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1. | Department of Mathematics, School of Science, Civil Aviation University of China, Tianjin 300300, China |
2. | School of Mathematics, Tianjin University, Tianjin 300072, China |
$X$ |
${x_t} + Ax = f\left( {x, h\left( t \right)} \right), \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {0.1} \right)$ |
$A$ |
$X$ |
$f \in C({X^\alpha } \times X,X)$ |
$h \in C(\mathbb{R},X)$ |
$\mu $ |
$\mu >0$ |
$h$ |
References:
[1] |
L. Arnold,
Random Dynamical Systems, Springer, New York, 1998. |
[2] |
P. Brune and B. Schmalfuss,
Inertial manifolds for stochastic pde with dynamical boundary conditions, Commun. Pure Appl. Anal., 10 (2011), 831-846.
|
[3] |
A. N. Carvalho, J. A. Langa and J. C. Robinson,
Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Appl. Math. Sci. 182, 2013. |
[4] |
A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez,
Characterization of nonautonomous attractors of a perturbed gradient system, J. Differential Equations, 236 (2007), 570-603.
|
[5] |
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.
|
[6] |
T. Caraballo, P. Kloeden and B. Schmalfuss,
Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.
|
[7] |
T. Caraballo, J. A. Langa and Z. X. Liu,
Gradient infinite-dimensional random dynamical system, SIAM J. Appl. Dyn. Syst., 11 (2012), 1817-1847.
|
[8] |
X. Chen and J. Duan,
State space decomposition for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 957-974.
|
[9] |
V. V. Chepyzhov and M. I. Vishik,
Attractors of Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002. |
[10] |
D. Cheban, P. Kloeden and B. Schmalfuss,
The relation between pullback and global attractors for nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 125-144.
|
[11] |
E. R. Arag$\tilde{a}$o-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa,
Stability of gradient semigroups under perturbation, Nonlinearity, 24 (2011), 2099-2117.
|
[12] |
H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
|
[13] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
|
[14] |
G. Da Prato and J. Zabczyk,
Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. |
[15] |
J. Duan, K. Lu and B. Schmalfuss,
Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.
|
[16] |
J. Duan, K. Lu and B. Schmalfuss,
Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.
|
[17] |
J. Duan and W. Wang,
Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, 2014. |
[18] |
J. K. Hale,
Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. 25, Amer. Math. Soc., R. I., 1989. |
[19] |
D. Henry,
Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, 1981. |
[20] |
D. S. Li and X. X. Zhang,
On dynamical properties of general dynamical systems and differential inclusions, J. Appl. Math. Anal. Appl., 274 (2002), 705-724.
|
[21] |
D. S. Li and J. Duan,
Structure of the set of bounded solutions for a class of nonautonomous second-order differential equations, J. Differential Equations, 246 (2009), 1754-1773.
|
[22] |
D. S. Li, Morse theory of attractors via Lyapunov functions, preprint, arXiv: 1003.0305v1. |
[23] |
K. Lu and B. Schmalfuss,
Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492.
|
[24] |
J. C. Robinson,
Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001. |
[25] |
M. I. Vishik, S. V. Zelik and V. V. Chepyzhov,
Regular attractors and nonautonomous perturbations of them, Mat. Sb., 204 (2013), 1-42.
|
show all references
References:
[1] |
L. Arnold,
Random Dynamical Systems, Springer, New York, 1998. |
[2] |
P. Brune and B. Schmalfuss,
Inertial manifolds for stochastic pde with dynamical boundary conditions, Commun. Pure Appl. Anal., 10 (2011), 831-846.
|
[3] |
A. N. Carvalho, J. A. Langa and J. C. Robinson,
Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Appl. Math. Sci. 182, 2013. |
[4] |
A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez,
Characterization of nonautonomous attractors of a perturbed gradient system, J. Differential Equations, 236 (2007), 570-603.
|
[5] |
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.
|
[6] |
T. Caraballo, P. Kloeden and B. Schmalfuss,
Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.
|
[7] |
T. Caraballo, J. A. Langa and Z. X. Liu,
Gradient infinite-dimensional random dynamical system, SIAM J. Appl. Dyn. Syst., 11 (2012), 1817-1847.
|
[8] |
X. Chen and J. Duan,
State space decomposition for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 957-974.
|
[9] |
V. V. Chepyzhov and M. I. Vishik,
Attractors of Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002. |
[10] |
D. Cheban, P. Kloeden and B. Schmalfuss,
The relation between pullback and global attractors for nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 125-144.
|
[11] |
E. R. Arag$\tilde{a}$o-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa,
Stability of gradient semigroups under perturbation, Nonlinearity, 24 (2011), 2099-2117.
|
[12] |
H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
|
[13] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
|
[14] |
G. Da Prato and J. Zabczyk,
Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. |
[15] |
J. Duan, K. Lu and B. Schmalfuss,
Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.
|
[16] |
J. Duan, K. Lu and B. Schmalfuss,
Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.
|
[17] |
J. Duan and W. Wang,
Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, 2014. |
[18] |
J. K. Hale,
Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. 25, Amer. Math. Soc., R. I., 1989. |
[19] |
D. Henry,
Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, 1981. |
[20] |
D. S. Li and X. X. Zhang,
On dynamical properties of general dynamical systems and differential inclusions, J. Appl. Math. Anal. Appl., 274 (2002), 705-724.
|
[21] |
D. S. Li and J. Duan,
Structure of the set of bounded solutions for a class of nonautonomous second-order differential equations, J. Differential Equations, 246 (2009), 1754-1773.
|
[22] |
D. S. Li, Morse theory of attractors via Lyapunov functions, preprint, arXiv: 1003.0305v1. |
[23] |
K. Lu and B. Schmalfuss,
Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492.
|
[24] |
J. C. Robinson,
Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001. |
[25] |
M. I. Vishik, S. V. Zelik and V. V. Chepyzhov,
Regular attractors and nonautonomous perturbations of them, Mat. Sb., 204 (2013), 1-42.
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