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Random attractors for non-autonomous stochastic wave equations with multiplicative noise
1. | Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801 |
References:
[1] |
L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. |
[2] |
J. M. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent, Communications in Partial Differential Equations, 17 (1992), 841-866.
doi: 10.1080/03605309208820866. |
[3] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992. |
[4] |
J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Continuous Dynamical Systems, 10 (2004), 31-52.
doi: 10.3934/dcds.2004.10.31. |
[5] |
P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[6] |
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. |
[7] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Continuous Dynamical Systems, 21 (2008), 415-443.
doi: 10.3934/dcds.2008.21.415. |
[8] |
T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Continuous Dynamical Systems B, 9 (2008), 525-539.
doi: 10.3934/dcdsb.2008.9.525. |
[9] |
T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513. |
[10] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.
doi: 10.3934/dcdsb.2010.14.439. |
[11] |
T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.
doi: 10.1016/j.na.2011.02.047. |
[12] |
T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Analysis, 11 (2003), 153-201.
doi: 10.1023/A:1022902802385. |
[13] |
A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds, J. Differential Equations, 233 (2007), 622-653.
doi: 10.1016/j.jde.2006.08.009. |
[14] |
A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668.
doi: 10.1016/j.jde.2009.01.007. |
[15] |
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. |
[16] |
I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144.
doi: 10.1080/1468936042000207792. |
[17] |
I. Chueshow, "Monotone Random Systems-Theory and Applications," Lecture Notes in Mathematics, 1779, Springer, Berlin, 2001.
doi: 10.1007/b83277. |
[18] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[19] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[20] |
J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151. |
[21] |
E. Feireisl, Attractors for semilinear damped wave equations on $\mathbb{R}^3$, Nonlinear Analysis TMA, 23 (1994), 187-195.
doi: 10.1016/0362-546X(94)90041-8. |
[22] |
E. Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent, J. Differential Equations, 116 (1995), 431-447.
doi: 10.1006/jdeq.1995.1042. |
[23] |
E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent, Communications in Partial Differential Equations, 18 (1993), 1539-1555.
doi: 10.1080/03605309308820985. |
[24] |
F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. |
[25] |
M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681.
doi: 10.1007/s10884-011-9222-5. |
[26] |
M. J. Garrido-Atienza, A. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388.
doi: 10.1142/S0219493711003358. |
[27] |
M. J. Garrido-Atienza, B. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion, International J. Bifurcation and Chaos, 20 (2010), 2761-2782.
doi: 10.1142/S0218127410027349. |
[28] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, RI, 1988. |
[29] |
J. K. Hale, X. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and PDE's, Math. Comp., 50 (1988), 89-123.
doi: 10.1090/S0025-5718-1988-0917820-X. |
[30] |
J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214.
doi: 10.1016/0022-0396(88)90104-0. |
[31] |
J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete and Continuous Dynamical Systems, 24 (2009), 855-882.
doi: 10.3934/dcds.2009.24.855. |
[32] |
A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations, 230 (2006), 702-719.
doi: 10.1016/j.jde.2006.06.001. |
[33] |
P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A., 463 (2007), 163-181.
doi: 10.1098/rspa.2006.1753. |
[34] |
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. |
[35] |
M. Prizzi and K. P. Rybakowski, Attractors for semilinear damped wave equations on arbitrary unbounded domains, Topol. Methods Nonlinear Anal., 31 (2008), 49-82. |
[36] |
M. Prizzi and K. P. Rybakowski, Attractors for singularly perturbed damped wave equations on unbounded domains, Topol. Methods Nonlinear Anal., 32 (2008), 1-20. |
[37] |
M. Prizzi, Regularity of invariant sets in semilinear damped wave equations, J. Differential Equations, 247 (2009), 3315-3337.
doi: 10.1016/j.jde.2009.08.011. |
[38] |
B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in "International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior," Dresden, (1992), 185-192. |
[39] |
R. Sell and Y. You, "Dynamics of Evolutionary Equations," Springer-Verlag, New York, 2002. |
[40] |
Z. Shen, S. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.
doi: 10.1016/j.jde.2009.10.007. |
[41] |
C. Sun, D. Cao and J. 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. |
[42] |
C. Sun, M. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping, J. Differential Equation, 227 (2006), 427-443.
doi: 10.1016/j.jde.2005.09.010. |
[43] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1997. |
[44] |
B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52.
doi: 10.1016/S0167-2789(98)00304-2. |
[45] |
B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Transactions of American Mathematical Society, 363 (2011), 3639-3663.
doi: 10.1090/S0002-9947-2011-05247-5. |
[46] |
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. |
[47] |
B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, arXiv:1205.4658v1. |
[48] |
B. Wang, Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations, arXiv:1304.4884v1. |
[49] |
B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electronic Journal of Differential Equations, 2009 (2009), 1-18. |
[50] |
Z. Wang, S. Zhou and A. Gu, Random attractors for a stochastic damped wave equation with multiplicative noise on unbounded domains, Nonlinear Analysis, Real World Applications, 12 (2011), 3468-3482.
