Comptes Rendus
no. 23-24
Partial Differential Equations/Optimal Control
Determination of source terms in a degenerate parabolic equation from a locally distributed observation
[Détermination d'un terme source dans une équation parabolique dégénérée à partir d'une observation interne localisée]

Présenté par : Gilles Lebeau

Jacques Tort1
1 Institut de mathématiques de Toulouse, UMR CNRS 5219, Université Paul-Sabatier Toulouse III, 118, route de Narbonne, 31 062 Toulouse cedex 4, France
Comptes Rendus. Mathématique, Volume 348 (2010) no. 23-24, pp. 1287-1291.

Le but de cette Note est de montrer un résultat d'unicité et stabilité pour un problème inverse consistant à déterminer un terme source dans une équation parabolique dégénérée en dimension 1. On reprend la méthode introduite par Imanuvilov et Yamamoto en 1998 en précisant une inégalité de Carleman récente obtenue par Cannarsa, Martinez et Vancostenoble.

The aim of this Note is to prove a Lipschitz stability and uniqueness result for an inverse source problem relative to a one-dimensional degenerate parabolic equation. We use the method introduced by Imanuvilov and Yamamoto in 1998, with the help of some recent Carleman estimate for degenerate equations obtained by Cannarsa, Martinez and Vancostenoble.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.10.031
Jacques Tort 1

1 Institut de mathématiques de Toulouse, UMR CNRS 5219, Université Paul-Sabatier Toulouse III, 118, route de Narbonne, 31 062 Toulouse cedex 4, France 
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Jacques Tort. Determination of source terms in a degenerate parabolic equation from a locally distributed observation. Comptes Rendus. Mathématique, Volume 348 (2010) no. 23-24, pp. 1287-1291. doi : 10.1016/j.crma.2010.10.031. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.10.031/

[1] L. Baudouin; J.P. Puel An inverse problem for the Schrödinger equation, Inverse Problems, Volume 18 (2002) no. 6, pp. 1537-1554

[2] A. Benabdallah; P. Gaitan; J. Le Rousseau Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation, SIAM J. Control Optim., Volume 46 (2007) no. 5, pp. 1849-1881

[3] P. Cannarsa; P. Martinez; J. Vancostenoble Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., Volume 47 (2008) no. 1, pp. 1-19

[4] P. Cannarsa; P. Martinez; J. Vancostenoble Null controllability of degenerate heat equations, Adv. Differential Equations, Volume 10 (2005) no. 2, pp. 153-190

[5] P. Cannarsa, J. Tort, M. Yamamoto, Determination of source terms in a degenerate parabolic equation, in preparation.

[6] M. Cristofol; P. Gaitan; H. Ramoul Inverse problems for a two by two reaction diffusion system using a Carleman estimate with one observation, Inverse Problems, Volume 22 (2006) no. 5, pp. 1561-1573

[7] A.V. Fursikov; O.Yu. Imanuvilov Controllability of Evolution Equations, Lecture Notes Ser., vol. 34, Seoul National University, Seoul, Korea, 1996

[8] O.Y. Imanuvilov; M. Yamamoto Lipschitz stability in inverse parabolic problems by the Carleman estimates, Inverse Problems, Volume 14 (1998) no. 5, pp. 1229-1245

[9] P. Martinez; J. Vancostenoble Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Eq., Volume 6 (2006) no. 2, pp. 325-362

[10] J.P. Puel; M. Yamamoto Applications of exact controllability to some inverse problems for the wave equation, Laredo, 1994 (Lecture Notes in Pure and Appl. Math.), Volume vol. 174, Dekker, New York (1996), pp. 241-249

[11] J. Vancostenoble Sharp Carleman estimates for singular parabolic equations and application to Lipschitz stability in inverse source problems, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010), pp. 801-805

[12] M. Yamamoto; J. Zou Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Problems, Volume 17 (2001) no. 4, pp. 1181-1202

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