American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise
  • DCDS-B Home
  • This Issue
  • Next Article
    Backward bifurcation and global stability in an epidemic model with treatment and vaccination
June  2014, 19(4): 1027-1045. doi: 10.3934/dcdsb.2014.19.1027

On the stochastic beam equation driven by a Non-Gaussian Lévy process

Hongjun Gao 1, and  Fei Liang 2,

1. 

Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, School of Mathematical Science, Nanjing Normal University, Nanjing 210023, China
2. 

Department of Mathematics, Northwest University, Xi An 710069, China

Received  June 2012 Revised  November 2013 Published  April 2014

A damped stochastic beam equation driven by a Non-Gaussian Lévy process is studied. Under appropriate conditions, the existence theorem for a unique global weak solution is given. Moreover, we also show the existence of a unique invariant measure associated with the transition semigroup under mild conditions.
Keywords: invariant measure., global weak solution, Lévy process, Stochastic beam equation, transition semigroup.
Mathematics Subject Classification: Primary: 35L05, 35L70; Secondary: 60H15, 36R6.
Citation: Hongjun Gao, Fei Liang. On the stochastic beam equation driven by a Non-Gaussian Lévy process. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1027-1045. doi: 10.3934/dcdsb.2014.19.1027
References:
[1]

D. Applebaum, Lévy Process and Stochastic Calculus, 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.

[2]

V. Barbu and G. D. Prato, The stochastic nonlinear damped wave equation, Appl. Math. Optim., 46 (2002), 125-141. doi: 10.1007/s00245-002-0744-4.

[3]

V. Barbu, G. D. Prato and L. Tubaro, Stochastic wave equations with dissipative damping, Stochastic Process. Appl., 117 (2007), 1001-1013. doi: 10.1016/j.spa.2006.11.006.

[4]

L. J. Bo, K. H. Shi and Y. J. Wang, ON a stochastic wave equation driven by a non-Gaussian Lévy process, J. Theor. Probab, 23 (2010), 328-343. doi: 10.1007/s10959-009-0228-4.

[5]

L. J. Bo, D. Tang and Y. J. Wang, Explosive solutions of stochastic wave equations with damping on $\mathbbmathbb{R}^{d}$, J. Differential Equations, 244 (2008), 170-187. doi: 10.1016/j.jde.2007.10.016.

[6]

Z. Brzeźniak, B. Maslowski and J. Seidler, Nonlinear stochstic wave equations: blow-up of second moments in $L^2$-norm, Probab. Theory Related Fields, 132 (2005), 119-149. doi: 10.1007/s00440-004-0392-5.

[7]

Z. Brzeźniak and J. H. Zhu, Stochastic nonlinear beam equations driven by compensated Poisson random measures, preprint, arXiv:math/1011.5377v1.

[8]

T. Caraballo, P. E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl Math Optim, 50 (2004), 183-207. doi: 10.1007/s00245-004-0802-1.

[9]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation, Commun. Contemp. Math., 6 (2004), 705-731. doi: 10.1142/S0219199704001483.

[10]

P. L. Chow, Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002), 1-382. doi: 10.1214/aoap/1015961168.

[11]

P. L. Chow, Asymptotics of solutions to semilinear stochastic wave equations, Ann. Appl. Probab., 16 (2006), 475-1058. doi: 10.1214/105051606000000141.

[12]

P. L. Chow, Asymptotic solutions of a nonlinear stochastic beam equation, Discrete Contin. Dyn. Syst. Ser. B., 6 (2006), 735-749. doi: 10.3934/dcdsb.2006.6.735.

[13]

P. L. Chow, Nonlinear stochstic wave equations: blow-up of second moments in $L^2$-norm, Ann. Appl. Probab., 19 (2009), 2039-2045. doi: 10.1214/09-AAP602.

[14]

P. L. Chow and J. L. Menaldi, Stochastic PDE for nonlinear vibration of elastic panels, Differential Integral Equations, 12 (1999), 419-434.

[15]

I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, AKTA, Kharkiv, 1999.

