Partial differential equations
On parabolic final value problems and well-posedness
Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 301-305.

We prove that a large class of parabolic final value problems is well posed. This results via explicit Hilbert spaces that characterise the data yielding existence, uniqueness and stability of solutions. This data space is the graph normed domain of an unbounded operator, which represents a new compatibility condition pertinent for final value problems. The framework is that of evolution equations for Lax–Milgram operators in vector distribution spaces. The final value heat equation on a smooth open set is also covered, and for non-zero Dirichlet data, a non-trivial extension of the compatibility condition is obtained by addition of an improper Bochner integral.

Nous prouvons que les problèmes à valeur finale sont bien posés pour une large classe d'opérateurs differentiels paraboliques. Ceci est obtenu via un espace de Hilbert qui caractérise l'existence des données impliquant l'existence, l'unicité et la stabilité des solutions. Cet espace de données est le domaine d'un opérateur non borné muni de la norme du graphe, qui représente une nouvelle condition de compatibilité pertinente pour les problèmes à valeur finale. Le cadre est celui des équations d'évolution pour des opérateurs de Lax–Milgram dans des espaces de distributions vectorielles. Nous étudions aussi le problème à valeur finale pour l'équation de la chaleur sur un ouvert lisse ; pour des données de Dirichlet non nulles, nous obtenons une extension non triviale de la condition de compatibilité par l'addition d'une intégrale de Bochner impropre.

Published online:
DOI: 10.1016/j.crma.2018.01.019
Christensen, Ann-Eva 1; Johnsen, Jon 1

1 Department of Mathematics, Aalborg University, Skjernvej 4A, DK-9220 Aalborg Øst, Denmark
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     title = {On parabolic final value problems and well-posedness},
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Christensen, Ann-Eva; Johnsen, Jon. On parabolic final value problems and well-posedness. Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 301-305. doi : 10.1016/j.crma.2018.01.019.

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