Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Structure and bifurcation of pullback attractors in a non-autonomous Chafee-Infante equation

Authors: A. N. Carvalho, J. A. Langa and J. C. Robinson
Journal: Proc. Amer. Math. Soc. 140 (2012), 2357-2373
MSC (2010): Primary 35B32, 35B40, 35B41, 37L30
Posted: October 26, 2011
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Chafee-Infante equation is one of the canonical infinite-dimensional dynamical systems for which a complete description of the global attractor is available. In this paper we study the structure of the pullback attractor for a non-autonomous version of this equation, $u_t=u_{xx}+\lambda u-\beta (t)u^3$ , and investigate the bifurcations that this attractor undergoes as $\lambda$ is varied. We are able to describe these in some detail, despite the fact that our model is truly non-autonomous; i.e., we do not restrict to `small perturbations' of the autonomous case.

References

1. A. V. Babin and M. I. Vishik, Attractors of evolution equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992. Translated and revised from the 1989 Russian original by Babin. MR 1156492 (93d:58090)
2. Alexandre N. Carvalho and José A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: continuity of local stable and unstable manifolds, J. Differential Equations 233 (2007), no. 2, 622–653. MR 2292521 (2008f:37174), http://dx.doi.org/10.1016/j.jde.2006.08.009
3. Alexandre N. Carvalho, José A. Langa, James C. Robinson, and Antonio Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations 236 (2007), no. 2, 570–603. MR 2322025 (2008e:37075), http://dx.doi.org/10.1016/j.jde.2007.01.017
4. Alexandre N. Carvalho and José A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations 246 (2009), no. 7, 2646–2668. MR 2503016 (2010a:37159), http://dx.doi.org/10.1016/j.jde.2009.01.007
5. N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal. 4 (1974/75), 17–37. MR 0440205 (55 #13084)
6. Vladimir V. Chepyzhov and Mark I. Vishik, Attractors for equations of mathematical physics, American Mathematical Society Colloquium Publications, vol. 49, American Mathematical Society, Providence, RI, 2002. MR 1868930 (2003f:37001c)
7. Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371 (89g:58059)
8. Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin, 1981. MR 610244 (83j:35084)
9. José A. Langa, James C. Robinson, Antonio Suárez, and Alejandro Vidal-López, The stability of attractors for non-autonomous perturbations of gradient-like systems, J. Differential Equations 234 (2007), no. 2, 607–625. MR 2300669 (2008c:37125), http://dx.doi.org/10.1016/j.jde.2006.11.016
10. José A. Langa, James C. Robinson, and Antonio Suárez, Stability, instability, and bifurcation phenomena in non-autonomous differential equations, Nonlinearity 15 (2002), no. 3, 887–903. MR 1901112 (2003a:37063), http://dx.doi.org/10.1088/0951-7715/15/3/322
11. José A. Langa, James C. Robinson, Aníbal Rodríguez-Bernal, and Antonio Suárez, Permanence and asymptotically stable complete trajectories for nonautonomous Lotka-Volterra models with diffusion, SIAM J. Math. Anal. 40 (2009), no. 6, 2179–2216. MR 2481291 (2010i:35151), http://dx.doi.org/10.1137/080721790
12. José A. Langa, James C. Robinson, Antonio Suárez, and Alejandro Vidal-López, The stability of attractors for non-autonomous perturbations of gradient-like systems, J. Differential Equations 234 (2007), no. 2, 607–625. MR 2300669 (2008c:37125), http://dx.doi.org/10.1016/j.jde.2006.11.016
13. José A. Langa, Aníbal Rodríguez-Bernal, and Antonio Suárez, On the long time behavior of non-autonomous Lotka-Volterra models with diffusion via the sub-supertrajectory method, J. Differential Equations 249 (2010), no. 2, 414–445. MR 2644122 (2011g:37239), http://dx.doi.org/10.1016/j.jde.2010.04.001
14. Jose A. Langa and Antonio Suárez, Pullback permanence for non-autonomous partial differential equations, Electron. J. Differential Equations (2002), No. 72, 20 pp. (electronic). MR 1921145 (2004a:35105)
15. Tian Ma and Shouhong Wang, Attractor bifurcation theory and its applications to Rayleigh-Bénard convection, Commun. Pure Appl. Anal. 2 (2003), no. 4, 591–599. MR 2019070 (2004i:37153), http://dx.doi.org/10.3934/cpaa.2003.2.591
16. Hiroshi Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 2, 401–441. MR 672070 (84m:35060)
17. James C. Robinson, Infinite-dimensional dynamical systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. An introduction to dissipative parabolic PDEs and the theory of global attractors. MR 1881888 (2003f:37001a)
18. Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. MR 953967 (89m:58056)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35B32, 35B40, 35B41, 37L30

Retrieve articles in all journals with MSC (2010): 35B32, 35B40, 35B41, 37L30

Additional Information

A. N. Carvalho
Affiliation: Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil
Email: andcarva@icmc.usp.br

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

J. C. Robinson
Affiliation: Mathematical Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: j.c.robinson@warwick.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11071-2
PII: S 0002-9939(2011)11071-2
Received by editor(s): September 14, 2010
Received by editor(s) in revised form: February 15, 2011
Posted: October 26, 2011
Additional Notes: The first author was partially supported by CNPq 302022/2008-2, CAPES/DGU 267/2008 and FAPESP 2008/55516-3, Brazil
The second author was partially supported by Ministerio de Ciencia e Innovación grants #MTM2008-0088, #PBH2006-0003-PC, and Junta de Andalucía grants #P07-FQM-02468, #FQM314 and #HF2008-0039, Spain
The third author is currently an EPSRC Leadership Fellow, grant #EP/G007470/1.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2011 American Mathematical Society

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|