[16]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations., 9 (1997), 307-341. doi: 10.1007/BF02219225.

[17]

G. Da Prato and J. Zabczyk,, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[18]

R. W. Dickey, Free vibrations and dynamic buckling of the extensible beam, J. Math. Anal. Appl., 29 (1970), 443-454. doi: 10.1016/0022-247X(70)90094-6.

[19]

J. G. Eisley, Nonlinear vibration of beams and rectangular plates, Z. Angew. Math. Phys., 15 (1964), 167-175. doi: 10.1007/BF01602658.

[20]

W. E. Fitzgibbon, Global existence and boundedness of solutions to the extensible beam equation, SIAM J. Math. Anal., 13 (1982), 739-745. doi: 10.1137/0513050.

[21]

P. Holmes and J. Marsden, A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam, Arch. Ration. Mech. Anal., 76 (1981), 135-165. doi: 10.1007/BF00251249.

[22]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981.

[23]

J. U. Kim, On the stochastic wave equation with nonlinear damping, Appl. Math. Optim., 58 (2008), 29-67. doi: 10.1007/s00245-007-9029-2.

[24]

S. Kouémou Patcheu, On a global solution and asymptotic behaviour for the generalized damped extensible beam equation, J. Differential Equations, 135 (1997), 229-314. doi: 10.1006/jdeq.1996.3231.

[25]

F. Liang, Explosive solutions of stochastic nonlinear beam equations with damping, accepted by J. Math. Anal. Appl.

[26]

A. Millet and P. L. Morien, On a nonlinear stochastic wave equation in the plane: Existence and uniqueness of the solution, Ann. Appl. Probab., 11 (2001), 922-951. doi: 10.1214/aoap/1015345353.

[27]

S. Peszat and J. Zabczyk, Stochastic heat and wave equations driven by an impulsive noise, In Da Prato, G., Tubaro, L. (eds.) Stochastic Partial Differential Equations and Applications, VII.(eds. G. Da Prato and L. Tubaro), Lect. Notes Pure Appl., 245, Chapman Hall/CRC, Boca Raton, 2006, 229-242. doi: 10.1201/9781420028720.ch19.

[28]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise. An Evolution Equation Approach, Encyclopedia of Mathematics and Its Applications, 113, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511721373.

[29]

E. L. Reiss and B. J. Matkowsky, Nonlinear dynamic buckling of a compressed elastic column, Quart. Appl. Math., 29 (1971), 245-260.

[30]

K. Sato, Lévy Process and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999.

[31]

L. Soraya and T. Nasser-eddine, Blow-up of solutions for a nonlinear beam equation with fractional feedback, Nonlinear Anal., 74 (2011), 1402-1409. doi: 10.1016/j.na.2010.10.012.

[32]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, New York, 1997.

[33]

A. Unai, Abstract nonlinear beam equations， SUT J. Math., 29 (1993), 323-336.

[34]

C. F. Vasconcellos and L. M. Teixeira, Existence, uniqueness and stabilization for a nonlinear plate system with nonlinear damping, Ann. Fac. Sci. ToulouseMath., 8 (1999), 173-193. doi: 10.5802/afst.928.

[35]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech.,17 (1950), 35-36.

[36]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B, Nonlinear Monotone Operators, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

References:
[1]

D. Applebaum, Lévy Process and Stochastic Calculus, 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.

[2]

V. Barbu and G. D. Prato, The stochastic nonlinear damped wave equation, Appl. Math. Optim., 46 (2002), 125-141. doi: 10.1007/s00245-002-0744-4.

[3]

V. Barbu, G. D. Prato and L. Tubaro, Stochastic wave equations with dissipative damping, Stochastic Process. Appl., 117 (2007), 1001-1013. doi: 10.1016/j.spa.2006.11.006.

[4]

L. J. Bo, K. H. Shi and Y. J. Wang, ON a stochastic wave equation driven by a non-Gaussian Lévy process, J. Theor. Probab, 23 (2010), 328-343. doi: 10.1007/s10959-009-0228-4.

