American Institute of Mathematical Sciences

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July  2012, 17(5): 1441-1453. doi: 10.3934/dcdsb.2012.17.1441

Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions

Guanggan Chen 1, and  Jian Zhang 2,

1. 

College of Mathematics and Software Science, Sichuan Normal University, Chengdu, 610068, China
2. 

College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068

Received  June 2011 Revised  February 2012 Published  March 2012

This work is concerned with the asymptotic dynamical behavior for a weakly damped stochastic nonlinear wave equation with dynamical boundary conditions. The white noises appear both in the model and in the dynamical boundary condition. Since the energy relation of this stochastic system does not directly imply the a priori estimate of the solution, we propose a pseudo energy equation to infer almost sure boundedness of the solution. Then a unique invariant measure is shown to exist for the system.
Keywords: dynamical boundary condition, Stochastic wave equation, invariant measure., bounded solution orbits.
Mathematics Subject Classification: Primary: 60H15, 37H05, 37L55, 37L25; Secondary: 37D1.
Citation: Guanggan Chen, Jian Zhang. Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1441-1453. doi: 10.3934/dcdsb.2012.17.1441
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show all references

References:
[1]

Indiana Univ. Math. J., 25 (1976), 895-917. doi: 10.1512/iumj.1976.25.25071.  Google Scholar

[2]

Indiana Univ. Math. J., 26 (1977), 199-222. doi: 10.1512/iumj.1977.26.26015.  Google Scholar

[3]

Stochastic Analysis and Applications, 27 (2009), 1096-1116. doi: 10.1080/07362990902976546.  Google Scholar

[4]

Ann. Appl. Probab., 12 (2002), 361-381. doi: 10.1214/aoap/1015961168.  Google Scholar

[5]

Ann. Appl. Probab., 16 (2006), 757-780. doi: 10.1214/105051606000000141.  Google Scholar

[6]

Differential Integral Equations, 17 (2004), 751-780.  Google Scholar

[7]

Funkcial. Ekvac., 44 (2001), 471-485.  Google Scholar

[8]

Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.  Google Scholar

[9]

London Mathematical Society Lecture Note Series, 229, Cambridge Univ. Press, Cambridge, 1996.  Google Scholar

[10]

Stoch. Anal. Appl., 25 (2007), 381-396. doi: 10.1080/07362990601139602.  Google Scholar

[11]

J. Evol. Equ., 10 (2010), 29-58. doi: 10.1007/s00028-009-0039-1.  Google Scholar

[12]

Journal of Differential Equations, 164 (2000), 92-109. doi: 10.1006/jdeq.1999.3743.  Google Scholar

[13]

Advances in Differential Equations, 13 (2008), 1051-1074.  Google Scholar

[14]

Math. Z., 189 (1985), 487-505. doi: 10.1007/BF01168155.  Google Scholar

[15]

Trans. Amer. Math. Soc., 192 (1974), 1-21. doi: 10.2307/1996814.  Google Scholar

[16]

J. Differential Equations, 236 (2007), 460-492. doi: 10.1016/j.jde.2006.09.024.  Google Scholar

[17]

J. Differential Equations, 244 (2008), 1-23. doi: 10.1016/j.jde.2007.10.009.  Google Scholar

[18]

Ann. Probab., 25 (1997), 133-151. doi: 10.1214/aop/1024404282.  Google Scholar

[19]

Math. Nachr., 279 (2006), 299-318. doi: 10.1002/mana.200310362.  Google Scholar

[20]

Israel J. Math.,22 (1975), 273-303. doi: 10.1007/BF02761595.  Google Scholar

[21]

Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 389-413. doi: 10.1017/S0308210500003279.  Google Scholar

[22]

Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.  Google Scholar

[23]

CBMS Regional Conference Series in Math., 73, Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the Amer. Math. Soc., Providence, RI, 1989.  Google Scholar

[24]

J. Differential Equations, 248 (2010), 1269-1296. doi: 10.1016/j.jde.2009.10.003.  Google Scholar

[25]

Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.  Google Scholar

[26]

Commun. Math. Phys., 275 (2007), 163-186. doi: 10.1007/s00220-007-0301-8.  Google Scholar

[27]

Stoch. Anal. Appl., 25 (2007), 613-639. doi: 10.1080/07362990701282963.  Google Scholar

[28]

SIAM J. Appl. Dyn. Syst., 4 (2005), 883-903. doi: 10.1137/050623097.  Google Scholar

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