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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

 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.
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.  Google Scholar [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.  Google Scholar [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.  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.  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.  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.  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.  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.  doi: 10.1142/S0219199704001483.  Google Scholar [10] P. L. Chow, Stochastic wave equations with polynomial nonlinearity,, Ann. Appl. Probab., 12 (2002), 1.  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.  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.  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.  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.   Google Scholar [15] I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, University Lectures in Contemporary Mathematics, (1999).   Google Scholar [16] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations., 9 (1997), 307.  doi: 10.1007/BF02219225.  Google Scholar [17] G. Da Prato and J. Zabczyk,, Stochastic Equations in Infinite Dimensions,, Cambridge Univ. Press, (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.  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.  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.  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.  doi: 10.1007/BF00251249.  Google Scholar [22] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, North-Holland Publishing Co., (1981).   Google Scholar [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.  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.  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.  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, (2006), 229.  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, (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.   Google Scholar [30] K. Sato, Lévy Process and Infinitely Divisible Distributions,, Cambridge University Press, (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.  doi: 10.1016/j.na.2010.10.012.  Google Scholar [32] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, 2nd edn. Springer, (1997).   Google Scholar [33] A. Unai, Abstract nonlinear beam equations，, SUT J. Math., 29 (1993), 323.   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.  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.   Google Scholar [36] E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B, Nonlinear Monotone Operators,, Springer, (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, (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.  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.  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.  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.  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.  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.  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.  doi: 10.1142/S0219199704001483.  Google Scholar [10] P. L. Chow, Stochastic wave equations with polynomial nonlinearity,, Ann. Appl. Probab., 12 (2002), 1.  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.  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.  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.  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.   Google Scholar [15] I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, University Lectures in Contemporary Mathematics, (1999).   Google Scholar [16] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations., 9 (1997), 307.  doi: 10.1007/BF02219225.  Google Scholar [17] G. Da Prato and J. Zabczyk,, Stochastic Equations in Infinite Dimensions,, Cambridge Univ. Press, (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.  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.  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.  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.  doi: 10.1007/BF00251249.  Google Scholar [22] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, North-Holland Publishing Co., (1981).   Google Scholar [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.  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.  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.  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, (2006), 229.  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, (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.   Google Scholar [30] K. Sato, Lévy Process and Infinitely Divisible Distributions,, Cambridge University Press, (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.  doi: 10.1016/j.na.2010.10.012.  Google Scholar [32] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, 2nd edn. Springer, (1997).   Google Scholar [33] A. Unai, Abstract nonlinear beam equations，, SUT J. Math., 29 (1993), 323.   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.  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.   Google Scholar [36] E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B, Nonlinear Monotone Operators,, Springer, (1990).  doi: 10.1007/978-1-4612-0985-0.  Google Scholar
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