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

Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions

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.
Citation: Guanggan Chen, Jian Zhang. Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1441-1453. doi: 10.3934/dcdsb.2012.17.1441
References:
[1]

J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25 (1976), 895-917. doi: 10.1512/iumj.1976.25.25071.

[2]

J. T. Beale, Acoustic scattering from locally reacting surfaces, Indiana Univ. Math. J., 26 (1977), 199-222. doi: 10.1512/iumj.1977.26.26015.

[3]

P. Brune, J. Duan and B. Schmalfuss, Random dynamics of the Boussinesq system with dynamical boundary conditions, Stochastic Analysis and Applications, 27 (2009), 1096-1116. doi: 10.1080/07362990902976546.

[4]

P.-L. Chow, Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002), 361-381. doi: 10.1214/aoap/1015961168.

[5]

P.-L. Chow, Asymptotics of solutions to semilinear stochastic wave equations, Ann. Appl. Probab., 16 (2006), 757-780. doi: 10.1214/105051606000000141.

[6]

I. Chueshov and B. Schmalfuss, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equations, 17 (2004), 751-780.

[7]

A. T. Cousin, C. L. Frota and N. A. Larkin, Global solvability and asymptotic behavior of a hyperbolic problem with acoustic boundary condition, Funkcial. Ekvac., 44 (2001), 471-485.

[8]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.

[9]

G. Da Prato and J. Zabczyk, "Ergodicity for Infinite-Dimensional Systems," London Mathematical Society Lecture Note Series, 229, Cambridge Univ. Press, Cambridge, 1996.

[10]

X. Fan and Y. Wang, Fractal dimensional of attractors for a stochastic wave equation with nonlinear damping and white noise, Stoch. Anal. Appl., 25 (2007), 381-396. doi: 10.1080/07362990601139602.

[11]

S. Frigeri, Attractors for semilinear damped wave equations with an acoustic boundary condition, J. Evol. Equ., 10 (2010), 29-58. doi: 10.1007/s00028-009-0039-1.

[12]

C. L. Frota and J. A. Goldstein, Some nonlinear wave equations with acoustic boundary conditions, Journal of Differential Equations, 164 (2000), 92-109. doi: 10.1006/jdeq.1999.3743.

[13]

S. Gerbi and B. Said-Houari, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions, Advances in Differential Equations, 13 (2008), 1051-1074.

[14]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505. doi: 10.1007/BF01168155.

[15]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{t t} =-Au +F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21. doi: 10.2307/1996814.

[16]

K. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492. doi: 10.1016/j.jde.2006.09.024.

[17]

Y. Lv, W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23. doi: 10.1016/j.jde.2007.10.009.

[18]

C. Mueller, Long time existence for the wave equation with a noise term, Ann. Probab., 25 (1997), 133-151. doi: 10.1214/aop/1024404282.

[19]

D. Mugnolo, Abstract wave equations with acoustic boundary conditions, Math. Nachr., 279 (2006), 299-318. doi: 10.1002/mana.200310362.

[20]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math.,22 (1975), 273-303. doi: 10.1007/BF02761595.

[21]

L. Popescu and A. Rodriguez-Bernal, On a singularly perturbed wave equation with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 389-413. doi: 10.1017/S0308210500003279.

[22]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness," Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[23]

W. A. Strauss, "Nonlinear Wave Equations," 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.

[24]

C. Sun, H. Gao, J. Duan and B. Schmalfuss, Rare events in the Boussinesq system with fluctuating dynamical boundary conditions, J. Differential Equations, 248 (2010), 1269-1296. doi: 10.1016/j.jde.2009.10.003.

[25]

G. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

[26]

W. Wang and J. Duan, Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions, Commun. Math. Phys., 275 (2007), 163-186. doi: 10.1007/s00220-007-0301-8.

[27]

D. Yang and J. Duan, An impact of stochastic dynamic boundary conditions on the evolution of the Cahn-Hilliard system, Stoch. Anal. Appl., 25 (2007), 613-639. doi: 10.1080/07362990701282963.

