Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators

Dalila Azzam-Laouir; Warda Belhoula; Charles Castaing; M. D. P. Monteiro Marques

doi:10.3934/eect.2020004

Article Contents

2020, Volume 9, Issue 1: 219-254. Doi: 10.3934/eect.2020004

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Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators

1.
Laboratoire LAOTI, FSEI, Université Mohammed Seddik Benyahia de Jijel, Algérie
2.
IMAG, Université Montpellier, CNRS, Montpellier Cedex 5, France
3.
CMAFcIO, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Portugal

^* Corresponding author: Dalila Azzam-Laouir
^* Corresponding author: Dalila Azzam-Laouir

Received: December 2018

Revised: May 2019

Early access: August 2019

Published: March 2020

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Abstract

In this paper, we study the existence of solutions for evolution problems of the form $ -\frac{du}{dr}(t) \in A(t)u(t) + F(t, u(t))+f(t, u(t)) $, where, for each $ t $, $ A(t) : D(A(t)) \to 2 ^H $ is a maximal monotone operator in a Hilbert space $ H $ with continuous, Lipschitz or absolutely continuous variation in time. The perturbation $ f $ is separately integrable on $ [0, T] $ and separately Lipschitz on $ H $, while $ F $ is scalarly measurable and separately scalarly upper semicontinuous on $ H $, with convex and weakly compact values. Several new applications are provided.

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Mathematics Subject Classification: Primary: 34A60, 28C20; Secondary: 28A25.

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References

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