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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 |
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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