[28]

S. F. Zhou, F. Q. Yin and Z. G. Ouyang, Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst., 4 (2005), 883-903. doi: 10.1137/050623097.

show all references

References:
[1]

J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25 (1976), 895-917. doi: 10.1512/iumj.1976.25.25071.

[2]

J. T. Beale, Acoustic scattering from locally reacting surfaces, Indiana Univ. Math. J., 26 (1977), 199-222. doi: 10.1512/iumj.1977.26.26015.

[3]

P. Brune, J. Duan and B. Schmalfuss, Random dynamics of the Boussinesq system with dynamical boundary conditions, Stochastic Analysis and Applications, 27 (2009), 1096-1116. doi: 10.1080/07362990902976546.

[4]

P.-L. Chow, Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002), 361-381. doi: 10.1214/aoap/1015961168.

[5]

P.-L. Chow, Asymptotics of solutions to semilinear stochastic wave equations, Ann. Appl. Probab., 16 (2006), 757-780. doi: 10.1214/105051606000000141.

[6]

I. Chueshov and B. Schmalfuss, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equations, 17 (2004), 751-780.

[7]

A. T. Cousin, C. L. Frota and N. A. Larkin, Global solvability and asymptotic behavior of a hyperbolic problem with acoustic boundary condition, Funkcial. Ekvac., 44 (2001), 471-485.

[8]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.

[9]

G. Da Prato and J. Zabczyk, "Ergodicity for Infinite-Dimensional Systems," London Mathematical Society Lecture Note Series, 229, Cambridge Univ. Press, Cambridge, 1996.

[10]

X. Fan and Y. Wang, Fractal dimensional of attractors for a stochastic wave equation with nonlinear damping and white noise, Stoch. Anal. Appl., 25 (2007), 381-396. doi: 10.1080/07362990601139602.

[11]

S. Frigeri, Attractors for semilinear damped wave equations with an acoustic boundary condition, J. Evol. Equ., 10 (2010), 29-58. doi: 10.1007/s00028-009-0039-1.

[12]

C. L. Frota and J. A. Goldstein, Some nonlinear wave equations with acoustic boundary conditions, Journal of Differential Equations, 164 (2000), 92-109. doi: 10.1006/jdeq.1999.3743.

[13]

S. Gerbi and B. Said-Houari, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions, Advances in Differential Equations, 13 (2008), 1051-1074.

[14]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505. doi: 10.1007/BF01168155.

[15]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{t t} =-Au +F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21. doi: 10.2307/1996814.

[16]

K. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492. doi: 10.1016/j.jde.2006.09.024.

[17]

Y. Lv, W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23. doi: 10.1016/j.jde.2007.10.009.

[18]

C. Mueller, Long time existence for the wave equation with a noise term, Ann. Probab., 25 (1997), 133-151. doi: 10.1214/aop/1024404282.

[19]

D. Mugnolo, Abstract wave equations with acoustic boundary conditions, Math. Nachr., 279 (2006), 299-318. doi: 10.1002/mana.200310362.

[20]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math.,22 (1975), 273-303. doi: 10.1007/BF02761595.

[21]

L. Popescu and A. Rodriguez-Bernal, On a singularly perturbed wave equation with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 389-413. doi: 10.1017/S0308210500003279.

[22]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness," Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[23]

W. A. Strauss, "Nonlinear Wave Equations," 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.

[24]

C. Sun, H. Gao, J. Duan and B. Schmalfuss, Rare events in the Boussinesq system with fluctuating dynamical boundary conditions, J. Differential Equations, 248 (2010), 1269-1296. doi: 10.1016/j.jde.2009.10.003.

[25]

G. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

[26]

W. Wang and J. Duan, Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions, Commun. Math. Phys., 275 (2007), 163-186. doi: 10.1007/s00220-007-0301-8.

[27]

D. Yang and J. Duan, An impact of stochastic dynamic boundary conditions on the evolution of the Cahn-Hilliard system, Stoch. Anal. Appl., 25 (2007), 613-639. doi: 10.1080/07362990701282963.

[28]

S. F. Zhou, F. Q. Yin and Z. G. Ouyang, Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst., 4 (2005), 883-903. doi: 10.1137/050623097.

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