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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.
|
[3] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Colloquium Publications, (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.
doi: 10.3934/dcds.2012.32.2079. |
[5] |
I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta Scientific Publishing House, (2002). Google Scholar |
[6] |
I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, J. Dynam. Differential Equations, 16 (2004), 469.
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 (2007), 153.
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.
|
[9] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195 (2008).
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.
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.
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.
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.
doi: 10.1007/BF01193705. |
[14] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307.
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.
|
[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.
doi: 10.1080/03605309308820985. |
[17] |
E. Feireisl, Global attractors for damped wave equations with supercritical exponent,, J. Differential Equations, 116 (1995), 431.
doi: 10.1006/jdeq.1995.1042. |
[18] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems,, AMS, (1988).
|
[19] |
A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear damping and critical exponents,, J. Differential Equations, 230 (2006), 702.
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.
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.
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. Google Scholar |
[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.
doi: 10.1016/S0022-247X(02)00006-9. |
[24] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge University Prss, (1991).
doi: 10.1017/CBO9780511569418. |
[25] |
J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites N-linéaires,, Dunod, (1969).
|
[26] |
I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations,, Discrete Contin. Dyn. Syst., 10 (2004), 473.
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.
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.
doi: 10.3934/dcds.2011.29.141. |
[29] |
V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495.
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.
|
[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. Google Scholar |
[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.
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.
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.
doi: 10.1137/060663805. |
[35] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Second edition, 68 (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.
|
[3] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Colloquium Publications, (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.
doi: 10.3934/dcds.2012.32.2079. |
[5] |
I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta Scientific Publishing House, (2002). Google Scholar |
[6] |
I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, J. Dynam. Differential Equations, 16 (2004), 469.
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 (2007), 153.
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.
|
[9] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195 (2008).
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.
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.
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.
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.
doi: 10.1007/BF01193705. |
[14] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307.
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.
|
[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.
doi: 10.1080/03605309308820985. |
[17] |
E. Feireisl, Global attractors for damped wave equations with supercritical exponent,, J. Differential Equations, 116 (1995), 431.
doi: 10.1006/jdeq.1995.1042. |
[18] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems,, AMS, (1988).
|
[19] |
A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear damping and critical exponents,, J. Differential Equations, 230 (2006), 702.
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.
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.
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. Google Scholar |
[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.
doi: 10.1016/S0022-247X(02)00006-9. |
[24] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge University Prss, (1991).
doi: 10.1017/CBO9780511569418. |
[25] |
J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites N-linéaires,, Dunod, (1969).
|
[26] |
I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations,, Discrete Contin. Dyn. Syst., 10 (2004), 473.
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.
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.
doi: 10.3934/dcds.2011.29.141. |
[29] |
V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495.
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.
|
[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. Google Scholar |
[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.
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.
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.
doi: 10.1137/060663805. |
[35] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Second edition, 68 (1997).
doi: 10.1007/978-1-4612-0645-3. |
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