-
Previous Article
Realization of arbitrary hysteresis by a low-dimensional gradient flow
- DCDS-B Home
- This Issue
-
Next Article
The optimal mean variance problem with inflation
Attractors for wave equations with nonlinear damping on time-dependent space
1. | School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China |
2. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China |
3. | Department of Mathematics, Nanjing University, Nanjing 210093 |
References:
[1] |
A. N. Carvaho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[2] |
V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333. |
[3] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloquium Publications, vol. 49, American Mathematical Society, Providence, RI, 2002. |
[4] |
V. V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractor, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088.
doi: 10.3934/dcds.2012.32.2079. |
[5] |
I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta Scientific Publishing House, Kharkiv, 2002. |
[6] |
I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512.
doi: 10.1007/s10884-004-4289-x. |
[7] |
I. Chueshov and I. Lasiecka, Long-time dynamics of semilinear wave equation with nonlinear interior-boundary damping and sources of critical exponents, Contemp. Math., 426, 153-192. Amer. Math. Soc., Providence, RI, 2007.
doi: 10.1090/conm/426/08188. |
[8] |
I. Chueshov, I. Lasiecka and D. Toundykov, Long-term dynamics of semilinear wave equations with nonlinear localized interior damping and a source term of critical exponent, Discrete. Contin. Dyn. Syst., 20 (2008), 459-509. |
[9] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp.
doi: 10.1090/memo/0912. |
[10] |
M. Conti, V. Pata and R. Temam, Attractors for processes on time-dependent space. Application to Wave equation, J. Differential Equations, 255 (2013), 1254-1277.
doi: 10.1016/j.jde.2013.05.013. |
[11] |
M. Conti and V. Pata, Asymptotic structure of the attractor for processes on time-dependent spaces, Nonlinear Analysis RWA, 19 (2014), 1-10.
doi: 10.1016/j.nonrwa.2014.02.002. |
[12] |
M. Conti and V. Pata, On the time-dependent cattaneo law in space dimension one, Applied Mathematic and Computation, 259 (2015), 32-44.
doi: 10.1016/j.amc.2015.02.039. |
[13] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[14] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[15] |
E. Feireisl, Attractors for wave equations with nonlinear dissipation and critical exponent, C. R. Acad. Sci. Paris, 315 (1992), 551-555. |
[16] |
E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent, Commun. PDE., 18 (1993), 1539-1555.
doi: 10.1080/03605309308820985. |
[17] |
E. Feireisl, Global attractors for damped wave equations with supercritical exponent, J. Differential Equations, 116 (1995), 431-447.
doi: 10.1006/jdeq.1995.1042. |
[18] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. |
[19] |
A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear damping and critical exponents, J. Differential Equations, 230 (2006), 702-719.
doi: 10.1016/j.jde.2006.06.001. |
[20] |
A. Kh. 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. |
[21] |
A. Kh. Khanmamedov, Remark on the regularity of the global attractor for the wave equation with nonlinear damping, Nonlinear Anal., 72 (2010), 1993-1999.
doi: 10.1016/j.na.2009.09.041. |
[22] |
P. S. Landahl, O. H. Soerensen and P. L. Christiansen, Soliton excitations in Josephson tunnel junctions Phys. Rev.B, 25 (1982), 5737-5348. |
[23] |
I. Lasiecka and A. R. Ruzmaikina, Finite dimensionality and regularity of attractors for 2-D semilinear wave equation with nonlinear dissipation, J. Math. Anal. Appl., 270 (2002), 16-50.
doi: 10.1016/S0022-247X(02)00006-9. |
[24] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Prss, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
[25] |
J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites N-linéaires, Dunod, Paris, 1969. |
[26] |
I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst., 10 (2004), 473-496.
doi: 10.3934/dcds.2004.10.473. |
[27] |
M. Nakao, Global attractors for nonlinear wave equations with nonlinear dissipative terms, J. Differential Equations, 227 (2006), 204-229.
