.021.

[4]

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.

[5]

M. ContiV. Pata and M. Squassina, Strongly damped wave equations on $\mathbb{R}^3$ with critical nonlinearities, Commun. Appl. Anal., 9 (2005), 161-176. 

[6]

X. Fan and S. Zhou, Kernel sections for non-autonomous strongly damped wave equations of non-degenerate Kirchhoff-type, Appl. Math. Comput., 158 (2004), 253-266.  doi: 10.1016/j.amc.2003.08.147.

[7]

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

[8]

M. Ghisi and M. Gobbino, Kirchhoff equations with strong damping, J. Evol. Equ., 16 (2016), 441-482.  doi: 10.1007/s00028-015-0308-0.

[9]

M. Ghisi, Global solutions for dissipative Kirchhoff strings with non-Lipschitz nonlinear term, J. Differential Equations, 230 (2006), 128-139.  doi: 10.1016/j.jde.2006.07.020.

[10]

H. Hashimoto and T. Yamazaki, Hyperbolic-parabolic singular perturbation for quasilinear equations of Kirchhoff type, J. Differential Equations, 237 (2007), 491-525.  doi: 10.1016/j.jde.2007.02.005.

[11]

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.

[12]

G. Kirchhoff, Vorlesungen $\ddot{u}$ber Mechanik, Teubner, Sluttgart, 1883.

[13]

S. Kolbasin, Attractors for Kirchhoff's equation with a nonlinear damping coefficient, Nonlinear Anal., 71 (2009), 2361-2371.  doi: 10.1016/j.na.2009.01.187.

[14]

J. Lions, Quelques M$\acute{e}$thodes de R$\acute{e}$solution des Probl$\grave{e}$mes Aux Limites Non Lin$\acute{e}$aires, Dunod, Paris, 1969.

[15]

Q. MaS. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana University Mathematics Journal, 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.

[16]

H. Ma and C. Zhong, Attractors for the Kirchhoff equations with strong nonlinear damping, Appl. Math. Lett., 74 (2017), 127-133.  doi: 10.1016/j.aml.2017.06.002.

[17]

T. Matsuyama and R. Ikehata, On global solution and energy decay for the wave equation of Kirchhoff type with nonlinear damping term, J. Math. Anal. Appl., 204 (1996), 729-753.  doi: 10.1006/jmaa.1996.0464.

[18]

M. Nakao, An attractor for a nonlinear dissipative wave equation of Kirchhoff type, J. Math. Anal. Appl., 353 (2009), 652-659.  doi: 10.1016/j.jmaa.2008.09.010.

[19]

M. Nakao and Z. Yang, Global attractors for some quasi-linear wave equations with a strong dissipation, Adv. Math. Sci. Appl., 17 (2007), 89-105. 

[20]

K. Nishihara, Degenerate quasilinear hyperbolic equation with strong damping, Funkcial. Ekvac., 27 (1984), 125-145. 

[21]

K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci., 20 (1997), 151-177.  doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.0.CO;2-0.

[22]

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.

[23]

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.

[24]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, USA, 2nd edition, 1997. doi: 10.1007/978-1-4612-0645-3.

[25]

M. Yang and C. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity, Transactions of the American Mathematical Society, 361 (2009), 1069-1101.  doi: 10.1090/S0002-9947-08-04680-1.

[26]

Z. Yang, Long-time behavior of the Kirchhoff type equation with strong damping in $R^N$, J. Differential Equations, 242 (2007), 269-286.  doi: 10.1016/j.jde.2007.08.004.

[27]

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.

[28]

Z. Yang and P. Ding, Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on $R^N$, J. Math. Anal. Appl., 434 (2016), 1826-1851.  doi: 10.1016/j.jmaa.2015.10.013.

[29]

Z. YangP. Ding and Z. Liu, Global attractor for the Kirchhoff type equations with strong nonlinear damping and supercritical nonlinearity, Applied Mathematics Letters, 33 (2014), 12-17.  doi: 10.1016/j.aml.2014.02.014.

[30]

Z. Yang and Y. Wang, Global attractor for the Kirchhoff equation with a strong dissipation, J. Differential Equations, 249 (2010), 3258-3278.  doi: 10.1016/j.jde.2010.09.024.

show all references

References:
[1]

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

[2]

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.

