American Institute of Mathematical Sciences

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

Semilinear stochastic equations with bilinear fractional noise

 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.
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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