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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.

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