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 JAMP  Vol.3 No.3 , March 2015
Asymptotic Behavior of Stochastic Strongly Wave Equation on Unbounded Domains
Abstract: We study the asymptotic behavior of solutions to the stochastic strongly damped wave equation with additive noise defined on unbounded domains. We first prove the uniform estimates of solutions, and then establish the existence of a random attractor.
Cite this paper: Wang, Z. and Zhou, S. (2015) Asymptotic Behavior of Stochastic Strongly Wave Equation on Unbounded Domains. Journal of Applied Mathematics and Physics, 3, 338-357. doi: 10.4236/jamp.2015.33046.
References

[1]   Arnold, L. (1998) Random Dynamical Systems. Springer-Verlag, New York and Berlin.
http://dx.doi.org/10.1007/978-3-662-12878-7

[2]   Belleri, V. and Pata, V. (2001) Attractors for Semilinear Strongly Damped Wave Equations on . Discrete and Continuous Dynamical Systems, 7, 719-735.
http://dx.doi.org/10.3934/dcds.2001.7.719

[3]   Conti, M., Pata, V. and Squassina, M. (2005) Strongly Damped Wave Equations on with Critical Nonlinearities. Communications on Pure and Applied Analysis, 9, 161-176.

[4]   Chen, F., Guo, B. and Wang, P. (1998) Long Time Behavior of Strongly Damped Nonlinear Wave Equations. Journal of Differential Equations, 147, 339-352.
http://dx.doi.org/10.1006/jdeq.1998.3447

[5]   Li, H. and Zhou, S. (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

[6]   Temam, R. (1998) Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York.

[7]   Yang, M. and Sun, C. (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

[8]   Yang, M. and Sun, C. (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

[9]   Zhou, S. (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

[10]   Zhou, S. and Fan, X. (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

[11]   Zhou, S. (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

[12]   Chow, P. (2002) Stochastic Wave Equation with Polynomial Nonlinearity. Annals of Applied Probability, 12, 361-381.
http://dx.doi.org/10.1214/aoap/1015961168

[13]   Fan, X. (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

[14]   Fan, X. 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

[15]   Fan, X. (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

[16]   Fan, X. (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

[17]   Jones, R. and Wang, B. (2013) Asymptotic Behavior of a Class of Stochastic Nonlinear Wave Equations with Dispersive and Dissipative Terms. Nonlinear Analysis: Real World Applications, 14, 1308-1322.
http://dx.doi.org/10.1016/j.nonrwa.2012.09.019

[18]   Lu, K. 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

[19]   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

[20]   Wang, B.X. and Gao, X.L. (2009) Random Attractors for Wave Equations on Unbounded Domains. Discrete and Continuous Dynamical Systems, Special, 800-809.

[21]   Wang, B. (2011) Asymptotic Behavior of Stochastic Wave Equations with Critical Exponents on . Transactions of the American Mathematical Society, 363, 3639-3663.
http://dx.doi.org/10.1090/S0002-9947-2011-05247-5

[22]   Wang, Z., Zhou, S. and Gu, A. (2012) Random Attractor of the Stochastic Strongly Damped Wave Equation. Communications in Nonlinear Science and Numerical Simulation, 17, 1649-1658.
http://dx.doi.org/10.1016/j.cnsns.2011.09.001

[23]   Wang, Z., Zhou, S. and Gu, A. (2011) Random Attractor for a Stochastic Damped Wave Equation with Multiplicative Noise on Unbounded Domains. Nonlinear Analysis: Real World Applications, 12, 3468-3482.
http://dx.doi.org/10.1016/j.nonrwa.2011.06.008

[24]   Yang, M., Duan, J. 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

[25]   Zhou, S., Yin, F. and Ou Yang, Z. (2005) Random Attractor for Damped Nonlinear Wave Equations with White Noise. The SIAM Journal on Applied Dynamical Systems, 4, 883-903.
http://dx.doi.org/10.1137/050623097

[26]   Crauel, H. (2002) Random Probability Measure on Polish Spaces. Taylor & Francis, London.

[27]   Crauel, H., Debussche, A. and Flandoli, F. (1997) Random Attractors. Journal of Dynamics and Differential Equations, 9, 307-341.
http://dx.doi.org/10.1007/BF02219225

[28]   Crauel, H. and Flandoli, F. (1994) Attractors for Random Dynamical Systems. Probability Theory and Related Fields, 100, 365-393.
http://dx.doi.org/10.1007/BF01193705

[29]   Flandoli, F. and Schmalfuß, B. (1996) Random Attractors for the 3D Stochastic Navier-Stokes Equation with Multiplicative Noise. Stochastics and Stochastic Reports, 59, 21-45.
http://dx.doi.org/10.1080/17442509608834083

[30]   Shen, Z., Zhou, S. and Shen, W. (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

[31]   Ball, J.M. (1997) Continuity Properties and Global Attractors of Generalized Semiflows and the Navier-Stokes Equations. Journal of Nonlinear Science, 7, 475-502.
http://dx.doi.org/10.1007/s003329900037

[32]   Ball, J.M. (2004) Global Attractors for Damped Semilinear Wave Equations. Discrete and Continuous Dynamical Systems, 10, 31-52.
http://dx.doi.org/10.3934/dcds.2004.10.31

[33]   Ghidaglia, J.M. (1994) A Note on the Strong Convergence towards Attractors for Damped Forced KdV Equations. Journal of Differential Equations, 110, 356-359.
http://dx.doi.org/10.1006/jdeq.1994.1071

[34]   Goubet, O. and Rosa, R. (2002) Asymptotic Smoothing and the Global Attractor of a Weakly Damped KdV Equation on the Real Line. Journal of Differential Equations, 185, 25-53.
http://dx.doi.org/10.1006/jdeq.2001.4163

[35]   Ju, N. (2000) The H1-Compact Global Attractor for the Solutions to the Navier-Stokes Equations in Two-Dimensional Unbounded Domains. Nonlinearity, 13, 1227-1238.
http://dx.doi.org/10.1088/0951-7715/13/4/313

[36]   Moise, I. and Rosa, R. (1997) On the Regularity of the Global Attractor of a Weakly Damped, Forced Korteweg-de Vries Equation. Advances in Differential Equations, 2, 257-296.

[37]   Moise, I., Rosa, R. and Wang, X. (1998) Attractors for Non-Compact Semigroups via Energy Equations. Nonlinearity, 11, 1369-1393.
http://dx.doi.org/10.1088/0951-7715/11/5/012

[38]   Rosa, R. (1998) The Global Attractor for the 2D Navier-Stokes Flow on Some Unbounded Domains. Nonlinear Analysis, 32, 71-85.
http://dx.doi.org/10.1016/S0362-546X(97)00453-7

[39]   Wang, X. (1995) An Energy Equation for the Weakly Damped Driven Nonlinear Schrodinger Equations and Its Applications. Physica D, 88, 167-175.
http://dx.doi.org/10.1016/0167-2789(95)00196-B

[40]   Bates, P.W., Lu, K. and Wang, B. (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

[41]   Chueshov, I. (2002) Monotone Random Systems Theory and Applications. Springer-Verlag, New York.
http://dx.doi.org/10.1007/b83277

[42]   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

 
 
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