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

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Carvalho, Alexandre Nolasco de ; Langa, José A. ; Robinson, James C. (2012)

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, ut = uxx + λu - β(t)u3, and investigate the bifurcations that this attractor undergoes as λ 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 (18)
    18 references, page 1 of 2

    [1] A. V. Babin and M. I. Vishik, Attractors in Evolutionary Equations, Studies in Mathematics and its Applications 25, North-Holland Publishing Co., Amsterdam, 1992. MR1156492 (93d:58090)

    [2] A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds, J. Differential Equations 233 (2007), 622-653. MR2292521 (2008f:37174)

    [3] A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Sua´rez, Characterization of nonautonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations 236 (2007), no. 2, 570-603. MR2322025 (2008e:37075)

    [4] A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations 246 (2009), 2646-2668. MR2503016 (2010a:37159)

    [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. MR0440205 (55:13084)

    [6] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Providence: AMS Colloquium Publications, vol. 49, 2002. MR1868930 (2003f:37001c)

    [7] J. Hale, Asymptotic Behavior of Dissipative Systems, Providence: Math. Surveys and Monographs, A.M.S., 1998. MR941371 (89g:58059)

    [8] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840, Berlin: Springer, 1981. MR610244 (83j:35084)

    [9] J. A. Langa, J. C. Robinson, A. Sua´rez and A. Vidal-Lo´pez, Structural stability of gradientlike attractors under non-autonomous perturbations, J. Differential Equations 234 (2007), 607-625. MR2300669 (2008c:37125)

    [10] J. A. Langa, J. C. Robinson and A. Sua´rez, Stability, instability, and bifurcation phenomena in non-autonomous differential equations, Nonlinearity 15 (2002), 887-903. MR1901112 (2003a:37063)

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