[3]

J. W. Cholewa and T. Doltko, Strongly damped wave equation in uniform spaces, Nonlinear Anal. TMA, 64 (2006), 174-187.  doi: 10.1016/j.na.2005.06.021.

[4]

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.

[5]

M. ContiV. Pata and M. Squassina, Strongly damped wave equations on $\mathbb{R}^3$ with critical nonlinearities, Commun. Appl. Anal., 9 (2005), 161-176. 

[6]

X. Fan and S. Zhou, Kernel sections for non-autonomous strongly damped wave equations of non-degenerate Kirchhoff-type, Appl. Math. Comput., 158 (2004), 253-266.  doi: 10.1016/j.amc.2003.08.147.

[7]

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

[8]

M. Ghisi and M. Gobbino, Kirchhoff equations with strong damping, J. Evol. Equ., 16 (2016), 441-482.  doi: 10.1007/s00028-015-0308-0.

[9]

M. Ghisi, Global solutions for dissipative Kirchhoff strings with non-Lipschitz nonlinear term, J. Differential Equations, 230 (2006), 128-139.  doi: 10.1016/j.jde.2006.07.020.

[10]

H. Hashimoto and T. Yamazaki, Hyperbolic-parabolic singular perturbation for quasilinear equations of Kirchhoff type, J. Differential Equations, 237 (2007), 491-525.  doi: 10.1016/j.jde.2007.02.005.

[11]

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.

[12]

G. Kirchhoff, Vorlesungen $\ddot{u}$ber Mechanik, Teubner, Sluttgart, 1883.

[13]

S. Kolbasin, Attractors for Kirchhoff's equation with a nonlinear damping coefficient, Nonlinear Anal., 71 (2009), 2361-2371.  doi: 10.1016/j.na.2009.01.187.

[14]

J. Lions, Quelques M$\acute{e}$thodes de R$\acute{e}$solution des Probl$\grave{e}$mes Aux Limites Non Lin$\acute{e}$aires, Dunod, Paris, 1969.

[15]

Q. MaS. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana University Mathematics Journal, 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.

[16]

H. Ma and C. Zhong, Attractors for the Kirchhoff equations with strong nonlinear damping, Appl. Math. Lett., 74 (2017), 127-133.  doi: 10.1016/j.aml.2017.06.002.

[17]

T. Matsuyama and R. Ikehata, On global solution and energy decay for the wave equation of Kirchhoff type with nonlinear damping term, J. Math. Anal. Appl., 204 (1996), 729-753.  doi: 10.1006/jmaa.1996.0464.

[18]

M. Nakao, An attractor for a nonlinear dissipative wave equation of Kirchhoff type, J. Math. Anal. Appl., 353 (2009), 652-659.  doi: 10.1016/j.jmaa.2008.09.010.

[19]

M. Nakao and Z. Yang, Global attractors for some quasi-linear wave equations with a strong dissipation, Adv. Math. Sci. Appl., 17 (2007), 89-105. 

[20]

K. Nishihara, Degenerate quasilinear hyperbolic equation with strong damping, Funkcial. Ekvac., 27 (1984), 125-145. 

[21]

K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci., 20 (1997), 151-177.  doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.0.CO;2-0.

[22]

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.

[23]

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.

[24]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, USA, 2nd edition, 1997. doi: 10.1007/978-1-4612-0645-3.

[25]

M. Yang and C. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity, Transactions of the American Mathematical Society, 361 (2009), 1069-1101.  doi: 10.1090/S0002-9947-08-04680-1.

[26]

Z. Yang, Long-time behavior of the Kirchhoff type equation with strong damping in $R^N$, J. Differential Equations, 242 (2007), 269-286.  doi: 10.1016/j.jde.2007.08.004.

[27]

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.

[28]

Z. Yang and P. Ding, Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on $R^N$, J. Math. Anal. Appl., 434 (2016), 1826-1851.  doi: 10.1016/j.jmaa.2015.10.013.

[29]

Z. YangP. Ding and Z. Liu, Global attractor for the Kirchhoff type equations with strong nonlinear damping and supercritical nonlinearity, Applied Mathematics Letters, 33 (2014), 12-17.  doi: 10.1016/j.aml.2014.02.014.

[30]

Z. Yang and Y. Wang, Global attractor for the Kirchhoff equation with a strong dissipation, J. Differential Equations, 249 (2010), 3258-3278.  doi: 10.1016/j.jde.2010.09.024.

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