# American Institute of Mathematical Sciences

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 & 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.  doi: 10.1512/iumj.1976.25.25071.  Google Scholar [2] J. T. Beale, Acoustic scattering from locally reacting surfaces,, Indiana Univ. Math. J., 26 (1977), 199.  doi: 10.1512/iumj.1977.26.26015.  Google Scholar [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.  doi: 10.1080/07362990902976546.  Google Scholar [4] P.-L. Chow, Stochastic wave equations with polynomial nonlinearity,, Ann. Appl. Probab., 12 (2002), 361.  doi: 10.1214/aoap/1015961168.  Google Scholar [5] P.-L. Chow, Asymptotics of solutions to semilinear stochastic wave equations,, Ann. Appl. Probab., 16 (2006), 757.  doi: 10.1214/105051606000000141.  Google Scholar [6] I. Chueshov and B. Schmalfuss, Parabolic stochastic partial differential equations with dynamical boundary conditions,, Differential Integral Equations, 17 (2004), 751.   Google Scholar [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.   Google Scholar [8] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Encyclopedia of Mathematics and its Applications, 44 (1992).   Google Scholar [9] G. Da Prato and J. Zabczyk, "Ergodicity for Infinite-Dimensional Systems,", London Mathematical Society Lecture Note Series, 229 (1996).   Google Scholar [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.  doi: 10.1080/07362990601139602.  Google Scholar [11] S. Frigeri, Attractors for semilinear damped wave equations with an acoustic boundary condition,, J. Evol. Equ., 10 (2010), 29.  doi: 10.1007/s00028-009-0039-1.  Google Scholar [12] C. L. Frota and J. A. Goldstein, Some nonlinear wave equations with acoustic boundary conditions,, Journal of Differential Equations, 164 (2000), 92.  doi: 10.1006/jdeq.1999.3743.  Google Scholar [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.   Google Scholar [14] J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation,, Math. Z., 189 (1985), 487.  doi: 10.1007/BF01168155.  Google Scholar [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.  doi: 10.2307/1996814.  Google Scholar [16] K. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations,, J. Differential Equations, 236 (2007), 460.  doi: 10.1016/j.jde.2006.09.024.  Google Scholar [17] Y. Lv, W. Wang, Limiting dynamics for stochastic wave equations,, J. Differential Equations, 244 (2008), 1.  doi: 10.1016/j.jde.2007.10.009.  Google Scholar [18] C. Mueller, Long time existence for the wave equation with a noise term,, Ann. Probab., 25 (1997), 133.  doi: 10.1214/aop/1024404282.  Google Scholar [19] D. Mugnolo, Abstract wave equations with acoustic boundary conditions,, Math. Nachr., 279 (2006), 299.  doi: 10.1002/mana.200310362.  Google Scholar [20] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273.  doi: 10.1007/BF02761595.  Google Scholar [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.  doi: 10.1017/S0308210500003279.  Google Scholar [22] M. Reed and B. Simon, "Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness,", Academic Press [Harcourt Brace Jovanovich, (1975).   Google Scholar [23] W. A. Strauss, "Nonlinear Wave Equations,", CBMS Regional Conference Series in Math., 73 (1989).   Google Scholar [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.  doi: 10.1016/j.jde.2009.10.003.  Google Scholar [25] G. Whitham, "Linear and Nonlinear Waves,", Pure and Applied Mathematics, (1974).   Google Scholar [26] W. Wang and J. Duan, Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions,, Commun. Math. Phys., 275 (2007), 163.  doi: 10.1007/s00220-007-0301-8.  Google Scholar [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.  doi: 10.1080/07362990701282963.  Google Scholar [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.  doi: 10.1137/050623097.  Google Scholar

