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Attractors for a Nonlinear Parabolic Problem with Terms Concentrating on the Boundary

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Abstract

We analyze the dynamics of the flow generated by a nonlinear parabolic problem when some reaction and potential terms are concentrated in a neighborhood of the boundary. We assume that this neighborhood shrinks to the boundary as a parameter \(\epsilon \) goes to zero. Also, we suppose that the “inner boundary” of this neighborhood presents a highly oscillatory behavior. Our main goal here is to show the continuity of the family of attractors with respect to \(\epsilon \). Indeed, we prove upper semicontinuity under the usual properties of regularity and dissipativeness and, assuming hyperbolicity of the equilibria, we also show the lower semicontinuity of the attractors at \(\epsilon =0\).

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Acknowledgments

The authors thank Professor Sérgio M. Oliva for their suggestions and remarks. We also would like to thank the anonymous referee whose comments have considerably improved the writing of the paper. G. S. Aragão partially supported by FAPESP 2010/51829-7, Brazil. A. L. Pereira partially supported by CNPq 308696/2006-9, FAPESP 2008/55516-3, Brazil. M. C. Pereira partially supported by CNPq 302847/2011-1 and 471210/2013-7, FAPESP 2008/53094-4 and 2010/18790-0, Brazil

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Authors and Affiliations

  1. Instituto de Ciências Ambientais, Químicas e Farmacêuticas, Universidade Federal de São Paulo, Diadema, SP, Brazil

    Gleiciane S. Aragão

  2. Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP, Brazil

    Antônio L. Pereira

  3. Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP, Brazil

    Marcone C. Pereira

Authors
  1. Gleiciane S. Aragão
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  2. Antônio L. Pereira
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  3. Marcone C. Pereira
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Correspondence to Marcone C. Pereira.

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Aragão, G.S., Pereira, A.L. & Pereira, M.C. Attractors for a Nonlinear Parabolic Problem with Terms Concentrating on the Boundary. J Dyn Diff Equat 26, 871–888 (2014). https://doi.org/10.1007/s10884-014-9412-z

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  • DOI: https://doi.org/10.1007/s10884-014-9412-z

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