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

Previous Article
Backward bifurcation and global stability in an epidemic model with treatment and vaccination
DCDS-B Home
This Issue
Next Article
Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise

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

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

Abstract
Related Papers

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 & 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, (2009). doi: 10.1017/CBO9780511809781.
[2]	V. Barbu and G. D. Prato, The stochastic nonlinear damped wave equation,, Appl. Math. Optim., 46 (2002), 125. 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. 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. 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 $\mathbbR^d$,, J. Differential Equations, 244* (2008), 170. 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. 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, ().
[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. 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. doi: 10.1142/S0219199704001483.
[10]	P. L. Chow, Stochastic wave equations with polynomial nonlinearity,, Ann. Appl. Probab., 12 (2002), 1. doi: 10.1214/aoap/1015961168.
[11]	P. L. Chow, Asymptotics of solutions to semilinear stochastic wave equations,, Ann. Appl. Probab., 16 (2006), 475. 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. 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. 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.
[15]	I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, University Lectures in Contemporary Mathematics, (1999).
[16]	H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations., 9 (1997), 307. doi: 10.1007/BF02219225.
[17]	G. Da Prato and J. Zabczyk,, Stochastic Equations in Infinite Dimensions,, Cambridge Univ. Press, (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. 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. 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. 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. doi: 10.1007/BF00251249.
[22]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, North-Holland Publishing Co., (1981).
[23]	J. U. Kim, On the stochastic wave equation with nonlinear damping,, Appl. Math. Optim., 58 (2008), 29. 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. 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. doi: 10.1214/aoap/1015345353.
[27]	S. Peszat and J. Zabczyk, Stochastic heat and wave equations driven by an impulsive noise,, In Da Prato, (2006), 229. 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, (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.
[30]	K. Sato, Lévy Process and Infinitely Divisible Distributions,, Cambridge University Press, (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. doi: 10.1016/j.na.2010.10.012.
[32]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, 2nd edn. Springer, (1997).
[33]	A. Unai, Abstract nonlinear beam equations，, SUT J. Math., 29 (1993), 323.
[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. 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]	E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B, Nonlinear Monotone Operators,, Springer, (1990). doi: 10.1007/978-1-4612-0985-0.

show all references

References:

[1]	D. Applebaum, Lévy Process and Stochastic Calculus,, 2nd edition, (2009). doi: 10.1017/CBO9780511809781.
[2]	V. Barbu and G. D. Prato, The stochastic nonlinear damped wave equation,, Appl. Math. Optim., 46 (2002), 125. 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. 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. 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 $\mathbbR^d$,, J. Differential Equations, 244* (2008), 170. 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. 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, ().
[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. 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. doi: 10.1142/S0219199704001483.
[10]	P. L. Chow, Stochastic wave equations with polynomial nonlinearity,, Ann. Appl. Probab., 12 (2002), 1. doi: 10.1214/aoap/1015961168.
[11]	P. L. Chow, Asymptotics of solutions to semilinear stochastic wave equations,, Ann. Appl. Probab., 16 (2006), 475. 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. 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. 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.
[15]	I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, University Lectures in Contemporary Mathematics, (1999).
[16]	H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations., 9 (1997), 307. doi: 10.1007/BF02219225.
[17]	G. Da Prato and J. Zabczyk,, Stochastic Equations in Infinite Dimensions,, Cambridge Univ. Press, (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. 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. 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. 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. doi: 10.1007/BF00251249.
[22]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, North-Holland Publishing Co., (1981).
[23]	J. U. Kim, On the stochastic wave equation with nonlinear damping,, Appl. Math. Optim., 58 (2008), 29. 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. 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. doi: 10.1214/aoap/1015345353.
[27]	S. Peszat and J. Zabczyk, Stochastic heat and wave equations driven by an impulsive noise,, In Da Prato, (2006), 229. 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, (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.
[30]	K. Sato, Lévy Process and Infinitely Divisible Distributions,, Cambridge University Press, (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. doi: 10.1016/j.na.2010.10.012.
[32]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, 2nd edn. Springer, (1997).
[33]	A. Unai, Abstract nonlinear beam equations，, SUT J. Math., 29 (1993), 323.
[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. 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]	E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B, Nonlinear Monotone Operators,, Springer, (1990). doi: 10.1007/978-1-4612-0985-0.

[1]	Yong-Kum Cho. On the Boltzmann equation with the symmetric stable Lévy process. Kinetic & Related Models, 2015, 8 (1) : 53-77. doi: 10.3934/krm.2015.8.53
[2]	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
[3]	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
[4]	Yongxia Zhao, Rongming Wang, Chuancun Yin. Optimal dividends and capital injections for a spectrally positive Lévy process. Journal of Industrial & Management Optimization, 2017, 13 (1) : 1-21. doi: 10.3934/jimo.2016001
[5]	Badr-eddine Berrhazi, Mohamed El Fatini, Tomás Caraballo, Roger Pettersson. A stochastic SIRI epidemic model with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2415-2431. doi: 10.3934/dcdsb.2018057
[6]	Linghua Chen, Espen R. Jakobsen. L¹ semigroup generation for Fokker-Planck operators associated to general Lévy driven SDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5735-5763. doi: 10.3934/dcds.2018250
[7]	Adam Andersson, Felix Lindner. Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-24. doi: 10.3934/dcdsb.2019081
[8]	Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221
[9]	Justin Cyr, Phuong Nguyen, Roger Temam. Stochastic one layer shallow water equations with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-54. doi: 10.3934/dcdsb.2018331
[10]	Wen Chen, Song Wang. A finite difference method for pricing European and American options under a geometric Lévy process. Journal of Industrial & Management Optimization, 2015, 11 (1) : 241-264. doi: 10.3934/jimo.2015.11.241
[11]	Pao-Liu Chow. Asymptotic solutions of a nonlinear stochastic beam equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 735-749. doi: 10.3934/dcdsb.2006.6.735
[12]	Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022
[13]	András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks & Heterogeneous Media, 2012, 7 (1) : 43-58. doi: 10.3934/nhm.2012.7.43
[14]	Jiangyan Peng, Dingcheng Wang. Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns. Journal of Industrial & Management Optimization, 2017, 13 (1) : 155-185. doi: 10.3934/jimo.2016010
[15]	Markus Riedle, Jianliang Zhai. Large deviations for stochastic heat equations with memory driven by Lévy-type noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1983-2005. doi: 10.3934/dcds.2018080
[16]	Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations & Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355
[17]	Jiahui Zhu, Zdzisław Brzeźniak. Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3269-3299. doi: 10.3934/dcdsb.2016097
[18]	Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282
[19]	Min Niu, Bin Xie. Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2989-3009. doi: 10.3934/dcdsb.2018296
[20]	Yanfang Li, Zhuangyi Liu, Yang Wang. Weak stability of a laminated beam. Mathematical Control & Related Fields, 2018, 8 (3&4) : 789-808. doi: 10.3934/mcrf.2018035

2017 Impact Factor: 0.972

American Institute of Mathematical Sciences