An example of noncontinuous attractors

Abstract

In this paper, we investigate the continuity with respect to the diffusion parameter \({\epsilon}\) of the asymptotic states of an one-dimensional p-Laplacian problem which is a nonlinear version of the well-known Chafee–Infante problem. We obtain the upper semicontinuity at any value \({\epsilon > 0}\) and exhibit cases where the lower semicontinuity of the set of equilibria with respect to the diffusion is always true. Finally, we study a particular case where the lower semicontinuity of the equilibrium set occurs for almost every \({\epsilon > 0}\) but fails when \({\epsilon}\) belongs to the bifurcation sequence. As a consequence, we show that the attractor cannot be continuous at these points.

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Correspondence to Ana Claudia Pereira.

Additional information

A. C. Pereira was partially supported by CAPES (Brazil).

O. H. Miyagaki was partially supported by CNPq (Brazil).

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Pereira, A.C., Gentile Moussa, C.B. & Miyagaki, O.H. An example of noncontinuous attractors. J. Evol. Equ. 15, 979–1000 (2015). https://doi.org/10.1007/s00028-015-0289-z

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Mathematics Subject Classification

  • 35B40
  • 35B41
  • 35K92

Keywords

  • Continuity of attractor
  • p-Laplacian
  • Asymptotic behavior