[5]

L. J. Bo, D. Tang and Y. J. Wang, Explosive solutions of stochastic wave equations with damping on $\mathbbmathbb{R}^{d}$, J. Differential Equations, 244 (2008), 170-187. doi: 10.1016/j.jde.2007.10.016.

[6]

Z. Brzeźniak, B. Maslowski and J. Seidler, Nonlinear stochstic wave equations: blow-up of second moments in $L^2$-norm, Probab. Theory Related Fields, 132 (2005), 119-149. doi: 10.1007/s00440-004-0392-5.

[7]

Z. Brzeźniak and J. H. Zhu, Stochastic nonlinear beam equations driven by compensated Poisson random measures, preprint, arXiv:math/1011.5377v1.

[8]

T. Caraballo, P. E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl Math Optim, 50 (2004), 183-207. doi: 10.1007/s00245-004-0802-1.

[9]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation, Commun. Contemp. Math., 6 (2004), 705-731. doi: 10.1142/S0219199704001483.

[10]

P. L. Chow, Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002), 1-382. doi: 10.1214/aoap/1015961168.

[11]

P. L. Chow, Asymptotics of solutions to semilinear stochastic wave equations, Ann. Appl. Probab., 16 (2006), 475-1058. doi: 10.1214/105051606000000141.

[12]

P. L. Chow, Asymptotic solutions of a nonlinear stochastic beam equation, Discrete Contin. Dyn. Syst. Ser. B., 6 (2006), 735-749. doi: 10.3934/dcdsb.2006.6.735.

[13]

P. L. Chow, Nonlinear stochstic wave equations: blow-up of second moments in $L^2$-norm, Ann. Appl. Probab., 19 (2009), 2039-2045. doi: 10.1214/09-AAP602.

[14]

P. L. Chow and J. L. Menaldi, Stochastic PDE for nonlinear vibration of elastic panels, Differential Integral Equations, 12 (1999), 419-434.

[15]

I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, AKTA, Kharkiv, 1999.

[16]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations., 9 (1997), 307-341. doi: 10.1007/BF02219225.

[17]

G. Da Prato and J. Zabczyk,, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[18]

R. W. Dickey, Free vibrations and dynamic buckling of the extensible beam, J. Math. Anal. Appl., 29 (1970), 443-454. doi: 10.1016/0022-247X(70)90094-6.

[19]

J. G. Eisley, Nonlinear vibration of beams and rectangular plates, Z. Angew. Math. Phys., 15 (1964), 167-175. doi: 10.1007/BF01602658.

[20]

W. E. Fitzgibbon, Global existence and boundedness of solutions to the extensible beam equation, SIAM J. Math. Anal., 13 (1982), 739-745. doi: 10.1137/0513050.

[21]

P. Holmes and J. Marsden, A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam, Arch. Ration. Mech. Anal., 76 (1981), 135-165. doi: 10.1007/BF00251249.

[22]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981.

[23]

J. U. Kim, On the stochastic wave equation with nonlinear damping, Appl. Math. Optim., 58 (2008), 29-67. doi: 10.1007/s00245-007-9029-2.

[24]

S. Kouémou Patcheu, On a global solution and asymptotic behaviour for the generalized damped extensible beam equation, J. Differential Equations, 135 (1997), 229-314. doi: 10.1006/jdeq.1996.3231.

[25]

F. Liang, Explosive solutions of stochastic nonlinear beam equations with damping, accepted by J. Math. Anal. Appl.

[26]

A. Millet and P. L. Morien, On a nonlinear stochastic wave equation in the plane: Existence and uniqueness of the solution, Ann. Appl. Probab., 11 (2001), 922-951. doi: 10.1214/aoap/1015345353.

[27]

S. Peszat and J. Zabczyk, Stochastic heat and wave equations driven by an impulsive noise, In Da Prato, G., Tubaro, L. (eds.) Stochastic Partial Differential Equations and Applications, VII.(eds. G. Da Prato and L. Tubaro), Lect. Notes Pure Appl., 245, Chapman Hall/CRC, Boca Raton, 2006, 229-242. doi: 10.1201/9781420028720.ch19.

