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Asymptotic behavior for a nonlocal model of neural fields

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Abstract

In this paper we consider the non local evolution problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u =- u + K(f\circ u ) \ \ in \ \ \Omega ,\\ u = 0 \ \ in \ \ \mathbb {R}^N\backslash \Omega , \end{array}\right. } \end{aligned}$$

where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^N\), \(f: \mathbb {R}\rightarrow \mathbb {R}\) and K is an integral operator with a symmetric kernel. We prove existence and some regularity properties of the global attractor. We also characterize the global attractor, using the properties of a Lyapunov functional for this model.

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Acknowledgments

The authors would like to thank the anonymous referee for his careful reading of the manuscript and his helpful suggestions.

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Authors and Affiliations

  1. Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB, 58051-900, Brazil

    Severino H. da Silva

  2. Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP, 13565-905, Brazil

    Antônio L. Pereira

Authors
  1. Severino H. da Silva
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  2. Antônio L. Pereira
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Correspondence to Antônio L. Pereira.

Additional information

Severino H. da Silva: Research partially supported by CAPES/CNPq.

Antônio L. Pereira: Research partially supported by CNPq.

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da Silva, S.H., Pereira, A.L. Asymptotic behavior for a nonlocal model of neural fields. São Paulo J. Math. Sci. 9, 181–194 (2015). https://doi.org/10.1007/s40863-015-0018-0

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