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Lower Semicontinuity of Global Attractors for a Class of Evolution Equations of Neural Fields Type in a Bounded Domain

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Abstract

In this work we consider the nonlocal evolution equation

$$\displaystyle \frac{\partial u(w,t)}{\partial t}=-u(w,t)+ \int _{S^{1}}J(wz^{-1})f(u(z,t))\mathrm{d}z+ h, \quad h > 0,$$

which arises in models of neuronal activity, in \(L^{2}(S^{1})\), where \(S^{1}\) denotes the unit sphere. We obtain more interesting results on existence of global attractors and the associate Lypaunov functional than the already existing in the literature. Furthermore, we prove the result, not yet known in the literature, of lower semicontinuity of global attractors with respect to connectivity function J.

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Acknowledgments

The author would like to thank the anonymous referees for his/her reading of the manuscript and valuable suggestions. He also wishes to thank the professors Antônio Luiz Pereira (USP) and Flank Bezerra (UFPB) for discussions on this model. Our gratitude goes also to Oxford International English (Campina Grande) that checked the English usage this paper.

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Authors and Affiliations

  1. Unidade Acadêmica de Matemática UAMat/CCT/UFCG, Rua Aprígio Veloso, 882, Bairro Universitário, Campina Grande, PB, CEP 58429-900, Brazil

    Severino Horácio da Silva

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  1. Severino Horácio da Silva
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Correspondence to Severino Horácio da Silva.

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The author dedicates this work to Arthur and Luana.

Partially supported by CAPES/CNPq-Brazil Grant Casadinho/Procad 552.464/2011-2 and INCTMat 5733523/2008-8.

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da Silva, S.H. Lower Semicontinuity of Global Attractors for a Class of Evolution Equations of Neural Fields Type in a Bounded Domain. Differ Equ Dyn Syst 26, 371–391 (2018). https://doi.org/10.1007/s12591-015-0258-6

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