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Axioms 2018, 7(2), 31; https://doi.org/10.3390/axioms7020031

Final Value Problems for Parabolic Differential Equations and Their Well-Posedness

1,†
and
2,†,*
1
Unit of Epidemiology and Biostatistics, Aalborg University Hospital, Hobrovej 18-22, DK-9000 Aalborg, Denmark
2
Department of Mathematics, Aalborg University, Skjernvej 4A, DK-9220 Aalborg Øst, Denmark
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Received: 29 March 2018 / Revised: 24 April 2018 / Accepted: 28 April 2018 / Published: 9 May 2018
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Abstract

This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of the corresponding solutions. The data space is given as the graph normed domain of an unbounded operator occurring naturally in the theory. It induces a new compatibility condition, which relies on the fact, shown here, that analytic semigroups always are invertible in the class of closed operators. The general set-up is evolution equations for Lax–Milgram operators in spaces of vector distributions. As a main example, the final value problem of the heat equation on a smooth open set is treated, and non-zero Dirichlet data are shown to require a non-trivial extension of the compatibility condition by addition of an improper Bochner integral. View Full-Text
Keywords: parabolic boundary problem; final value; compatibility condition; well posed; non-selfadjoint; hyponormal parabolic boundary problem; final value; compatibility condition; well posed; non-selfadjoint; hyponormal
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Christensen, A.-E.; Johnsen, J. Final Value Problems for Parabolic Differential Equations and Their Well-Posedness. Axioms 2018, 7, 31.

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