show all references

##### References:
 [1] J. T. Beale, Spectral properties of an acoustic boundary condition,, Indiana Univ. Math. J., 25 (1976), 895.  doi: 10.1512/iumj.1976.25.25071.  Google Scholar [2] J. T. Beale, Acoustic scattering from locally reacting surfaces,, Indiana Univ. Math. J., 26 (1977), 199.  doi: 10.1512/iumj.1977.26.26015.  Google Scholar [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.  doi: 10.1080/07362990902976546.  Google Scholar [4] P.-L. Chow, Stochastic wave equations with polynomial nonlinearity,, Ann. Appl. Probab., 12 (2002), 361.  doi: 10.1214/aoap/1015961168.  Google Scholar [5] P.-L. Chow, Asymptotics of solutions to semilinear stochastic wave equations,, Ann. Appl. Probab., 16 (2006), 757.  doi: 10.1214/105051606000000141.  Google Scholar [6] I. Chueshov and B. Schmalfuss, Parabolic stochastic partial differential equations with dynamical boundary conditions,, Differential Integral Equations, 17 (2004), 751.   Google Scholar [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.   Google Scholar [8] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Encyclopedia of Mathematics and its Applications, 44 (1992).   Google Scholar [9] G. Da Prato and J. Zabczyk, "Ergodicity for Infinite-Dimensional Systems,", London Mathematical Society Lecture Note Series, 229 (1996).   Google Scholar [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.  doi: 10.1080/07362990601139602.  Google Scholar [11] S. Frigeri, Attractors for semilinear damped wave equations with an acoustic boundary condition,, J. Evol. Equ., 10 (2010), 29.  doi: 10.1007/s00028-009-0039-1.  Google Scholar [12] C. L. Frota and J. A. Goldstein, Some nonlinear wave equations with acoustic boundary conditions,, Journal of Differential Equations, 164 (2000), 92.  doi: 10.1006/jdeq.1999.3743.  Google Scholar [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.   Google Scholar [14] J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation,, Math. Z., 189 (1985), 487.  doi: 10.1007/BF01168155.  Google Scholar [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.  doi: 10.2307/1996814.  Google Scholar [16] K. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations,, J. Differential Equations, 236 (2007), 460.  doi: 10.1016/j.jde.2006.09.024.  Google Scholar [17] Y. Lv, W. Wang, Limiting dynamics for stochastic wave equations,, J. Differential Equations, 244 (2008), 1.  doi: 10.1016/j.jde.2007.10.009.  Google Scholar [18] C. Mueller, Long time existence for the wave equation with a noise term,, Ann. Probab., 25 (1997), 133.  doi: 10.1214/aop/1024404282.  Google Scholar [19] D. Mugnolo, Abstract wave equations with acoustic boundary conditions,, Math. Nachr., 279 (2006), 299.  doi: 10.1002/mana.200310362.  Google Scholar [20] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273.  doi: 10.1007/BF02761595.  Google Scholar [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.  doi: 10.1017/S0308210500003279.  Google Scholar [22] M. Reed and B. Simon, "Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness,", Academic Press [Harcourt Brace Jovanovich, (1975).   Google Scholar [23] W. A. Strauss, "Nonlinear Wave Equations,", CBMS Regional Conference Series in Math., 73 (1989).   Google Scholar [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.  doi: 10.1016/j.jde.2009.10.003.  Google Scholar [25] G. Whitham, "Linear and Nonlinear Waves,", Pure and Applied Mathematics, (1974).   Google Scholar [26] W. Wang and J. Duan, Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions,, Commun. Math. Phys., 275 (2007), 163.  doi: 10.1007/s00220-007-0301-8.  Google Scholar [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.  doi: 10.1080/07362990701282963.  Google Scholar [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.  doi: 10.1137/050623097.  Google Scholar
 [1] Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140 [2] Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191 [3] Lu Yang, Meihua Yang. Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2627-2650. doi: 10.3934/dcdsb.2017102 [4] Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057 [5] Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085 [6] Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305 [7] Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601 [8] Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285 [9] Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861 [10] Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179 [11] Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343 [12] Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325 [13] Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194 [14] Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099 [15] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 [16] Imen Benabbas, Djamel Eddine Teniou. Observability of wave equation with Ventcel dynamic condition. Evolution Equations & Control Theory, 2018, 7 (4) : 545-570. doi: 10.3934/eect.2018026 [17] Vyacheslav A. Trofimov, Evgeny M. Trykin. A new way for decreasing of amplitude of wave reflected from artificial boundary condition for 1D nonlinear Schrödinger equation. Conference Publications, 2015, 2015 (special) : 1070-1078. doi: 10.3934/proc.2015.1070 [18] Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017 [19] Peter Brune, Björn Schmalfuss. Inertial manifolds for stochastic pde with dynamical boundary conditions. Communications on Pure & Applied Analysis, 2011, 10 (3) : 831-846. doi: 10.3934/cpaa.2011.10.831 [20] Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101

2018 Impact Factor: 1.008