doi: 10.1016/j.jde.2005.09.013. |
[28] |
F. Di Plinio, G. S.Duane and R. Temam, Time dependent attractor for the oscillon equation, Discrete Contin. Dyn. Syst., 29 (2011), 141-167.
doi: 10.3934/dcds.2011.29.141. |
[29] |
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. |
[30] |
G. Raugel, Une equation des ondes avec amortissment non lineaire dans le cas critique en dimensions trois, C. R. Acad. Sci. Paris, 314 (1992), 177-182. |
[31] |
B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics and Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch), Dresden, 73(1992), 185-192. |
[32] |
C. Y. Sun, D. M. Cao and J. Q. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665.
doi: 10.1088/0951-7715/19/11/008. |
[33] |
C. Y. Sun, M. H. Yang and C. K. 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. |
[34] |
C. Y. Sun, D. M. Cao and J. Q. Duan, Uniform attractors for nonautonomous wave equations with nonlinear damping, SIAM J. Appl. Dyn. Syst., 6 (2007), 293-318.
doi: 10.1137/060663805. |
[35] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
show all references
References:
[1] |
A. N. Carvaho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[2] |
V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333. |
[3] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloquium Publications, vol. 49, American Mathematical Society, Providence, RI, 2002. |
[4] |
V. V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractor, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088.
doi: 10.3934/dcds.2012.32.2079. |
[5] |
I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta Scientific Publishing House, Kharkiv, 2002. |
[6] |
I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512.
doi: 10.1007/s10884-004-4289-x. |
[7] |
I. Chueshov and I. Lasiecka, Long-time dynamics of semilinear wave equation with nonlinear interior-boundary damping and sources of critical exponents, Contemp. Math., 426, 153-192. Amer. Math. Soc., Providence, RI, 2007.
doi: 10.1090/conm/426/08188. |
[8] |
I. Chueshov, I. Lasiecka and D. Toundykov, Long-term dynamics of semilinear wave equations with nonlinear localized interior damping and a source term of critical exponent, Discrete. Contin. Dyn. Syst., 20 (2008), 459-509. |
[9] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp.
doi: 10.1090/memo/0912. |
[10] |
M. Conti, V. Pata and R. Temam, Attractors for processes on time-dependent space. Application to Wave equation, J. Differential Equations, 255 (2013), 1254-1277.
doi: 10.1016/j.jde.2013.05.013. |
[11] |
M. Conti and V. Pata, Asymptotic structure of the attractor for processes on time-dependent spaces, Nonlinear Analysis RWA, 19 (2014), 1-10.
doi: 10.1016/j.nonrwa.2014.02.002. |
[12] |
M. Conti and V. Pata, On the time-dependent cattaneo law in space dimension one, Applied Mathematic and Computation, 259 (2015), 32-44.
doi: 10.1016/j.amc.2015.02.039. |
[13] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[14] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[15] |
E. Feireisl, Attractors for wave equations with nonlinear dissipation and critical exponent, C. R. Acad. Sci. Paris, 315 (1992), 551-555. |
[16] |
E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent, Commun. PDE., 18 (1993), 1539-1555.
doi: 10.1080/03605309308820985. |
[17] |
E. Feireisl, Global attractors for damped wave equations with supercritical exponent, J. Differential Equations, 116 (1995), 431-447.
doi: 10.1006/jdeq.1995.1042. |
[18] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. |
[19] |
A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear damping and critical exponents, J. Differential Equations, 230 (2006), 702-719.
doi: 10.1016/j.jde.2006.06.001. |
[20] |
A. Kh. 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. |
[21] |
A. Kh. Khanmamedov, Remark on the regularity of the global attractor for the wave equation with nonlinear damping, Nonlinear Anal., 72 (2010), 1993-1999.
