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Exponential stability of stochastic generalized porous media equations with jump

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Abstract

Stochastic generalized porous media equation with jump is considered. The aim is to show the moment exponential stability and the almost certain exponential stability of the stochastic equation.

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References

  1. Peszat, S. and Zabczyk, J. Stochastic Partial Differential Equation with Lévy Noise: an Evolution Equation Approach, Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  2. Da Prato, G. and Zabczyk, J. Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  3. Liu, K. Stability of Infinite Dimensional Stochastic Differential Equations with Applications, CRC Press, Boca Raton (2006)

    MATH  Google Scholar 

  4. Mariton, M. Jump Linear Systems in Automatic Control, CRC Press, Boca Raton (1990)

    Google Scholar 

  5. Basak, G. K., Bisi, A., and Ghosh, M. K. Stability of a random diffusion with linear drift. Journal of Mathematical Analysis and Applications, 202, 604–622 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  6. Mao, X. Stability of stochastic differential equations with Markovian switching. Stochastic Processes and Their Applications, 79, 45–67 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Mao, X. Asymptotic stability for stochastic differential delay equations with Markovian switching. Functional Differential Equations, 9, 201–220 (2002)

    MATH  MathSciNet  Google Scholar 

  8. Mao, X., Matasov, A., and Piunovskiy, A. Stochastic differential delay equations with Markovian switching. Bernoulli, 6(1), 73–90 (2002)

    Article  MathSciNet  Google Scholar 

  9. Luo, J.W. and Liu, K. Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps. Stochastic Processes and Their Applications, 118, 864–895 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Aronson, D. G. The porous medium equation. Lecture Notes in Mathematics, 1224, 1–46 (1986)

    Article  MathSciNet  Google Scholar 

  11. Aronson, D. G. and Peletier, L. A. Large time behaviour of solutions of the porous medium equation in bounded domains. Journal of Differential Equations, 39, 378–412 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  12. Beyn, W. J., Gess, B., Lescot, P., and Röckner, M. The global random attractors for a class of stochastic porous media equations. Communications in Partial Differential Equations, 36, 446–469 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Da Prato, G. and Röckner, M. Weak solutions to stochastic porous media equations. Journal of Evolution Equations, 4, 249–271 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Da Prato, G. and Röckner, M. Invariant measures for a stochastic porous media equation. Advances in Pure Mathematics, 41, 13–29 (2004)

    Google Scholar 

  15. Da Prato, G., Röckner, M., Rozovskii, B. L., and Wang, F. Y. Strong solutions of stochastic generalized porous media equations: existence, uniqueness and ergodicity. Commnications in Partial Differential Equations, 31, 277–291 (2006)

    Article  MATH  Google Scholar 

  16. Ebmeye, C. Regularrity in Sobolev spaces for the fast diffusion and the porous medium equation. Journal of Mathematical Analysis and Applications, 307, 134–152 (2005)

    Article  MathSciNet  Google Scholar 

  17. Ren, J., Röckner, M., and Wang, F. Y. Stochastic generalized porous meadia and fast diffusion equation. Journal of Differential Equations, 238, 118–152 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kim, J. U. On the stochastic porous medium equation. Journal of Differential Equations, 206, 163–194 (2006)

    Article  Google Scholar 

  19. Röckner, M. and Wang, F. Y. Non-monotone stochastic generalized porous media equations. Journal of Differential Equations, 245, 3898–3935 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Röckner, M., Wang, F. Y., and Wu, L. M. Large deviations for stochastic generalized porous media equations. Stochastic Processes and Their Applications, 116, 1677–1689 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Wang, F. Y. Harnack inequality and applications for stochastic generalized porous media equations. Annals of Probability, 35(4), 1333–1350 (2006)

    Article  Google Scholar 

  22. Caraballo, T. and Liu, K. On the exponential stability criteria of stochastic partial differential equations. Stochastic Processes and Their Applications, 83, 289–301 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  23. Zhou, G. L. and Hou, Z. T. Stochastic generalized porous media equations with L`evy jump. Acta Mathematica Sinica, 27, 1671–1696 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Zhou, G. L. and Hou, Z. T. The ergodicity of stochastic generalized porous media equations with Lévy jump. Acta Mathematica Scientia, 31, 925–933 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, P. R. China

    Bo-ling Guo  (郭柏灵)

  2. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P. R. China

    Guo-li Zhou  (周国立)

Authors
  1. Bo-ling Guo  (郭柏灵)
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  2. Guo-li Zhou  (周国立)
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Corresponding author

Correspondence to Guo-li Zhou  (周国立).

Additional information

Project supported by the Tianyuan Foundation of National Natural Science of China (No. 11126079), the China Postdoctoral Science Foundation (No. 2013M530559), and the Fundamental Research Funds for the Central Universities (No.CDJRC10100011)

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Guo, Bl., Zhou, Gl. Exponential stability of stochastic generalized porous media equations with jump. Appl. Math. Mech.-Engl. Ed. 35, 1067–1078 (2014). https://doi.org/10.1007/s10483-014-1845-7

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  • DOI: https://doi.org/10.1007/s10483-014-1845-7

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

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