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Robustness of nonautonomous attractors for a family of nonlocal reaction–diffusion equations without uniqueness

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Abstract

In this paper, we consider a nonautonomous nonlocal reaction–diffusion equation with a small perturbation in the nonlocal diffusion term and the nonautonomous force. Under the assumptions imposed on the viscosity function, the uniqueness of weak solutions cannot be guaranteed. In this multi-valued framework, the existence of weak solutions and minimal pullback attractors in the \(L^2\)-norm is analysed. In addition, some relationships between the attractors of the universe of fixed bounded sets and those associated to a universe given by a tempered condition are established. Finally, the upper semicontinuity property of pullback attractors w.r.t. the parameter is proved. Indeed, under suitable assumptions, we prove that the family of pullback attractors converges to the corresponding global compact attractor associated with the autonomous nonlocal limit problem when the parameter goes to zero.

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Acknowledgments

We would like to thank the referees for a careful reading and suggestions on a previous version of this manuscript. This work has been partially supported by FEDER and Ministerio de Economía y Competitividad (Spain) Grant MTM2011-22411 and by Junta de Andalucía Grant P12-FQM-1492. M.H.-C. is a fellow of Programa de FPI del Ministerio de Economía y Competitividad, reference BES-2012-053398.

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  1. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080, Seville, Spain

    Tomás Caraballo, Marta Herrera-Cobos & Pedro Marín-Rubio

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  1. Tomás Caraballo
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  2. Marta Herrera-Cobos
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  3. Pedro Marín-Rubio
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Correspondence to Marta Herrera-Cobos.

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Caraballo, T., Herrera-Cobos, M. & Marín-Rubio, P. Robustness of nonautonomous attractors for a family of nonlocal reaction–diffusion equations without uniqueness. Nonlinear Dyn 84, 35–50 (2016). https://doi.org/10.1007/s11071-015-2200-4

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