-
Previous Article
Delay-induced spiking dynamics in integrate-and-fire neurons
- DCDS-B Home
- This Issue
-
Next Article
Density function analysis for a stochastic SEIS epidemic model with non-degenerate diffusion
Limiting behavior of unstable manifolds for spdes in varying phase spaces
1. | School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China |
2. | Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA |
In this paper, we study a class of singularly perturbed stochastic partial differential equations in terms of the phase spaces. We establish the smooth convergence of unstable manifolds of these equations. As an example, we study the stochastic reaction-diffusion equations on thin domains.
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer, New York, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
J. M. Arrieta and E. Santamaría,
Estimates on the distance of inertial manifolds, Discrete Contin. Dyn. Syst., 34 (2014), 3921-3944.
doi: 10.3934/dcds.2014.34.3921. |
[3] |
P. W. Bates, K. Lu and C. Zeng,
Persistence of overflowing manifolds for semiflow, Comm. Pure Appl. Math., 52 (1999), 983-1046.
doi: 10.1002/(SICI)1097-0312(199908)52:8<983::AID-CPA4>3.0.CO;2-O. |
[4] |
P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc., 135 (1998), 645.
doi: 10.1090/memo/0645. |
[5] |
P. W. Bates, K. Lu and C. Zeng,
Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433.
doi: 10.1007/s00222-008-0141-y. |
[6] |
A. Bensoussan and F. Flandoli,
Stochastic inertial manifold, Stochastics Rep., 53 (1995), 13-39.
doi: 10.1080/17442509508833981. |
[7] |
T. Caraballo, I. Chueshov and J. A. Langa,
Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations, Nonlinearity, 18 (2005), 747-767.
doi: 10.1088/0951-7715/18/2/015. |
[8] |
T. Caraballo, I. D. Chueshov and P. E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal. 38 (2007) 1489–1507.
doi: 10.1137/050647281. |
[9] |
S.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite-dimensional spaces, J. Diff. Eqs., 94 (1991) 266–291.
doi: 10.1016/0022-0396(91)90093-O. |
[10] |
T. V. Girya and I. D. Chueshov, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems, Sb. Math., 186 (1995) 29–45.
doi: 10.1070/SM1995v186n01ABEH000002. |
[11] |
I. Chueshov and S. Kuksin, Random kick-forced 3D Navier-Stokes equations in a thin domain, Arch. Ration. Mech. Anal., 188 (2008) 117–153.
doi: 10.1007/s00205-007-0068-2. |
[12] |
I. Chueshov and S. Kuksin, Stochastic 3D Navier-Stokes equations in a thin domain and its $\alpha$-approximation, Phys. D, 237 (2008) 1352–1367.
doi: 10.1016/j.physd.2008.03.012. |
[13] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimension, University Press, Cambridge, 1992.
doi: 10.1017/CBO9780511666223.![]() ![]() |
[14] |
G. Da Prato and A. Debussche, Construction of stochastic inertial manifolds using backward integration, Stochastics Rep., 59 (1996) 305–324.
doi: 10.1080/17442509608834094. |
[15] |
J. Duan, K. Lu and B. Schmalfuß, Invariant manifolds for stochastic partial differential equations, Ann. Prob., 31 (2003) 2109–2135.
doi: 10.1214/aop/1068646380. |
[16] |
J. Duan, K. Lu and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Diff. Eqns., 16 (2004) 949–972.
doi: 10.1007/s10884-004-7830-z. |
[17] |
J. K. Hale and G. Raugel, Reaction-diffusion equation on the thin domain, J. Math. pures et. appl., 71 (1992) 33–95. |
[18] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, New York, 1981. |
[19] |
D. A. Jones, A. M. Stuart and E. S. Titi, Persistence of invariant sets for dissipative evolution equations, J. Math. Anal. Appl., 219 (1998) 479–502.
doi: 10.1006/jmaa.1997.5847. |
[20] |
D. Li, B. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Diff. Eqs., 262 (2017) 1575–1602.
doi: 10.1016/j.jde.2016.10.024. |
[21] |
D. Li, K. Lu, B. Wang, and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018) 187–208.
doi: 10.3934/dcds.2018009. |
[22] |
K. Lu and B. Schmalfuß, Invariant foliations for stochastic partial differential equations, Stoch. Dyn., 8 (2008) 505–518.
