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Time-dependent attractors for non-autonomous non-local reaction–diffusion equations

Part of: Infinite-dimensional dissipative dynamical systems Partial differential equations Equations of mathematical physics and other areas of application Parabolic equations and systems

Published online by Cambridge University Press:  22 April 2018

Tomás Caraballo ,
Marta Herrera-Cobos  and
Pedro Marín-Rubio
Tomás Caraballo
Affiliation:
Departamento Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain (caraball@us.es; mhc@us.es; pmr@us.es)
Marta Herrera-Cobos
Affiliation:
Departamento Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain (caraball@us.es; mhc@us.es; pmr@us.es)
Pedro Marín-Rubio
Affiliation:
Departamento Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain (caraball@us.es; mhc@us.es; pmr@us.es)
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Abstract

In this paper the existence and uniqueness of weak and strong solutions for a non-autonomous non-local reaction–diffusion equation is proved. Furthermore, the existence of minimal pullback attractors in the L2-norm in the frameworks of universes of fixed bounded sets and those given by a tempered growth condition is established, along with some relationships between them. Finally, we prove the existence of minimal pullback attractors in the H1-norm and study relationships among these new families and those given previously in the L2 context. We also present new results in the autonomous framework that ensure the existence of global compact attractors as a particular case.

Keywords

non-autonomous non-local reaction–diffusion equationspullback attractorsasymptotic compactnessregularity of attractors

MSC classification

Primary: 35B41: Attractors 35B65: Smoothness and regularity of solutions 35K57: Reaction-diffusion equations 35Q92: PDEs in connection with biology and other natural sciences 37L30: Attractors and their dimensions, Lyapunov exponents
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 5 , October 2018 , pp. 957 - 981
Copyright
Copyright © Royal Society of Edinburgh 2018 

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Footnotes

Dedicated to Karin Wahl, in memoriam