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November  2016, 21(9): 3075-3094. doi: 10.3934/dcdsb.2016088

Semilinear stochastic equations with bilinear fractional noise

María J. Garrido-Atienza 1, , Bohdan Maslowski 2, and  Jana  Šnupárková 2,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico , Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
2. 

Charles University in Prague, Faculty of Mathematics and Physics, Sokolovska 83, Prague 8, Czech Republic, Czech Republic

Received  February 2016 Revised  March 2016 Published  October 2016

In the paper, we study existence and uniqueness of solutions to semilinear stochastic evolution systems, driven by a fractional Brownian motion with bilinear noise term, and the long time behavior of solutions to such equations. For this purpose, we study at first the random evolution operator defined by the corresponding bilinear equation which is later used to define the mild solution of the semilinear equation. The mild solution is also shown to be weak in the PDE sense. Furthermore, the asymptotic behavior is investigated by using the Random Dynamical Systems theory. We show that the solution generates a random dynamical system that, under appropriate stability and compactness conditions, possesses a random attractor.
Keywords: fractional Brownian motion, Semilinear stochastic equations, random attractor..
Mathematics Subject Classification: Primary: 60H15; Secondary: 60G22, 37L5.
Citation: María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088
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show all references

References:
[1]

Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

Physica D: Nonlinear Phenomena, 320 (2016), 38-56. doi: 10.1016/j.physd.2016.01.008.  Google Scholar

[3]

Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin, 1977.  Google Scholar

[4]

Discrete Contin. Dyn. Syst., 34 (2014), 79-98. doi: 10.3934/dcds.2014.34.79.  Google Scholar

[5]

Czechoslovak Math. J., 54 (2004), 991-1013. doi: 10.1007/s10587-004-6447-z.  Google Scholar

[6]

Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.  Google Scholar

[7]

Stochastic Process. Appl., 115 (2005), 1357-1383. doi: 10.1016/j.spa.2005.03.011.  Google Scholar

[8]

Stochastics Stochastics Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.  Google Scholar

[9]

SIAM J. Math. Anal., 46 (2014), 2281-2309. doi: 10.1137/130930662.  Google Scholar

[10]

Applied Mathematics and Optimization, 60 (2009), 151-172. doi: 10.1007/s00245-008-9062-9.  Google Scholar

[11]

Discrete and Continuous Dynamical System, Series B, 14 (2010), 473-493. doi: 10.3934/dcdsb.2010.14.473.  Google Scholar

[12]

International Journal of Bifurcation and Chaos, 20 (2010), 2761-2782. doi: 10.1142/S0218127410027349.  Google Scholar

[13]

Journal of Dynamics and Differential Equations, 23 (2011), 671-681. doi: 10.1007/s10884-011-9222-5.  Google Scholar

[14]

C.R. Acad. Sci. Paris, Ser. I, 350 (2012), 299-302. doi: 10.1016/j.crma.2012.02.004.  Google Scholar

[15]

Journal of Differential Equations, 251 (2011), 1225-1253. doi: 10.1016/j.jde.2011.02.013.  Google Scholar

[16]

Cambridge Studies in Advanced Mathematics, 24, Cambridge University Press, Cambridge, 1990.  Google Scholar

[17]

Cambridge Studies in Advanced Mathematics, 100, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511617997.  Google Scholar

[18]

J. Funct. Anal., 202 (2003), 277-305. doi: 10.1016/S0022-1236(02)00065-4.  Google Scholar

[19]

Stochastic Anal. Appl., 22 (2004), 1577-1607. doi: 10.1081/SAP-200029498.  Google Scholar

[20]

B. Maslowski and J. Šnupárková, Stochastic equations with multiplicative fractional noise in Hilbert space,, preprint, ().   Google Scholar

[21]

Collect. Math., 53 (2002), 55-81.  Google Scholar

[22]

Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[23]

Gordon and Breach, 1993.  Google Scholar

[24]

in Int. Seminar on Applied Mathematics Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch), 1992, 185-192. Google Scholar

[25]

in Int. Conf. Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 684-689.  Google Scholar

[26]

Acta Univ. Carolin. Math. Phys., 51 (2010), 49-67.  Google Scholar

[27]

Probab. Theory Relat. Fields, 111 (1998), 333-374. doi: 10.1007/s004400050171.  Google Scholar

[28]

Math. Nachr., 225 (2001), 145-183. doi: 10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.0.CO;2-0.  Google Scholar

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