Parabolic quasi-variational diffusion problems with gradient constraints

Nobuyuki Kenmochi

doi:10.3934/dcdss.2013.6.423

Article Contents

2013, Volume 6, Issue 2: 423-438. Doi: 10.3934/dcdss.2013.6.423

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Parabolic quasi-variational diffusion problems with gradient constraints

Nobuyuki Kenmochi^1,

1.
Department of Education, School of Education, Bukkyo University, 96 Kitahananobo-cho, Murasakino, Kita-ku, Kyoto, 603-8301

Received: June 30, 2011

Revised: November 30, 2011

Published: March 2013

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Abstract

In this paper we consider some mechanical phenomena whose dynamics is described by a class of quasi-variational inequalities of parabolic type. Our system consists of a second-order parabolic variational inequality with gradient constraint depending on the temperature and the heat equation. Since the temperature is unknown in our problem, the constraint function is unknown as well. In this sense, our problem includes the quasi-variational structure, and in the mathamtical analysis one of main difficulties comes from it. Our approach to the problem is based on the abstract theory of quasi-variational inequalities with non-local constraint which has been developed in [6]. However the abstract theory is not directly used in the existence proof of a solution, since the mathematical situation of the problem is much nicer than that in the abstract theory [6]. In this paper we prove the existence of a weak solution of our system.

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Mathematics Subject Classification: 35K35, 35K55, 34K40, 35K87.

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References

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