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Pullback dynamics of a non-autonomous mixture problem in one dimensional solids with nonlinear damping
Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise
1. | School of Mathematics, South China University of Technology, Guangzhou 510640, China |
2. | College of Science, National University of Defense Technology, Changsha 410073, China |
The fBm-driving rough stochastic lattice dynamical system with a general diffusion term is investigated. First, an area element in space of tensor is desired to define the rough path integral using the Chen-equality and fractional calculus. Under certain conditions, the considered equation is proved to possess a unique local mild path-area solution.
References:
[1] |
P. W. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice dynamical systems, Stochastics and Dynamics, 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[2] |
P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internal Journal of Bifurcation and Chaos, 11 (2001) 143–153.
doi: 10.1142/S021812740100203. |
[3] |
P. W. Bates, K. Lu and B. Wang,
Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D: Nonlinear Phenomena, 289 (2014), 32-50.
doi: 10.1016/j.physd.2014.08.004. |
[4] |
H. Bessaih, M. J. Garrido-Atienza, X. Han and B. Schmalfuss,
Stochastic lattice dynamical systems with fractional noise, SIAM Journal on Mathematical Analysis, 49 (2017), 1495-1518.
doi: 10.1137/16M1085504. |
[5] |
T. Caraballo, X. Han, B. Schmalfuss and J. Valero,
Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Analysis-theory Methods & Applications, 130 (2016), 255-278.
doi: 10.1016/j.na.2015.09.025. |
[6] |
T. Caraballo and K. Lu,
Attractors for stochastic lattice dynamical systems with a multiplicative noise, Frontiers of Mathematics in China, 3 (2008), 317-335.
doi: 10.1007/s11464-008-0028-7. |
[7] |
T. Caraballo, F. Morillas and J. Valero,
Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, Journal of Differential Equations and Applications, 17 (2011), 161-184.
doi: 10.1080/10236198.2010.549010. |
[8] |
T. Caraballo, F. Morillas and J. Valero,
Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, Journal of Differential Equations, 253 (2012), 667-693.
doi: 10.1016/j.jde.2012.03.020. |
[9] |
B. Chen,
Some pinching and classification theorems for minimal submanifolds, Archiv der Mathematik, 60 (1993), 568-578.
doi: 10.1007/BF01236084. |
[10] |
Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuss,
Pathwise solutions of SPDEs driven by hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems, Discrete and Continuous Dynamical Systems, 34 (2014), 79-98.
doi: 10.3934/dcds.2014.34.79. |
[11] |
S.-N. Chow and J. Mallet-Paret,
Pattern formation and spatial chaos in lattice dynamical systems-I, IEEE Transactions on Circuits and Systems, 42 (1995), 746-751.
doi: 10.1109/81.473583. |
[12] |
S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, New York, 2003. |
[13] |
M. Garrido-Atienza, K. Lu and B. Schmalfuss, Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3, 1/2]$, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), pp. 2553–2581.
doi: 10.3934/dcdsb.2015.20.2553. |
[14] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuss,
Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H\in (1/3, 1/2]$, SIAM Journal on Applied Dynamical Systems, 15 (2016), 625-654.
doi: 10.1137/15M1030303. |
[15] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Lévy-areas of Ornstein-Uhlenbeck processes in Hilbert-spaces, in Continuous and Distributed Systems II: Theory and Applications (eds. V.A. Sadovnichiy and M.Z. Zgurovsky)
doi: 10.1007/978-3-319-19075-4_10. |
[16] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Pathwise solutions to stochastic partial differential equations driven by fractional Brownian motions with Hurst parameters in $(1/3, 1/2]$, preprint, arXiv: 1205.6735v3. |
[17] |
M. J. Garrido-Atienza, A. Neuenkirch and B. Schmalfus,
Asymptotical stability of differential equations driven by Hölder continuous paths, Journal of Dynamics and Differential Equations, 30 (2018), 1-19.
doi: 10.1007/s10884-017-9574-6. |
[18] |
M. J. Garrido-Atienza and B. Schmalfuß, Ergodicity of the infinite dimensional fractional Brownian motion, Journal of Dynamics and Differential Equations, 23 (2011), pp. 671–681.
doi: 10.1007/s10884-011-9222-5. |
[19] |
A. Gu, Random attractors of stochastic lattice dynamical systems driven by fractional Brownian motions, International Journal of Bifurcation and Chaos, 23 (2013), 1350041.
doi: 10.1142/S0218127413500417. |
[20] |
A. Gu and Y. Li,
Singleton sets random attractor for stochastic FitzHugh-Nagumo lattice equations driven by fractional Brownian motions, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3929-3937.
doi: 10.1016/j.cnsns.2014.04.005. |
[21] |
A. Gu, C. Zeng and Y. Li,
Synchronization of systems with fractional environmental noises on finite lattice, Fractional Calculus and Applied Analysis, 18 (2015), 891-910.
