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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 |
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.
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. Conti, V. 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. Ma, S. 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. Yang, P. 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. Yang, P. 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. Conti, V. 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. Ma, S. 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. Yang, P. 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. Yang, P. 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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