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A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds
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 |
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.
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Asymptotic behavior and changes in phase in one-dimensional nonlinear viscoelasticity, J.Dierential Equations, 44 (1982), 306-341.
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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.
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J. M. Ball,
Global attractors for damped semilinear wave equations, Discrete Continuous Dynam. Systems, 10 (2004), 31-52.
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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. |
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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. |
[8] |
I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, AKTA, Kharkiv, 1999. |
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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. 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. |
[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 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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