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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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), 170-187. doi: 10.1016/j.jde.2007.10.016.  Google Scholar

[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.  Google Scholar

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Z. Brzeźniak and J. H. Zhu, Stochastic nonlinear beam equations driven by compensated Poisson random measures,, preprint, ().   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[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.  Google Scholar

[32]

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

[33]

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

[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.  Google Scholar

[35]

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

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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), 170-187. doi: 10.1016/j.jde.2007.10.016.  Google Scholar

[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.  Google Scholar

[7]

Z. Brzeźniak and J. H. Zhu, Stochastic nonlinear beam equations driven by compensated Poisson random measures,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[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.  Google Scholar

[32]

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

[33]

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

[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.  Google Scholar

[35]

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

[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.  Google Scholar

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