doi: 10.1515/fca-2015-0054. |
[22] |
X. Han,
Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, Journal of Mathematical Analysis and Applications, 376 (2011), 481-493.
doi: 10.1016/j.jmaa.2010.11.032. |
[23] |
X. Han, W. Shen and S. Zhou,
Random attractors for stochastic lattice dynamical systems in weighted spaces, Journal of Differential Equations, 250 (2011), 1235-1266.
doi: 10.1016/j.jde.2010.10.018. |
[24] |
Y. Hu and D. Nualart,
Rough path analysis via fractional calculus, Transactions of the American Mathematical Society, 361 (2009), 2689-2718.
doi: 10.1090/S0002-9947-08-04631-X. |
[25] |
J. Huang,
The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises, Physica D: Nonlinear Phenomena, 233 (2007), 83-94.
doi: 10.1016/j.physd.2007.06.008. |
[26] |
Y. Lv and J. Sun,
Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Physica D: Nonlinear Phenomena, 221 (2006), 157-169.
doi: 10.1016/j.physd.2006.07.02. |
[27] |
Y. Lv and J. Sun,
Dynamical behavior for stochastic lattice systems, Chaos, Solitons & Fractals, 27 (2006), 1080-1090.
doi: 10.1016/j.chaos.2005.04.089. |
[28] |
J. Mallet-Paret and S.-N. Chow,
Pattern formation and spatial chaos in lattice dynamical systems-II, IEEE Transactions on Circuits and Systems, 42 (1995), 752-756.
doi: 10.1109/81.473583. |
[29] |
S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. |
[30] |
B. Wang,
Dynamics of systems on infinite lattices, Journal of Differential Equations, 221 (2006), 224-245.
doi: 10.1016/j.jde.2005.01.003. |
[31] |
X. Wang, S. Li and D. Xu,
Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 483-494.
doi: 10.1016/j.na.2009.06.094. |
[32] |
Y. Wang, J. Xu and P. E. Kloeden,
Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Analysis-Theory Methods & Applications, 135 (2016), 205-222.
doi: 10.1016/j.na.2016.01.020. |
[33] |
S. Zhou,
Attractors for second order lattice dynamical systems, Journal of Differential Equations, 179 (2002), 606-624.
doi: 10.1006/jdeq.2001.4032. |
[34] |
S. Zhou,
Attractors and approximations for lattice dynamical systems, Journal of Differential Equations, 200 (2004), 342-268.
doi: 10.1016/j.jde.2004.02.005. |
[35] |
S. Zhou and W. Shi,
Attractors and dimension of dissipative lattice systems, Journal of Differential Equations, 224 (2006), 172-204.
doi: 10.1016/j.jde.2005.06.024. |
show all references
References:
[1] |
P. W. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice dynamical systems, Stochastics and Dynamics, 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[2] |
P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internal Journal of Bifurcation and Chaos, 11 (2001) 143–153.
doi: 10.1142/S021812740100203. |
[3] |
P. W. Bates, K. Lu and B. Wang,
Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D: Nonlinear Phenomena, 289 (2014), 32-50.
doi: 10.1016/j.physd.2014.08.004. |
[4] |
H. Bessaih, M. J. Garrido-Atienza, X. Han and B. Schmalfuss,
Stochastic lattice dynamical systems with fractional noise, SIAM Journal on Mathematical Analysis, 49 (2017), 1495-1518.
doi: 10.1137/16M1085504. |
[5] |
T. Caraballo, X. Han, B. Schmalfuss and J. Valero,
Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Analysis-theory Methods & Applications, 130 (2016), 255-278.
doi: 10.1016/j.na.2015.09.025. |
[6] |
T. Caraballo and K. Lu,
Attractors for stochastic lattice dynamical systems with a multiplicative noise, Frontiers of Mathematics in China, 3 (2008), 317-335.
doi: 10.1007/s11464-008-0028-7. |
[7] |
T. Caraballo, F. Morillas and J. Valero,
Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, Journal of Differential Equations and Applications, 17 (2011), 161-184.
doi: 10.1080/10236198.2010.549010. |
[8] |
T. Caraballo, F. Morillas and J. Valero,
Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, Journal of Differential Equations, 253 (2012), 667-693.
doi: 10.1016/j.jde.2012.03.020. |
[9] |
B. Chen,
Some pinching and classification theorems for minimal submanifolds, Archiv der Mathematik, 60 (1993), 568-578.
doi: 10.1007/BF01236084. |
[10] |
Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuss,
Pathwise solutions of SPDEs driven by hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems, Discrete and Continuous Dynamical Systems, 34 (2014), 79-98.
doi: 10.3934/dcds.2014.34.79. |
[11] |
S.-N. Chow and J. Mallet-Paret,
Pattern formation and spatial chaos in lattice dynamical systems-I, IEEE Transactions on Circuits and Systems, 42 (1995), 746-751.
