Solvability of nonlinear evolution equations generated by subdifferentials and perturbations

Risei Kano; Yusuke Murase

doi:10.3934/dcdss.2014.7.75

Article Contents

2014, Volume 7, Issue 1: 75-93. Doi: 10.3934/dcdss.2014.7.75

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Solvability of nonlinear evolution equations generated by subdifferentials and perturbations

Risei Kano^1, and
Yusuke Murase^2,

1.
Faculty of Education, Kochi University, 2-5-1 Akebono-cho, Kochi 780-8520
2.
Department of Mathematics, Faculty of Science and Technology, Meijo University, 1-501 Shiogamaguchi, Tenpaku-ku, Nagoya, 468-8502

Received: March 2012

Revised: October 2012

Published: February 2014

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Abstract

The main objective of this paper is to discuss solvability of the Cauchy problem of an evolution equation with subdifferentials of convex functions which is generated by unknown functions and perturbations of the form:
$ u'(t) + ∂ \varphi^t(u;u(t)) + G(u(t)) \ni f(t) $ 0 < t < T, in H.
where H is a Hilbert space, $u'=\frac{du}{dt}$, and $∂ \varphi^t(u;\cdot )$ is a subdifferential operator of convex function $\varphi^t(u;\cdot )$. The evolution equation corresponds to parabolic quasi-variational inequalities.

Keywords:

Mathematics Subject Classification: Primary: 47J35; Secondary: 47J20.

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References

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