Comptes Rendus
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
Ann-Eva Christensen 1; Jon Johnsen 1

1 Department of Mathematics, Aalborg University, Skjernvej 4A, DK-9220 Aalborg Øst, Denmark
     author = {Ann-Eva Christensen and Jon Johnsen},
     title = {On parabolic final value problems and well-posedness},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {301--305},
     publisher = {Elsevier},
     volume = {356},
     number = {3},
     year = {2018},
     doi = {10.1016/j.crma.2018.01.019},
     language = {en},
AU  - Ann-Eva Christensen
AU  - Jon Johnsen
TI  - On parabolic final value problems and well-posedness
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 301
EP  - 305
VL  - 356
IS  - 3
PB  - Elsevier
DO  - 10.1016/j.crma.2018.01.019
LA  - en
ID  - CRMATH_2018__356_3_301_0
ER  - 
%0 Journal Article
%A Ann-Eva Christensen
%A Jon Johnsen
%T On parabolic final value problems and well-posedness
%J Comptes Rendus. Mathématique
%D 2018
%P 301-305
%V 356
%N 3
%I Elsevier
%R 10.1016/j.crma.2018.01.019
%G en
%F CRMATH_2018__356_3_301_0
Ann-Eva Christensen; Jon Johnsen. 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.

[1] Y. Almog; B. Helffer On the spectrum of non-selfadjoint Schrödinger operators with compact resolvent, Commun. Partial Differ. Equ., Volume 40 (2015) no. 8, pp. 1441-1466

[2] H. Amann Linear and Quasilinear Parabolic Problems, Vol. I, Abstract Linear Theory, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, USA, 1995

[3] A.-E. Christensen; J. Johnsen Final value problems for parabolic differential equations and their well-posedness, 2017 (preprint) | arXiv

[4] D.S. Grebenkov; B. Helffer On the spectral properties of the Bloch–Torrey operator in two dimensions, 2016 | arXiv

[5] D.S. Grebenkov; B. Helffer; R. Henry The complex Airy operator on the line with a semipermeable barrier, SIAM J. Math. Anal., Volume 49 (2017) no. 3, pp. 1844-1894

[6] G. Grubb; V.A. Solonnikov Solution of parabolic pseudo-differential initial-boundary value problems, J. Differ. Equ., Volume 87 (1990), pp. 256-304

[7] V. Isakov Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, vol. 127, Springer-Verlag, New York, 1998

[8] J. Janas On unbounded hyponormal operators III, Stud. Math., Volume 112 (1994), pp. 75-82

[9] F. John Numerical solution of the equation of heat conduction for preceding times, Ann. Mat. Pura Appl. (4), Volume 40 (1955), pp. 129-142

[10] J.-L. Lions; E. Magenes Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, vol. 181, Springer-Verlag, New York–Heidelberg, 1972 (translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181)

[11] J.-L. Lions; B. Malgrange Sur l'unicité rétrograde dans les problèmes mixtes parabolic, Math. Scand., Volume 8 (1960), pp. 227-286

[12] W.L. Miranker A well posed problem for the backward heat equation, Proc. Amer. Math. Soc., Volume 12 (1961), pp. 243-247

[13] R.E. Showalter The final value problem for evolution equations, J. Math. Anal. Appl., Volume 47 (1974), pp. 563-572

[14] H. Tanabe Equations of Evolution, Monographs and Studies in Mathematics, vol. 6, Pitman, Boston, Mass, 1979 (translated from the Japanese by N. Mugibayashi and H. Haneda)

[15] R. Temam Navier–Stokes Equations, Theory and Numerical Analysis, Elsevier Science Publishers B.V., Amsterdam, 1984

Cited by Sources:

Comments - Policy