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The existence and properties of the solution of a class of nonlinear differential equations with switching at variable times
Asymptotic $ H^2$ regularity of a stochastic reaction-diffusion equation
1. | School of Mathematics and Statistics, and Hubei Key Laboratory of Engineering Modeling and Science Computing, Huazhong University of Science & Technology, Wuhan 430074, China |
2. | School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
$ {\mathcal{O}} \subset {\mathbb{R}}^N $ |
$ N \leqslant 3 $ |
$ \begin{equation} \nonumber\begin{aligned} { {{\rm{d}}} u } +(-\Delta u + u ^3- \beta u ) {{\rm{d}}} t = g(x) {{\rm{d}}} t+h(x) {{\rm{d}}} W , \quad u|_{t = 0} = u_0\in H: = L^2( {\mathcal{O}}), \end{aligned} \end{equation} $ |
$ \beta>0 $ |
$ g\in H $ |
$ W $ |
$ h $ |
$ H^2 $ |
$ (H,H^2) $ |
$ H^2 $ |
$ H $ |
$ H^2 $ |
$ H^2 $ |
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7.![]() ![]() ![]() |
[2] |
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. |
[3] |
T. Caraballo, H. Crauel, J. A. Langa and J. C. Robinson,
The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proc. Amer. Math. Soc., 135 (2007), 373-382.
doi: 10.1090/S0002-9939-06-08593-5. |
[4] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[5] |
H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[6] |
H. Cui, A. C. Cunha and J. A. Langa, Finite-dimensionality of tempered random uniform attractors, Journal of Nonlinear Science, in press. |
[7] |
H. Cui, J. A. Langa and Y. Li,
Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness, Nonlinear Anal., 140 (2016), 208-235.
doi: 10.1016/j.na.2016.03.012. |
[8] |
H. Cui, J. A. Langa and Y. Li,
Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.
doi: 10.1007/s10884-017-9617-z. |
[9] |
H. Cui, Y. Li and J. Yin,
Long time behavior of stochastic MHD equations perturbed by multiplicative noises, J. Appl. Anal. Comput., 6 (2016), 1081-1104.
doi: 10.11948/2016071. |
[10] |
H. Cui, Y. Li and J. Yin,
Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324.
doi: 10.1016/j.na.2015.08.009. |
[11] |
A. Debussche,
On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491.
doi: 10.1080/07362999708809490. |
[12] |
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. |
[13] |
Y. Li, A. Gu and J. Li,
Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.
doi: 10.1016/j.jde.2014.09.021. |
[14] |
J. Mallet-Paret,
Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, J. Differential Equations, 22 (1976), 331-348.
doi: 10.1016/0022-0396(76)90032-2. |
[15] |
R. Mañé, On the Dimension of the Compact Invariant Sets of Certain Non-Linear Maps, in Dynamical Systems and Turbulence, Warwick 1980, D. Rand and L. -S. Young, eds., Berlin, Heidelberg, 1981, Springer Berlin Heidelberg, pp. 230–242. |
[16] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, vol. 28, Cambridge University Press, 2001. |
[17] |
J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge University Press, 2011.
![]() ![]() |
[18] |
A. Shirikyan and S. Zelik,
Exponential attractors for random dynamical systems and applications, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 241-281.
doi: 10.1007/s40072-013-0007-1. |
[19] |
C. Sun,
Asymptotic regularity for some dissipative equations, J. Differential Equations, 248 (2010), 342-362.
doi: 10.1016/j.jde.2009.08.007. |
[20] |
W. Zhao,
Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203.
doi: 10.1016/j.jmaa.2017.06.025. |
[21] |
C.-K. Zhong, M.-H. Yang and C.-Y. Sun,
The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399.
doi: 10.1016/j.jde.2005.06.008. |
[22] |
S. Zhou, Y. Tian and Z. Wang,
Fractal dimension of random attractors for stochastic non-autonomous reaction-diffusion equations, Appl. Math. Comput., 276 (2016), 80-95.
doi: 10.1016/j.amc.2015.12.009. |
show all references
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7.![]() ![]() ![]() |
[2] |
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. |
[3] |
T. Caraballo, H. Crauel, J. A. Langa and J. C. Robinson,
The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proc. Amer. Math. Soc., 135 (2007), 373-382.
doi: 10.1090/S0002-9939-06-08593-5. |
[4] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[5] |
H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[6] |
H. Cui, A. C. Cunha and J. A. Langa, Finite-dimensionality of tempered random uniform attractors, Journal of Nonlinear Science, in press. |
[7] |
H. Cui, J. A. Langa and Y. Li,
Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness, Nonlinear Anal., 140 (2016), 208-235.
doi: 10.1016/j.na.2016.03.012. |
[8] |
H. Cui, J. A. Langa and Y. Li,
Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.
doi: 10.1007/s10884-017-9617-z. |
[9] |
H. Cui, Y. Li and J. Yin,
Long time behavior of stochastic MHD equations perturbed by multiplicative noises, J. Appl. Anal. Comput., 6 (2016), 1081-1104.
doi: 10.11948/2016071. |
[10] |
H. Cui, Y. Li and J. Yin,
Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324.
doi: 10.1016/j.na.2015.08.009. |
[11] |
A. Debussche,
On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491.
doi: 10.1080/07362999708809490. |
[12] |
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. |
[13] |
Y. Li, A. Gu and J. Li,
Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.
doi: 10.1016/j.jde.2014.09.021. |
[14] |
J. Mallet-Paret,
Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, J. Differential Equations, 22 (1976), 331-348.
doi: 10.1016/0022-0396(76)90032-2. |
[15] |
R. Mañé, On the Dimension of the Compact Invariant Sets of Certain Non-Linear Maps, in Dynamical Systems and Turbulence, Warwick 1980, D. Rand and L. -S. Young, eds., Berlin, Heidelberg, 1981, Springer Berlin Heidelberg, pp. 230–242. |
[16] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, vol. 28, Cambridge University Press, 2001. |
[17] |
J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge University Press, 2011.
![]() ![]() |
[18] |
A. Shirikyan and S. Zelik,
Exponential attractors for random dynamical systems and applications, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 241-281.
doi: 10.1007/s40072-013-0007-1. |
[19] |
C. Sun,
Asymptotic regularity for some dissipative equations, J. Differential Equations, 248 (2010), 342-362.
doi: 10.1016/j.jde.2009.08.007. |
[20] |
W. Zhao,
Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203.
doi: 10.1016/j.jmaa.2017.06.025. |
[21] |
C.-K. Zhong, M.-H. Yang and C.-Y. Sun,
The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399.
doi: 10.1016/j.jde.2005.06.008. |
[22] |
S. Zhou, Y. Tian and Z. Wang,
Fractal dimension of random attractors for stochastic non-autonomous reaction-diffusion equations, Appl. Math. Comput., 276 (2016), 80-95.
doi: 10.1016/j.amc.2015.12.009. |
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