In this paper we study the asymptotic dynamics of the stochastic strongly damped wave equation with multiplicative noise under homogeneous Dirichlet boundary condition. We investigate the existence of a compact random attractor for the random dynamical system associated with the equation.
Lomdahl, P.S., Soerensen, O.H. and Christiansen, P. L. (1982) Soliton Excitations in Josephson Tunnel Junctions. Physical Review B, 25, 5337-5348.
Shen, Z.W., Zhou, S.F. and Shen, W.X. (2010) One-Dimensional Random Attractor and Rotation Number of the Stochastic Damped Sine-Gordon Equation. Journal of Differential Equations, 248, 1432-1457. http://dx.doi.org/10.1016/j.jde.2009.10.007
Chen, F.X., Guo, B.L. and Wang, P. (1998) Long Time Behavior of Strongly Damped Nonlinear Wave Equations. Journal of Differential Equations, 147, 231-241. http://dx.doi.org/10.1006/jdeq.1998.3447
Chow, P.-L. (2002) Stochastic Wave Equation with Polynomial Nonlinearity. The Annals of Applied Probability, 12, 361-381. http://dx.doi.org/10.1214/aoap/1015961168
Ghidaglia, J.M. and Marzocchi, A. (1991) Longtime Behavior of Strongly Damped Nonlinear Wave Equations, Global Attractors and Their Dimension. SIAM Journal on Mathematical Analysis, 22, 879-895. http://dx.doi.org/10.1137/0522057
Kalantarov, V. and Zelik, S. (2009) Finite-Dimensional Attractors for the Quasi-Linear Strongly-Damped Wave Equation. Journal of Differential Equations, 247, 1120-1155. http://dx.doi.org/10.1016/j.jde.2009.04.010
Fan, X.M. (2006) Attractors for a Damped Stochastic Wave Equation of Sine-Gordon Type with Sublinear Multiplicative Noise. Stochastic Analysis and Applications, 24, 767-793. http://dx.doi.org/10.1080/07362990600751860
Fan, X.M. (2004) Random Attractor for a Damped Sine-Gordon Equation with White Noise. Pacific Journal of Mathematics, 216, 63-76. http://dx.doi.org/10.2140/pjm.2004.216.63
Fan, X.M. and Wang, Y. (2007) Fractal Dimensional of Attractors for a Stochastic Wave Equation with Nonlinear Damping and White Noise. Stochastic Analysis and Applications, 25, 381-396. http://dx.doi.org/10.1080/07362990601139602
Fan, X.M. (2008) Random Attractors for Damped Stochastic Wave Equations with Multiplicative Noise. International Journal of Mathematics, 19, 421-437. http://dx.doi.org/10.1142/S0129167X08004741
Li, H.Y. and Zhou, S.F. (2007) One-Dimensional Global Attractor for Strongly Damped Wave Equations. Communications in Nonlinear Science and Numerical Simulation, 12, 784-793. http://dx.doi.org/10.1016/j.cnsns.2005.06.007
Li, H.Y. and Zhou, S.F. (2008) On Non-Autonomous Strongly Damped Wave Equations with a Uniform Attractor and Some Averaging. Journal of Mathematical Analysis and Applications, 341, 791-802. http://dx.doi.org/10.1016/j.jmaa.2007.10.051
Lu, K.N. and Schmalfuß, B. (2007) Invariant Manifolds for Stochastic Wave Equations. Journal of Differential Equations, 236, 460-492. http://dx.doi.org/10.1016/j.jde.2006.09.024
Lv, Y. and Wang, W. (2008) Limiting Dynamics for Stochastic Wave Equations. Journal of Differential Equations, 244, 1-23. http://dx.doi.org/10.1016/j.jde.2007.10.009
Massatt, P. (1983) Limiting Behavior for Strongly Damped Nonlinear Wave Equations. Journal of Differential Equations, 48, 334-349. http://dx.doi.org/10.1016/0022-0396(83)90098-0
Pata, V. and Zelik, S. (2006) Smooth Attractors for Strongly Damped Wave Equations. Nonlinearity, 19, 1495-1506. http://dx.doi.org/10.1088/0951-7715/19/7/001
Temam, R. (1988) Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin. http://dx.doi.org/10.1007/978-1-4684-0313-8
Wang, B.X. and Gao, X.L. (2009) Random Attractors for Wave Equations on Unbounded Domains. Discrete and Continuous Dynamical Systems, Special, 800-809.
Yang, M.H. and Sun, C.Y. (2009) Attractors for Strongly Damped Wave Equations. Nonlinear Analysis: Real World Applications, 10, 1097-1100. http://dx.doi.org/10.1016/j.nonrwa.2007.12.001
Yang, M.H. and Sun, C.Y. (2010) Exponential Attractors for the Strongly Damped Wave Equations. Nonlinear Analysis: Real World Applications, 11, 913-919. http://dx.doi.org/10.1016/j.nonrwa.2009.01.022
Yang, M.H. and Sun, C.Y. (2009) Dynamics of Strongly Damped Wave Equations in Locally Uniform Spaces: Attractors and Asymptotic Regularity. Transactions of the American Mathematical Society, 361, 1069-1101. http://dx.doi.org/10.1090/S0002-9947-08-04680-1
Yang, M.H., Duan, J.Q. and Kloeden, P. (2011) Asymptotic Behavior of Solutions for Random Wave Equations with Nonlinear Damping and White Noise. Nonlinear Analysis: Real World Applications, 12, 464-478. http://dx.doi.org/10.1016/j.nonrwa.2010.06.032
Zhou, S.F. (1999) Dimension of the Global Attractor for Strongly Damped Nonlinear Wave Equation. Journal of Mathematical Analysis and Applications, 233, 102-115. http://dx.doi.org/10.1006/jmaa.1999.6269
Zhou, S.F. (1999) Global Attractor for Strongly Damped Nonlinear Wave Equation. Functional Differential Equations, 6, 451-470.
Zhou, S.F. and Fan, X.M. (2002) Kernel Sections for Non-Autonomous Strongly Damped Wave Equations. Journal of Mathematical Analysis and Applications, 275, 850-869. http://dx.doi.org/10.1016/S0022-247X(02)00437-7
Zhou, S.F. (2003) Attractors for Strongly Damped Wave Equations with Critical Exponent. Applied Mathematics Letters, 16, 1307-1314. http://dx.doi.org/10.1016/S0893-9659(03)90134-0
Zhou, S.F., Yin, F.Q. and Ouyang, Z.G. (2005) Random Attractor for Damped Nonlinear Wave Equations with White Noise. SIAM Journal on Applied Dynamical Systems, 4, 883-903. http://dx.doi.org/10.1137/050623097
Arnold, L. (1998) Random Dynamical Systems. Springer-Verlag, New York and Berlin. http://dx.doi.org/10.1007/978-3-662-12878-7
Chueshov, I. (2002) Monotone Random Systems Theory and Applications. Springer-Verlag, New York. http://dx.doi.org/10.1007/b83277
Bates, P.W., Lu, K.N. and Wang, B.X. (2009) Random Attractors for Stochastic Reaction-Diffusion Equations on Unbounded Domains. Journal of Differential Equations, 246, 845-869. http://dx.doi.org/10.1016/j.jde.2008.05.017
Duan, J.Q., Lu, K.N. and Schmalfuss, B. (2003) Invariant Manifolds for Stochastic Partial Differential Equations. The Annals of Probability, 31, 2109-2135. http://dx.doi.org/10.1214/aop/1068646380
Pazy, A. (1983) Semigroup of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York. http://dx.doi.org/10.1007/978-1-4612-5561-1