doi: 10.1016/j.na.2009.09.041. |
[22] |
P. S. Landahl, O. H. Soerensen and P. L. Christiansen, Soliton excitations in Josephson tunnel junctions Phys. Rev.B, 25 (1982), 5737-5348. |
[23] |
I. Lasiecka and A. R. Ruzmaikina, Finite dimensionality and regularity of attractors for 2-D semilinear wave equation with nonlinear dissipation, J. Math. Anal. Appl., 270 (2002), 16-50.
doi: 10.1016/S0022-247X(02)00006-9. |
[24] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Prss, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
[25] |
J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites N-linéaires, Dunod, Paris, 1969. |
[26] |
I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst., 10 (2004), 473-496.
doi: 10.3934/dcds.2004.10.473. |
[27] |
M. Nakao, Global attractors for nonlinear wave equations with nonlinear dissipative terms, J. Differential Equations, 227 (2006), 204-229.
doi: 10.1016/j.jde.2005.09.013. |
[28] |
F. Di Plinio, G. S.Duane and R. Temam, Time dependent attractor for the oscillon equation, Discrete Contin. Dyn. Syst., 29 (2011), 141-167.
doi: 10.3934/dcds.2011.29.141. |
[29] |
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. |
[30] |
G. Raugel, Une equation des ondes avec amortissment non lineaire dans le cas critique en dimensions trois, C. R. Acad. Sci. Paris, 314 (1992), 177-182. |
[31] |
B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics and Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch), Dresden, 73(1992), 185-192. |
[32] |
C. Y. Sun, D. M. Cao and J. Q. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665.
doi: 10.1088/0951-7715/19/11/008. |
[33] |
C. Y. Sun, M. H. Yang and C. K. 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. |
[34] |
C. Y. Sun, D. M. Cao and J. Q. Duan, Uniform attractors for nonautonomous wave equations with nonlinear damping, SIAM J. Appl. Dyn. Syst., 6 (2007), 293-318.
doi: 10.1137/060663805. |
[35] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[1] |
Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307 |
[2] |
Xudong Luo, Qiaozhen Ma. The existence of time-dependent attractor for wave equation with fractional damping and lower regular forcing term. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021253 |
[3] |
Tingting Liu, Qiaozhen Ma, Ling Xu. Attractor of the Kirchhoff type plate equation with memory and nonlinear damping on the whole time-dependent space. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022046 |
[4] |
Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141 |
[5] |
A. Kh. Khanmamedov. Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 119-138. doi: 10.3934/dcds.2011.31.119 |
[6] |
Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459 |
[7] |
Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure and Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165 |
[8] |
Qingquan Chang, Dandan Li, Chunyou Sun. Random attractors for stochastic time-dependent damped wave equation with critical exponents. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2793-2824. doi: 10.3934/dcdsb.2020033 |
[9] |
Mohamed Jleli, Bessem Samet. Blow-up for semilinear wave equations with time-dependent damping in an exterior domain. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3885-3900. doi: 10.3934/cpaa.2020143 |
[10] |
Ahmad Z. Fino, Mohamed Ali Hamza. Blow-up of solutions to semilinear wave equations with a time-dependent strong damping. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022006 |
[11] |
Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022 |
[12] |
Fengjuan Meng, Chengkui Zhong. Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 217-230. doi: 10.3934/dcdsb.2014.19.217 |
[13] |
Jun-Ren Luo, Ti-Jun Xiao. Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping. Evolution Equations and Control Theory, 2020, 9 (2) : 359-373. doi: 10.3934/eect.2020009 |
[14] |
Qiwei Wu. Large-time behavior of solutions to the bipolar quantum Euler-Poisson system with critical time-dependent over-damping. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022008 |
[15] |
Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6207-6228. doi: 10.3934/dcdsb.2021015 |
[16] |
Ying Sui, Huimin Yu. Singularity formation for compressible Euler equations with time-dependent damping. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4921-4941. doi: 10.3934/dcds.2021062 |
[17] |
Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351 |
[18] |
Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407 |
[19] |
Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations and Control Theory, 2022, 11 (2) : 515-536. doi: 10.3934/eect.2021011 |
[20] |
Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure and Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]