An abstract existence theorem for parabolic systems

Hans Wilhelm Alt

doi:10.3934/cpaa.2012.11.2079

Article Contents

2012, Volume 11, Issue 5: 2079-2123. Doi: 10.3934/cpaa.2012.11.2079

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An abstract existence theorem for parabolic systems

Hans Wilhelm Alt^1,

1.
University Bonn, Institute for Applied Mathematics, Endenicher Allee 60, 53115 Bonn

Received: December 31, 2010

Revised: April 30, 2011

Published: August 2012

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Abstract

In this paper we prove an abstract existence theorem which can be applied to solve parabolic problems in a wide range of applications. It also applies to parabolic variational inequalities. The abstract theorem is based on a Gelfand triple $(V,H,V^*)$, where the standard realization for parabolic systems of second order is $(W^{1, 2}(\Omega),L^2(\Omega), W^{1,2}(\Omega)^*)$. But also realizations to other problems are possible, for example, to fourth order systems.
In all applications to boundary value problems the set $M\subset V$ is an affine subspace, whereas for variational inequalities the constraint $M$ is a closed convex set.
The proof is purely abstract and new.
The corresponding compactness theorem is based on [5].
The present paper is suitable for lectures, since it relays on the corresponding abstract elliptic theory.

Keywords:

Mathematics Subject Classification: 35D30, 35K59, 35K65, 35K90, 35R37, 47F05, 47J30, 47J35, 47H05.

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