doi: 10.1016/j.nonrwa.2011.06.008. |
show all references
References:
[1] |
L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. |
[2] |
J. M. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent, Communications in Partial Differential Equations, 17 (1992), 841-866.
doi: 10.1080/03605309208820866. |
[3] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992. |
[4] |
J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Continuous Dynamical Systems, 10 (2004), 31-52.
doi: 10.3934/dcds.2004.10.31. |
[5] |
P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[6] |
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. |
[7] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Continuous Dynamical Systems, 21 (2008), 415-443.
doi: 10.3934/dcds.2008.21.415. |
[8] |
T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Continuous Dynamical Systems B, 9 (2008), 525-539.
doi: 10.3934/dcdsb.2008.9.525. |
[9] |
T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513. |
[10] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.
doi: 10.3934/dcdsb.2010.14.439. |
[11] |
T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.
doi: 10.1016/j.na.2011.02.047. |
[12] |
T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Analysis, 11 (2003), 153-201.
doi: 10.1023/A:1022902802385. |
[13] |
A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds, J. Differential Equations, 233 (2007), 622-653.
doi: 10.1016/j.jde.2006.08.009. |
[14] |
A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668.
doi: 10.1016/j.jde.2009.01.007. |
[15] |
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. |
[16] |
I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144.
doi: 10.1080/1468936042000207792. |
[17] |
I. Chueshow, "Monotone Random Systems-Theory and Applications," Lecture Notes in Mathematics, 1779, Springer, Berlin, 2001.
doi: 10.1007/b83277. |
[18] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[19] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[20] |
J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151. |
[21] |
E. Feireisl, Attractors for semilinear damped wave equations on $\mathbb{R}^3$, Nonlinear Analysis TMA, 23 (1994), 187-195.
doi: 10.1016/0362-546X(94)90041-8. |
[22] |
E. Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent, J. Differential Equations, 116 (1995), 431-447.
doi: 10.1006/jdeq.1995.1042. |
[23] |
E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent, Communications in Partial Differential Equations, 18 (1993), 1539-1555.
doi: 10.1080/03605309308820985. |
[24] |
F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. |
[25] |
M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681.
doi: 10.1007/s10884-011-9222-5. |
[26] |
M. J. Garrido-Atienza, A. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388.
doi: 10.1142/S0219493711003358. |
[27] |
M. J. Garrido-Atienza, B. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion, International J. Bifurcation and Chaos, 20 (2010), 2761-2782.
doi: 10.1142/S0218127410027349. |
[28] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, RI, 1988. |
[29] |
J. K. Hale, X. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and PDE's, Math. Comp., 50 (1988), 89-123.
doi: 10.1090/S0025-5718-1988-0917820-X. |
[30] |
J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214.
doi: 10.1016/0022-0396(88)90104-0. |
[31] |
J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete and Continuous Dynamical Systems, 24 (2009), 855-882.
doi: 10.3934/dcds.2009.24.855. |
[32] |
A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations, 230 (2006), 702-719.
doi: 10.1016/j.jde.2006.06.001. |
[33] |
P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A., 463 (2007), 163-181.
doi: 10.1098/rspa.2006.1753. |
[34] |
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. |
[35] |
M. Prizzi and K. P. Rybakowski, Attractors for semilinear damped wave equations on arbitrary unbounded domains, Topol. Methods Nonlinear Anal., 31 (2008), 49-82. |
[36] |
M. Prizzi and K. P. Rybakowski, Attractors for singularly perturbed damped wave equations on unbounded domains, Topol. Methods Nonlinear Anal., 32 (2008), 1-20. |
[37] |
M. Prizzi, Regularity of invariant sets in semilinear damped wave equations, J. Differential Equations, 247 (2009), 3315-3337.
doi: 10.1016/j.jde.2009.08.011. |
[38] |
B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in "International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior," Dresden, (1992), 185-192. |
[39] |
R. Sell and Y. You, "Dynamics of Evolutionary Equations," Springer-Verlag, New York, 2002. |
[40] |
Z. Shen, S. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.
doi: 10.1016/j.jde.2009.10.007. |
[41] |
C. Sun, D. Cao and J. 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. |
[42] |
C. Sun, M. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping, J. Differential Equation, 227 (2006), 427-443.
doi: 10.1016/j.jde.2005.09.010. |
[43] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1997. |
[44] |
B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52.
doi: 10.1016/S0167-2789(98)00304-2. |
[45] |
B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Transactions of American Mathematical Society, 363 (2011), 3639-3663.
doi: 10.1090/S0002-9947-2011-05247-5. |
[46] |
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. |
[47] |
B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, arXiv:1205.4658v1. |
[48] |
B. Wang, Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations, arXiv:1304.4884v1. |
[49] |
B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electronic Journal of Differential Equations, 2009 (2009), 1-18. |
[50] |
Z. Wang, S. Zhou and A. Gu, Random attractors for a stochastic damped wave equation with multiplicative noise on unbounded domains, Nonlinear Analysis, Real World Applications, 12 (2011), 3468-3482.
doi: 10.1016/j.nonrwa.2011.06.008. |
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