In this paper, we study the long-time behaviors of solutions for the Berger equation with white noise. By applying the asymptotic a priori estimates technique and the operator decomposition technique, the existence of random attractors is shown.
SONG An
,
WANG Xuan
. Existence of random attractors for the Berger equation with white noise[J]. Journal of East China Normal University(Natural Science), 2020
, 2020(4)
: 51
-63
.
DOI: 10.3969/j.issn.1000-5641.201911011
[1] BERGER H M. A new approach to the analysis of large deflections of plates [J]. Appl Mech, 1955, 22: 465-472.
[2] DMITRIEVAŽN. On the theory of nonlinear oscillations of thin rectangular plates [J]. Izvestiya Vysshikh Uchebnykh Zavedeniǐ Matematika, 1978(8): 62-66.
[3] BALL J M. Stability theory for an extensible beam [J]. Journal of Differential Equations, 1973, 14(3): 399-418. DOI: 10.1016/0022-0396(73)90056-9.
[4] BIANCHI G, MARZOCCHI A. Asymptotic properties of solutions to semilinear damped equations for elastic beams [J]. Rivista di Matematica della Universitàdi Parma, 1996, 4: 241-258.
[5] SUN C, YANG L, DUAN J. Asymptotic behavior for a semilinear second order evolution equation [J]. Transactions of the American Mathematical Society, 2011, 363(11): 6085-6109. DOI: 10.1090/S0002-9947-2011-05373-0.
[6] CHOW P L. Asymptotic solutions of a nonlinear stochastic beam equation [J]. Discrete and Continuous Dynamical Systems (B), 2012, 6(4): 735-749.
[7] MARZOCCHI A, VUK E. Global attractor for damped semilinear elastic beam equations with memory [J]. Zeitschrift für Angewandte Mathematik und Physik, 2003, 54(2): 224-234. DOI: 10.1007/s000330300002.
[8] SELL G R, YOU Y C. Dynamics of Evolutionary Equations [M]. New York: Springer, 2002.
[9] CHUESHOV I, LASIECKA I. Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping [M]. Rhode Island: American Mathematical Society, 2008.
[10] NABOKA O. Synchronization of nonlinear oscillations of two coupling Berger plates [J]. Nonlinear Analysis Theory Methods and Applications, 2007, 67(4): 1015-1026. DOI: 10.1016/j.na.2006.06.034.
[11] YANG L, WANG X. Existence of attractors for the non-autonomous Berger equation with nonlinear damping [J]. Electronic Journal of Differential Equations, 2017, 2017(278): 1-14.
[12] MENG F J, WU J, ZHAO C. Time-dependent global attractor for extensible Berger equation [J]. Journal of Mathematical Analysis and Applications, 2019, 469(2): 1045-1069. DOI: 10.1016/j.jmaa.2018.09.050.
[13] CRAUEL H, FLANDOLI F. Attractors for random dynamical systems [J]. Probability Theory & Related Fields, 1994, 100(3): 365-393.
[14] CRAUEL H, DEBUSSCHE A, FLANDOLI F. Random attractors [J]. Journal of Dynamics & Differential Equations, 1997, 9(2): 307-341.
[15] ARNOLD L. Random Dynamical Systems [M]. New York: Springer-Verlag, 1998.
[16] CRAURL H. Random point attractors versus random set attractors [J]. Journal of the London Mathematical Society, 2001, 63(2): 413-427. DOI: 10.1017/S0024610700001915.
[17] LI Y R, GUO B L. Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations [J]. Journal of Differential Equations, 2008, 245: 1775-1800. DOI: 10.1016/j.jde.2008.06.031.
[18] PAZY A. Semigroups of Linear Operators and Applications to Partial Differential Equations [M]. New York: Springer-Verlag, 1983.
[19] MA Q, XU L. Random attractors for the extensible suspension bridge equation with white noise [J]. Computers and Mathematics with Applications, 2015, 70(12): 2895-2903. DOI: 10.1016/j.camwa.2015.09.029.
[20] XU L, MA Q Z. Existence of random attractors for the floating beam equation with strong damping and white noise [J]. Boundary Value Problems, 2015, 2015(1): 126. DOI: 10.1186/s13661-015-0391-8.