Carleman estimates for degenerate parabolic equations with first order terms and applications

Carmelo Flores; Luz de Teresa

doi:10.1016/j.crma.2010.01.007

Partial Differential Equations/Optimal Control

Carleman estimates for degenerate parabolic equations with first order terms and applications
[Une inégalitée de Carleman pour une équation parabolique dégenerée avec des termes de premier ordre et applications]

Présenté par : Gilles Lebeau

Carmelo Flores ¹ ; Luz de Teresa ²

¹ Universidad Autónoma de la Ciudad de México, Prolongacion San Isidro, 151 San Lorenzo Tezonco Delegac, 09790 Mexico, Distrito Federal, Mexico
² Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico

Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 391-396.

Résumé
Abstract

Dans cette Note on montre une inégalité de Carleman pour une équation unidimensionelle parabolique dégenerée

v_{t} + (x^{α} v_{x})_{x} - b (x, t) v + (x^{β / 2} c (x, t) v)_{x} = 0, (x, t) \in (0, 1) \times (0, T),

où

α \in [0, 2)

,

β ⩾ α

et

b (x, t), c (x, t) \in L^{\infty} ((0, 1) \times (0, T))

et on donne des conséquences en controllabilitée.

In this Note we prove a Carleman estimate for the one-dimensional degenerate parabolic equation

v_{t} + (x^{α} v_{x})_{x} - b (x, t) v + (x^{β / 2} c (x, t) v)_{x} = 0, (x, t) \in (0, 1) \times (0, T),

where

α \in [0, 2)

,

β ⩾ α

and

b (x, t), c (x, t) \in L^{\infty} ((0, 1) \times (0, T))

and give some controllability consequences.

Reçu le : 2009-11-03
Accepté le : 2009-12-16
Publié le : 2010-03-12

DOI : 10.1016/j.crma.2010.01.007

Affiliations des auteurs :

Carmelo Flores ¹ ; Luz de Teresa ²

¹ Universidad Autónoma de la Ciudad de México, Prolongacion San Isidro, 151 San Lorenzo Tezonco Delegac, 09790 Mexico, Distrito Federal, Mexico
² Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico

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     author = {Carmelo Flores and Luz de Teresa},
     title = {Carleman estimates for degenerate parabolic equations with first order terms and applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {391--396},
     publisher = {Elsevier},
     volume = {348},
     number = {7-8},
     year = {2010},
     doi = {10.1016/j.crma.2010.01.007},
     language = {en},
}

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Carmelo Flores; Luz de Teresa. Carleman estimates for degenerate parabolic equations with first order terms and applications. Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 391-396. doi : 10.1016/j.crma.2010.01.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.01.007/

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