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A Permutation Related to Non-compact Global Attractors for Slowly Non-dissipative Systems

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Abstract

We consider scalar reaction-diffusion equations with non-dissipative nonlinearities generating global semiflows which exhibit blow-up in infinite time. This type of equations was only recently approached and the corresponding dynamical systems are known as slowly non-dissipative systems. The existence of unbounded solutions, referred to as grow-up solutions, requires the introduction of some objects interpreted as equilibria at infinity. By extending known results, we are able to obtain a complete decomposition of the associated non-compact global attractor. The connecting orbit structure is determined based on the Sturm permutation method, which yields a simple criterion for the existence of heteroclinic connections.

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Acknowledgments

This work was partially supported by FCT/Portugal through the project PEst-OE/EEI/LA0009/2013. The first author was supported by SFRH/BD/51389/2011 (FCT/Portugal), and GDE/246318/2012-0 (CNPq-Brazil). The authors also acknowledge the useful comments of the referee.

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Authors and Affiliations

  1. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Av. Trabalhador Sancarlense, 13566-590, São Carlos, Brazil

    Juliana Pimentel

  2. Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Carlos Rocha

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  1. Juliana Pimentel
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  2. Carlos Rocha
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Correspondence to Juliana Pimentel.

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Pimentel, J., Rocha, C. A Permutation Related to Non-compact Global Attractors for Slowly Non-dissipative Systems. J Dyn Diff Equat 28, 1–28 (2016). https://doi.org/10.1007/s10884-014-9414-x

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  • DOI: https://doi.org/10.1007/s10884-014-9414-x

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