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Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise
1. | Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China, China |
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992. |
[3] |
F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2591-2636.
doi: 10.1142/S0218127410027246. |
[4] |
P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.
doi: 10.1016/j.jde.2008.05.017. |
[5] |
T. Caraballo, J. A. Langa and J. C. Robinson, Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Discrete Contin. Dynam. Systems, 6 (2000), 875-892.
doi: 10.3934/dcds.2000.6.875. |
[6] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. |
[7] |
I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002.
doi: 10.1007/b83277. |
[8] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[9] |
H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems, J. Dynam. Differential Equations, 10 (1998), 449-474.
doi: 10.1023/A:1022605313961. |
[10] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[11] |
A. Debussche, On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491.
doi: 10.1080/07362999708809490. |
[12] |
A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pures Appl., 77 (1998), 967-988.
doi: 10.1016/S0021-7824(99)80001-4. |
[13] |
X. Fan, Random attractor for a damped sine-Gordon equation with white noise, Pacific J. Math., 216 (2004), 63-76.
doi: 10.2140/pjm.2004.216.63. |
[14] |
X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437.
doi: 10.1142/S0129167X08004741. |
[15] |
X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767-793.
doi: 10.1080/07362990600751860. |
[16] |
H. Gao, M. J. Garrido-Atienza and B. Schmalfuss, Random attractors for stochastic evolution equations driven by fractional Brownian motion, SIAM J. Math. Anal., 46 (2014), 2281-2309.
doi: 10.1137/130930662. |
[17] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. |
[18] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
[19] |
J. A. Langa, Finite-dimensional limiting dynamics of random dynamical systems, Dyn. Syst., 18 (2003), 57-68.
doi: 10.1080/1468936031000080812. |
[20] |
J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set, J. Math. Pures Appl., 85 (2006), 269-294.
doi: 10.1016/j.matpur.2005.08.001. |
[21] |
Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.
doi: 10.1016/j.jde.2007.10.009. |
[22] |
T. Sauer, J. A. Yorke and M. Casdagli, Embedology, J. Statist. Phys., 65 (1991), 579-616.
doi: 10.1007/BF01053745. |
[23] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[24] |
P. Walters, Introduction to Ergodic Theory, Springer-Verlag, New York, 2000. |
[25] |
B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.
doi: 10.3934/dcds.2014.34.269. |
[26] |
B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.
doi: 10.1016/j.jde.2012.05.015. |
[27] |
B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbbN^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.
doi: 10.1090/S0002-9947-2011-05247-5. |
[28] |
B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron. J. Differential Equations, 2009 (2009), 1-18. |
[29] |
G. Wang and Y. Tang, Fractal dimension of a random invariant set and applications, J. Appl. Math., (2013), Art. ID 415764, 5 pp. |
[30] |
M. Yang, J. Duan and P. Kloeden, Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise, Nonlinear Anal. Real World Appl., 12 (2011), 464-478.
doi: 10.1016/j.nonrwa.2010.06.032. |
[31] |
S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921-934.
doi: 10.3934/cpaa.2004.3.921. |
[32] |
S. Zhou, F. Yin and Z. Ouyang, Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst., 4 (2005), 883-903.
doi: 10.1137/050623097. |
show all references
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992. |
[3] |
F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2591-2636.
doi: 10.1142/S0218127410027246. |
[4] |
P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.
doi: 10.1016/j.jde.2008.05.017. |
[5] |
T. Caraballo, J. A. Langa and J. C. Robinson, Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Discrete Contin. Dynam. Systems, 6 (2000), 875-892.
doi: 10.3934/dcds.2000.6.875. |
[6] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. |
[7] |
I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002.
doi: 10.1007/b83277. |
[8] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[9] |
H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems, J. Dynam. Differential Equations, 10 (1998), 449-474.
doi: 10.1023/A:1022605313961. |
[10] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[11] |
A. Debussche, On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491.
doi: 10.1080/07362999708809490. |
[12] |
A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pures Appl., 77 (1998), 967-988.
doi: 10.1016/S0021-7824(99)80001-4. |
[13] |
X. Fan, Random attractor for a damped sine-Gordon equation with white noise, Pacific J. Math., 216 (2004), 63-76.
doi: 10.2140/pjm.2004.216.63. |
[14] |
X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437.
doi: 10.1142/S0129167X08004741. |
[15] |
X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767-793.
doi: 10.1080/07362990600751860. |
[16] |
H. Gao, M. J. Garrido-Atienza and B. Schmalfuss, Random attractors for stochastic evolution equations driven by fractional Brownian motion, SIAM J. Math. Anal., 46 (2014), 2281-2309.
doi: 10.1137/130930662. |
[17] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. |
[18] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
[19] |
J. A. Langa, Finite-dimensional limiting dynamics of random dynamical systems, Dyn. Syst., 18 (2003), 57-68.
doi: 10.1080/1468936031000080812. |
[20] |
J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set, J. Math. Pures Appl., 85 (2006), 269-294.
doi: 10.1016/j.matpur.2005.08.001. |
[21] |
Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.
doi: 10.1016/j.jde.2007.10.009. |
[22] |
T. Sauer, J. A. Yorke and M. Casdagli, Embedology, J. Statist. Phys., 65 (1991), 579-616.
doi: 10.1007/BF01053745. |
[23] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[24] |
P. Walters, Introduction to Ergodic Theory, Springer-Verlag, New York, 2000. |
[25] |
B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.
doi: 10.3934/dcds.2014.34.269. |
[26] |
B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.
doi: 10.1016/j.jde.2012.05.015. |
[27] |
B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbbN^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.
doi: 10.1090/S0002-9947-2011-05247-5. |
[28] |
B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron. J. Differential Equations, 2009 (2009), 1-18. |
[29] |
G. Wang and Y. Tang, Fractal dimension of a random invariant set and applications, J. Appl. Math., (2013), Art. ID 415764, 5 pp. |
[30] |
M. Yang, J. Duan and P. Kloeden, Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise, Nonlinear Anal. Real World Appl., 12 (2011), 464-478.
doi: 10.1016/j.nonrwa.2010.06.032. |
[31] |
S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921-934.
doi: 10.3934/cpaa.2004.3.921. |
[32] |
S. Zhou, F. Yin and Z. Ouyang, Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst., 4 (2005), 883-903.
doi: 10.1137/050623097. |
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