doi: 10.1109/81.473583. |
[12] |
S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, New York, 2003. |
[13] |
M. Garrido-Atienza, K. Lu and B. Schmalfuss, Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3, 1/2]$, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), pp. 2553–2581.
doi: 10.3934/dcdsb.2015.20.2553. |
[14] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuss,
Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H\in (1/3, 1/2]$, SIAM Journal on Applied Dynamical Systems, 15 (2016), 625-654.
doi: 10.1137/15M1030303. |
[15] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Lévy-areas of Ornstein-Uhlenbeck processes in Hilbert-spaces, in Continuous and Distributed Systems II: Theory and Applications (eds. V.A. Sadovnichiy and M.Z. Zgurovsky)
doi: 10.1007/978-3-319-19075-4_10. |
[16] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Pathwise solutions to stochastic partial differential equations driven by fractional Brownian motions with Hurst parameters in $(1/3, 1/2]$, preprint, arXiv: 1205.6735v3. |
[17] |
M. J. Garrido-Atienza, A. Neuenkirch and B. Schmalfus,
Asymptotical stability of differential equations driven by Hölder continuous paths, Journal of Dynamics and Differential Equations, 30 (2018), 1-19.
doi: 10.1007/s10884-017-9574-6. |
[18] |
M. J. Garrido-Atienza and B. Schmalfuß, Ergodicity of the infinite dimensional fractional Brownian motion, Journal of Dynamics and Differential Equations, 23 (2011), pp. 671–681.
doi: 10.1007/s10884-011-9222-5. |
[19] |
A. Gu, Random attractors of stochastic lattice dynamical systems driven by fractional Brownian motions, International Journal of Bifurcation and Chaos, 23 (2013), 1350041.
doi: 10.1142/S0218127413500417. |
[20] |
A. Gu and Y. Li,
Singleton sets random attractor for stochastic FitzHugh-Nagumo lattice equations driven by fractional Brownian motions, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3929-3937.
doi: 10.1016/j.cnsns.2014.04.005. |
[21] |
A. Gu, C. Zeng and Y. Li,
Synchronization of systems with fractional environmental noises on finite lattice, Fractional Calculus and Applied Analysis, 18 (2015), 891-910.
doi: 10.1515/fca-2015-0054. |
[22] |
X. Han,
Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, Journal of Mathematical Analysis and Applications, 376 (2011), 481-493.
doi: 10.1016/j.jmaa.2010.11.032. |
[23] |
X. Han, W. Shen and S. Zhou,
Random attractors for stochastic lattice dynamical systems in weighted spaces, Journal of Differential Equations, 250 (2011), 1235-1266.
doi: 10.1016/j.jde.2010.10.018. |
[24] |
Y. Hu and D. Nualart,
Rough path analysis via fractional calculus, Transactions of the American Mathematical Society, 361 (2009), 2689-2718.
doi: 10.1090/S0002-9947-08-04631-X. |
[25] |
J. Huang,
The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises, Physica D: Nonlinear Phenomena, 233 (2007), 83-94.
doi: 10.1016/j.physd.2007.06.008. |
[26] |
Y. Lv and J. Sun,
Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Physica D: Nonlinear Phenomena, 221 (2006), 157-169.
doi: 10.1016/j.physd.2006.07.02. |
[27] |
Y. Lv and J. Sun,
Dynamical behavior for stochastic lattice systems, Chaos, Solitons & Fractals, 27 (2006), 1080-1090.
doi: 10.1016/j.chaos.2005.04.089. |
[28] |
J. Mallet-Paret and S.-N. Chow,
Pattern formation and spatial chaos in lattice dynamical systems-II, IEEE Transactions on Circuits and Systems, 42 (1995), 752-756.
doi: 10.1109/81.473583. |
[29] |
S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. |
[30] |
B. Wang,
Dynamics of systems on infinite lattices, Journal of Differential Equations, 221 (2006), 224-245.
doi: 10.1016/j.jde.2005.01.003. |
[31] |
X. Wang, S. Li and D. Xu,
Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 483-494.
doi: 10.1016/j.na.2009.06.094. |
[32] |
Y. Wang, J. Xu and P. E. Kloeden,
Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Analysis-Theory Methods & Applications, 135 (2016), 205-222.
doi: 10.1016/j.na.2016.01.020. |
[33] |
S. Zhou,
Attractors for second order lattice dynamical systems, Journal of Differential Equations, 179 (2002), 606-624.
doi: 10.1006/jdeq.2001.4032. |
[34] |
S. Zhou,
Attractors and approximations for lattice dynamical systems, Journal of Differential Equations, 200 (2004), 342-268.
doi: 10.1016/j.jde.2004.02.005. |
[35] |
S. Zhou and W. Shi,
Attractors and dimension of dissipative lattice systems, Journal of Differential Equations, 224 (2006), 172-204.
doi: 10.1016/j.jde.2005.06.024. |
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