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The effect of noise on the Chafee-Infante equation: A nonlinear case study


Authors: Tomás Caraballo, Hans Crauel, José A. Langa and James C. Robinson
Journal: Proc. Amer. Math. Soc. 135 (2007), 373-382
MSC (2000): Primary 37L55, 35K57; Secondary 60H15
DOI: https://doi.org/10.1090/S0002-9939-06-08593-5
Published electronically: August 1, 2006
MathSciNet review: 2255283
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Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the effect of perturbing the Chafee-Infante scalar reaction diffusion equation, $ u_t-\Delta u=\beta u-u^3$, by noise. While a single multiplicative Itô noise of sufficient intensity will stabilise the origin, its Stratonovich counterpart leaves the dimension of the attractor essentially unchanged. We then show that a collection of multiplicative Stratonovich terms can make the origin exponentially stable, while an additive noise of sufficient richness reduces the random attractor to a single point.


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  • 1. H. Allouba and J.A. Langa, Semimartingale attractor for Allen-Cahn SPDEs driven by space-time white noise I: Existence and finite dimensional asymptotic behaviour, Stoch. Dyn. 4 (2004), 223-244. MR 2069690 (2005e:60124)
  • 2. L. Arnold, Random Dynamical Systems, Springer, New York, 1998. MR 1723992 (2000m:37087)
  • 3. L. Arnold and I. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications, Dyn. Stab. Sys. 13 (1998), 265-280. MR 1645467 (99i:58096)
  • 4. L. Arnold, H. Crauel, and V. Wihstutz, Stabilization of linear systems by noise, SIAM J. Control Optim. 21 (1983), 451-461. MR 0696907 (84g:93080)
  • 5. A.V. Babin and M.I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992. MR 1156492 (93d:58090)
  • 6. Yu. Bakhtin and J.C. Mattingly, Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations, Preprint 2003.
  • 7. T. Caraballo, J.A. Langa, and J.C. Robinson, Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Discrete Cont. Dyn. Sys. 6 (2000), 875-892. MR 1788258 (2001j:37093)
  • 8. T. Caraballo, J.A. Langa, and J.C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, R. Soc. Lond. Proc. Ser. A 457 (2001), 2041-2061. MR 1857922 (2003a:60106)
  • 9. T. Caraballo, K. Liu, and X. Mao, On stabilization of partial differential equations by noise, Nagoya Math. J. 161 (2001), 155-170. MR 1820216 (2002b:60110)
  • 10. T. Caraballo and J.C. Robinson, Stabilisation of linear PDEs by Stratonovich noise, Systems Control Lett. 53 (2004), 41-50. MR 2077187 (2005e:60125)
  • 11. I. D. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dyn. Sys. 19 (2004), 127-144. MR 2060422 (2005c:37095)
  • 12. I. D. Chueshov and P.A. Vuillermot, Non-random invariant sets for some systems of parabolic stochastic partial differential equations, Stochastic Anal. Appl. 22 (2004), 1421-1486. MR 2095066 (2005h:37197)
  • 13. H. Crauel, White noise eliminates instability, Archiv der Mathematik 75 (2000), 472-480. MR 1799434 (2001k:37084)
  • 14. H. Crauel, Random Probability Measures on Polish Spaces, Taylor&Francis, London, 2002. MR 1993844 (2004e:60005)
  • 15. H. Crauel and F. Flandoli, Additive noise destroys a pitchfork bifurcation, J. Dyn. Diff. Eqn. 10 (1998), 259-274. MR 1623013 (99d:58115)
  • 16. H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems, J. Dyn. Diff. Eqn. 10 (1998), 449-474. MR 1646622 (99h:58117)
  • 17. G. Da Prato and A. Debussche, Construction of stochastic inertial manifolds using backward integration, Stoch. and Stoch. Reports 59 (1996), 305-324. MR 1427743 (98d:60121)
  • 18. G. Da Prato, A. Debussche, and B. Goldys, Some properties of invariant measures of non symmetric dissipative stochastic systems, Prob. Th. Rel. Fields 123 (2002), 355-380. MR 1918538 (2003e:60136)
  • 19. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. MR 1207136 (95g:60073)
  • 20. A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pures Appl. 10 (1998), 967-988. MR 1661029 (99k:60161)
  • 21. Weinan E and Di Liu, Gibbsian dynamics and invariant measures for stochastic dissipative PDEs, J. Stat. Phys. 108 (2002), 1125-1156. MR 1933448 (2003k:37136)
  • 22. J.-P. Eckmann and M. Hairer, Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise, Commun. Math. Phys. 219 (2001), 523-565. MR 1838749 (2002d:60054)
  • 23. J.K. Hale, Asymptotic behavior of dissipative dynamical systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, R.I., 1988. MR 0941371 (89g:58059)
  • 24. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Verlag, Berlin, 1984. MR 0610244 (83j:35084)
  • 25. P. Kotelenez, Positive solutions for a class of stochastic partial differential equations. Stochastic partial differential equations and applications (Trento, 1990), 235-238, Pitman Res. Notes Math. Ser., 268, Longman Sci. Tech., Harlow, 1992. MR 1222700 (94g:60114)
  • 26. J.A. Langa and J.C. Robinson, Fractal dimension of a random invariant set, J. Math. Pures Appl. 85 (2006), 269-294. MR 2199015
  • 27. Yuhong Li, Asymptotical behaviour of 2D stochastic Navier-Stokes equations, Ph.D. Thesis, University of Hull 2004.
  • 28. M. Marion, Attractors for reaction-diffusion equations: existence and estimate of their dimension, Appl. Anal. 25 (1987), 101-147. MR 0911962 (88m:35082)
  • 29. E. Pardoux, Équations aux Dérivées Partielles Stochastiques non Linéaires Monotones, Thesis, Univ. Paris XI, 1975.
  • 30. J.C. Robinson, Infinite-dimensional dynamical systems, Cambridge University Press, Cambridge, 2001. MR 1881888 (2003f:37001a)
  • 31. O.M. Tearne, Collapse of attractors for ODEs under small random perturbations, Submitted.
  • 32. R. Temam, Infinite dimensional dynamical systems in mechanics and physics, Springer, New York, 1997. MR 1441312 (98b:58056)

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Additional Information

Tomás Caraballo
Affiliation: Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain
Email: caraball@us.es

Hans Crauel
Affiliation: Institut für Mathematik, Technische Universität Ilmenau, Weimarer Straße 25, 98693 Ilmenau, Germany
Email: hans.crauel@tu-ilmenau.de

José A. Langa
Affiliation: Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain
Email: langa@us.es

James C. Robinson
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL United Kingdom
Email: jcr@maths.warwick.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-06-08593-5
Keywords: Chafee-Infante equation, stochastic stabilisation, random attractor, random equilibrium, one-point attractor, attractor collapse
Received by editor(s): August 12, 2005
Published electronically: August 1, 2006
Additional Notes: The first and third authors were supported by Ministerio de Ciencia y Tecnología (Spain) and FEDER (European Community), project BFM2002-03068.
The fourth author is a Royal Society University Research Fellow, and would like to thank the Society for all their support.
Communicated by: Walter Craig
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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