doi: 10.1142/S0219493708002421. |
[23] |
K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equations, J. Diff. Eqs., 236 (2007) 460–492.
doi: 10.1016/j.jde.2006.09.024. |
[24] |
S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for stochastic differential equations., The Annals of Probability, 27 (1999) 615–652, .
doi: 10.1214/aop/1022677380. |
[25] |
S. Mohammed, T. Zhang and H. Zhao, The stable manifold theorem for semilinear SPDEs, Memoirs of AMS, , 196 (2008) 1–105. Google Scholar |
[26] |
P. S. Ngiamsunthorn, Invariant manifolds for parabolic equations under perturbation of the domain, Nonlinear Analysis TMA, 80 (2013) 23–48.
doi: 10.1016/j.na.2012.12.001. |
[27] |
M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Diff. Eqs. 173 (2001) 271–320.
doi: 10.1006/jdeq.2000.3917. |
[28] |
M. Prizzi and K. P. Rybakowski, Inertial manifolds on squeezed domains, J. Dynam. Diff. Eqs., 15 (2003) 1–48.
doi: 10.1023/A:1026151910637. |
[29] |
M. Prizzi and K. P. Rybakowski, On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains, Studia Math., 154 (2003) 253–275.
doi: 10.4064/sm154-3-6. |
[30] |
E. Santamaría, Distance of Attractors of Evolutionary Equations, Universidad Complutense de Madrid, Ph.D thesis, 2014. Google Scholar |
[31] |
B. Schmalfuß, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl., 225 (1998) 91–113.
doi: 10.1006/jmaa.1998.6008. |
[32] |
N. Varchon, Domain perturbation and invariant manifolds, J. Evol. Equ., 12 (2012) 547–569.
doi: 10.1007/s00028-012-0144-4. |
[33] |
T. Wanner, Linearization of random dynamical systems, Dynamics Rep., 4 (1995) 203–269. |
show all references
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer, New York, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
J. M. Arrieta and E. Santamaría,
Estimates on the distance of inertial manifolds, Discrete Contin. Dyn. Syst., 34 (2014), 3921-3944.
doi: 10.3934/dcds.2014.34.3921. |
[3] |
P. W. Bates, K. Lu and C. Zeng,
Persistence of overflowing manifolds for semiflow, Comm. Pure Appl. Math., 52 (1999), 983-1046.
doi: 10.1002/(SICI)1097-0312(199908)52:8<983::AID-CPA4>3.0.CO;2-O. |
[4] |
P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc., 135 (1998), 645.
doi: 10.1090/memo/0645. |
[5] |
P. W. Bates, K. Lu and C. Zeng,
Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433.
doi: 10.1007/s00222-008-0141-y. |
[6] |
A. Bensoussan and F. Flandoli,
Stochastic inertial manifold, Stochastics Rep., 53 (1995), 13-39.
doi: 10.1080/17442509508833981. |
[7] |
T. Caraballo, I. Chueshov and J. A. Langa,
Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations, Nonlinearity, 18 (2005), 747-767.
doi: 10.1088/0951-7715/18/2/015. |
[8] |
T. Caraballo, I. D. Chueshov and P. E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal. 38 (2007) 1489–1507.
doi: 10.1137/050647281. |
[9] |
S.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite-dimensional spaces, J. Diff. Eqs., 94 (1991) 266–291.
doi: 10.1016/0022-0396(91)90093-O. |
[10] |
T. V. Girya and I. D. Chueshov, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems, Sb. Math., 186 (1995) 29–45.
doi: 10.1070/SM1995v186n01ABEH000002. |
[11] |
I. Chueshov and S. Kuksin, Random kick-forced 3D Navier-Stokes equations in a thin domain, Arch. Ration. Mech. Anal., 188 (2008) 117–153.
doi: 10.1007/s00205-007-0068-2. |
[12] |
I. Chueshov and S. Kuksin, Stochastic 3D Navier-Stokes equations in a thin domain and its $\alpha$-approximation, Phys. D, 237 (2008) 1352–1367.
doi: 10.1016/j.physd.2008.03.012. |
[13] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimension, University Press, Cambridge, 1992.
