Second order doubly nonlinear evolution inclusions –quasi-variational approach–

Nobuyuki Kenmochi; Ken Shirakawa; Noriaki Yamazaki

doi:10.3934/dcdss.2023203

Article Contents

2024, Volume 17, Issue 1: 304-325. Doi: 10.3934/dcdss.2023203

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Second order doubly nonlinear evolution inclusions –quasi-variational approach–

1.
Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-chō, Inage-Ku, Chiba, 263-8522, Japan
2.
Department of Applied Systems and Mathematics, Faculty of Informatics, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-Ku, Yokohama, 221-8686, Japan

^*Corresponding author: Noriaki Yamazaki
^*Corresponding author: Noriaki Yamazaki

Dedicated to Professor Pierluigi Colli on the occasion of his 65th birthday

Received: May 2023

Revised: October 2023

Early access: November 6, 2023

Published online: November 6, 2023

This work was supported by JSPS KAKENHI Grant Numbers 20K03672 (Ken Shirakawa), and 20K03665 (Noriaki Yamazaki).

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Abstract

In this paper we study a class of second order doubly nonlinear evolution inclusions in Banach spaces of the form

$ u''(t)+\partial_*\psi^t(\theta; u'(t))+\partial_*\varphi^t(\theta; u(t)) \ni f(t)\; \; {\rm in}\; V^*, \; 0<t<T,\; \; \theta = \Lambda u, $

where $ H $ is a (real) Hilbert space, $ V $ is a Banach space, and $ V^* $ is the dual space of $ V $ such that $ V $ is dense and compactly embedded in $ H $; in this case we have the usual triplet $ V\subset H \subset V^* $; $ \psi^t(\theta;u) $ and $ \varphi^t(\theta;u) $ are time-dependent convex functions with respect to $ u $, where $ \theta $ is a parameter; $ \Lambda $ is a feedback system from a subset of $ L^2(0,T;V) $ into the space of parameters $ \theta $. Under some assumptions on functions $ \psi^t(\theta;\cdot) $ and $ \varphi^t(\theta; \cdot) $, we shall give an abstract existence result and an application in parabolic quasi-variational inequalities.

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Mathematics Subject Classification: Primary: 47J35, 34G20; Secondary: 47H05, 47H14.

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References

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