Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions

Matheus C. Bortolan; José Manuel Uzal

doi:10.3934/dcds.2020150

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2020, Volume 40, Issue 5: 2791-2826. Doi: 10.3934/dcds.2020150

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Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions

Matheus C. Bortolan^1, , and
José Manuel Uzal^2,

1.
Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis - Brazil
2.
Departamento de Estatística, Análise Matemática e Optimización & Instituto de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela - Spain

^* Corresponding author: Matheus C. Bortolan
^* Corresponding author: Matheus C. Bortolan

Received: July 2019

Published: March 2020

The first author is by supported by CNPq, project # 407635/2016-5. The second author was partially supported by the predoctoral contact BES-2017-082334

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Abstract

We define the notions of impulsive evolution processes and their pullback attractors, and exhibit conditions under which a given impulsive evolution process has a pullback attractor. We apply our results to a nonautonomous ordinary differential equation describing an integrate-and-fire model of neuron membrane, as well as to a heat equation with nonautonomous impulse and a nonautonomous 2D Navier-Stokes equation. Finally, we introduce the notion of tube conditions to impulsive evolution processes, and use them as an alternative way to obtain pullback attractors.

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Mathematics Subject Classification: Primary: 37L05, 37L30. Secondary: 34H05.

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References

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