doi: 10.1017/CBO9780511666223.![]() ![]() |
[14] |
G. Da Prato and A. Debussche, Construction of stochastic inertial manifolds using backward integration, Stochastics Rep., 59 (1996) 305–324.
doi: 10.1080/17442509608834094. |
[15] |
J. Duan, K. Lu and B. Schmalfuß, Invariant manifolds for stochastic partial differential equations, Ann. Prob., 31 (2003) 2109–2135.
doi: 10.1214/aop/1068646380. |
[16] |
J. Duan, K. Lu and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Diff. Eqns., 16 (2004) 949–972.
doi: 10.1007/s10884-004-7830-z. |
[17] |
J. K. Hale and G. Raugel, Reaction-diffusion equation on the thin domain, J. Math. pures et. appl., 71 (1992) 33–95. |
[18] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, New York, 1981. |
[19] |
D. A. Jones, A. M. Stuart and E. S. Titi, Persistence of invariant sets for dissipative evolution equations, J. Math. Anal. Appl., 219 (1998) 479–502.
doi: 10.1006/jmaa.1997.5847. |
[20] |
D. Li, B. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Diff. Eqs., 262 (2017) 1575–1602.
doi: 10.1016/j.jde.2016.10.024. |
[21] |
D. Li, K. Lu, B. Wang, and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018) 187–208.
doi: 10.3934/dcds.2018009. |
[22] |
K. Lu and B. Schmalfuß, Invariant foliations for stochastic partial differential equations, Stoch. Dyn., 8 (2008) 505–518.
doi: 10.1142/S0219493708002421. |
[23] |
K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equations, J. Diff. Eqs., 236 (2007) 460–492.
doi: 10.1016/j.jde.2006.09.024. |
[24] |
S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for stochastic differential equations., The Annals of Probability, 27 (1999) 615–652, .
doi: 10.1214/aop/1022677380. |
[25] |
S. Mohammed, T. Zhang and H. Zhao, The stable manifold theorem for semilinear SPDEs, Memoirs of AMS, , 196 (2008) 1–105. Google Scholar |
[26] |
P. S. Ngiamsunthorn, Invariant manifolds for parabolic equations under perturbation of the domain, Nonlinear Analysis TMA, 80 (2013) 23–48.
doi: 10.1016/j.na.2012.12.001. |
[27] |
M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Diff. Eqs. 173 (2001) 271–320.
doi: 10.1006/jdeq.2000.3917. |
[28] |
M. Prizzi and K. P. Rybakowski, Inertial manifolds on squeezed domains, J. Dynam. Diff. Eqs., 15 (2003) 1–48.
doi: 10.1023/A:1026151910637. |
[29] |
M. Prizzi and K. P. Rybakowski, On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains, Studia Math., 154 (2003) 253–275.
doi: 10.4064/sm154-3-6. |
[30] |
E. Santamaría, Distance of Attractors of Evolutionary Equations, Universidad Complutense de Madrid, Ph.D thesis, 2014. Google Scholar |
[31] |
B. Schmalfuß, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl., 225 (1998) 91–113.
doi: 10.1006/jmaa.1998.6008. |
[32] |
N. Varchon, Domain perturbation and invariant manifolds, J. Evol. Equ., 12 (2012) 547–569.
doi: 10.1007/s00028-012-0144-4. |
[33] |
T. Wanner, Linearization of random dynamical systems, Dynamics Rep., 4 (1995) 203–269. |
[1] |
Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 |
[2] |
Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020047 |
[3] |
Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324 |
[4] |
Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020317 |
[5] |
Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391 |
[6] |
Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020352 |
[7] |
Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020399 |
[8] |
Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028 |
[9] |
Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020280 |
[10] |
Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-969. doi: 10.3934/dcdss.2019065 |
[11] |
Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 |
[12] |
The Editors. The 2019 Michael Brin Prize in Dynamical Systems. Journal of Modern Dynamics, 2020, 16: 349-350. doi: 10.3934/jmd.2020013 |
[13] |
Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021004 |
[14] |
Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345 |
[15] |
Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129 |
[16] |
Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331 |
[17] |
Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021012 |
[18] |
Stefan Siegmund, Petr Stehlík. Time scale-induced asynchronous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1011-1029. doi: 10.3934/dcdsb.2020151 |
[19] |
Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 |
[20] |
Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121 |
2019 Impact Factor: 1.27
Tools
Article outline
[Back to Top]