Asymptotic behavior of nonlocal partial differential equations with long time memory

Jiaohui Xu; Tomás Caraballo; José Valero

doi:10.3934/dcdss.2021140

Article Contents

2022, Volume 15, Issue 10: 3059-3078. Doi: 10.3934/dcdss.2021140

This issue Previous Article Random attractors for stochastic delay wave equations on $ \mathbb{R}^n $ with linear memory and nonlinear damping Next Article

Asymptotic behavior of nonlocal partial differential equations with long time memory

1.
Departmento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain
2.
Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avda. de la Universidad, s/n, 03202-Elche, Spain

^* Corresponding author: Tomás Caraballo
^* Corresponding author: Tomás Caraballo

Dedicated to Georg Hetzer on occasion of his 75th birthday

Received: May 31, 2021

Revised: August 31, 2021

Early access: November 2021

Published: September 2022

The research has been partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00, and by Junta de Andalucía (Consejería de Economía y Conocimiento) and FEDER under projects US-1254251 and P18-FR-4509

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Abstract

In this paper, it is first addressed the well-posedness of weak solutions to a nonlocal partial differential equation with long time memory, which is carried out by exploiting the nowadays well-known technique used by Dafermos in the early 70's. Thanks to this Dafermos transformation, the original problem with memory is transformed into a non-delay one for which the standard theory of autonomous dynamical system can be applied. Thus, some results about the existence of global attractors for the transformed problem are {proved}. Particularly, when the initial values have higher regularity, the solutions of both problems (the original and the transformed ones) are equivalent. Nevertheless, the equivalence of global attractors for both problems is still unsolved due to the lack of enough regularity of solutions in the transformed problem. It is therefore proved the existence of global attractors of the transformed problem. Eventually, it is highlighted how to proceed to obtain meaningful results about the original problem, without performing any transformation, but working directly with the original delay problem.

Keywords:

Nonlocal partial differential equations,
Long time memory,
Dafermos transformation,
Global attractors.

Mathematics Subject Classification: 35B40, 35K05, 37L30, 45K05.

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References

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