[28]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise. An Evolution Equation Approach, Encyclopedia of Mathematics and Its Applications, 113, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511721373.

[29]

E. L. Reiss and B. J. Matkowsky, Nonlinear dynamic buckling of a compressed elastic column, Quart. Appl. Math., 29 (1971), 245-260.

[30]

K. Sato, Lévy Process and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999.

[31]

L. Soraya and T. Nasser-eddine, Blow-up of solutions for a nonlinear beam equation with fractional feedback, Nonlinear Anal., 74 (2011), 1402-1409. doi: 10.1016/j.na.2010.10.012.

[32]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, New York, 1997.

[33]

A. Unai, Abstract nonlinear beam equations， SUT J. Math., 29 (1993), 323-336.

[34]

C. F. Vasconcellos and L. M. Teixeira, Existence, uniqueness and stabilization for a nonlinear plate system with nonlinear damping, Ann. Fac. Sci. ToulouseMath., 8 (1999), 173-193. doi: 10.5802/afst.928.

[35]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech.,17 (1950), 35-36.

[36]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B, Nonlinear Monotone Operators, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

[1]

Yong-Kum Cho. On the Boltzmann equation with the symmetric stable Lévy process. Kinetic and Related Models, 2015, 8 (1) : 53-77. doi: 10.3934/krm.2015.8.53
[2]

Justin Cyr, Phuong Nguyen, Sisi Tang, Roger Temam. Review of local and global existence results for stochastic pdes with Lévy noise. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5639-5710. doi: 10.3934/dcds.2020241
[3]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133
[4]

Karel Kadlec, Bohdan Maslowski. Ergodic boundary and point control for linear stochastic PDEs driven by a cylindrical Lévy process. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 4039-4055. doi: 10.3934/dcdsb.2020137
[5]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085
[6]

Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140
[7]

Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5641-5660. doi: 10.3934/dcdsb.2020371
[8]

Yongxia Zhao, Rongming Wang, Chuancun Yin. Optimal dividends and capital injections for a spectrally positive Lévy process. Journal of Industrial and Management Optimization, 2017, 13 (1) : 1-21. doi: 10.3934/jimo.2016001
[9]

Linghua Chen, Espen R. Jakobsen. L1 semigroup generation for Fokker-Planck operators associated to general Lévy driven SDEs. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5735-5763. doi: 10.3934/dcds.2018250
[10]

Badr-eddine Berrhazi, Mohamed El Fatini, Tomás Caraballo, Roger Pettersson. A stochastic SIRI epidemic model with Lévy noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2415-2431. doi: 10.3934/dcdsb.2018057
[11]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5421-5448. doi: 10.3934/dcdsb.2020352
[12]

Chuchu Chen, Jialin Hong, Yulan Lu. Stochastic differential equation with piecewise continuous arguments: Markov property, invariant measure and numerical approximation. Discrete and Continuous Dynamical Systems - B, 2023, 28 (1) : 765-807. doi: 10.3934/dcdsb.2022098
[13]

Adam Andersson, Felix Lindner. Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4271-4294. doi: 10.3934/dcdsb.2019081
[14]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209
[15]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221
[16]

Justin Cyr, Phuong Nguyen, Roger Temam. Stochastic one layer shallow water equations with Lévy noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3765-3818. doi: 10.3934/dcdsb.2018331
[17]

Wen Chen, Song Wang. A finite difference method for pricing European and American options under a geometric Lévy process. Journal of Industrial and Management Optimization, 2015, 11 (1) : 241-264. doi: 10.3934/jimo.2015.11.241
[18]

Hamza Ruzayqat, Ajay Jasra. Unbiased parameter inference for a class of partially observed Lévy-process models. Foundations of Data Science, 2022, 4 (2) : 299-322. doi: 10.3934/fods.2022008
[19]

Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022
[20]

Pao-Liu Chow. Asymptotic solutions of a nonlinear stochastic beam equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 735-749. doi: 10.3934/dcdsb.2006.6.735

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy