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

Flores, Carmelo; de Teresa, Luz

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

Flores, Carmelo ¹ ; de Teresa, Luz 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, Tome 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

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 :

Flores, Carmelo ¹ ; de Teresa, Luz ²

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     author = {Flores, Carmelo and de Teresa, Luz},
     title = {Carleman estimates for degenerate parabolic equations with first order terms and applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {391--396},
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     doi = {10.1016/j.crma.2010.01.007},
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Flores, Carmelo; de Teresa, Luz. Carleman estimates for degenerate parabolic equations with first order terms and applications. Comptes Rendus. Mathématique, Tome 348 (2010) no. 7-8, pp. 391-396. doi : 10.1016/j.crma.2010.01.007. http://www.numdam.org/articles/10.1016/j.crma.2010.01.007/

Bibliographie
Cité par

[1] Alabau-Boussouira, F.; Cannarsa, P.; Fragnelli, G. Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., Volume 6 (2006) no. 2, pp. 161-204

[2] Campiti, M.; Metafune, G.; Pallara, D. Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum, Volume 57 (1998), pp. 1-36

[3] Cannarsa, P.; Fragnelli, G. Null controllability of semilinear degenerate parabolic equations in bounded domains, EJDE, Volume 136 (2006), pp. 1-20

[4] Cannarsa, P.; Fragnelli, G.; Vancostenoble, J. Regional controllability of semilinear degenerate parabolic equations in bounded domains, J. Math. Anal. Appl., Volume 320 (2006) no. 2, pp. 804-818

[5] Cannarsa, P.; Fragnelli, G.; Rocchetti, D. Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, Volume 2 (2007) no. 4, pp. 695-715 (electronic)

[6] Cannarsa, P.; Martinez, P.; Vancostenoble, J. Persistent regional null controllability for a class of degenerate parabolic equations, Commun. Pure Appl. Anal., Volume 3 (2004), pp. 607-635

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

[8] Cannarsa, P.; de Teresa, L. Insensitizing controls for one-dimensional degenerate parabolic equations, EJDE, Volume 2009 (2009) no. 73, pp. 1-21

[9] C. Flores, L. de Teresa, Null controllability of one-dimensional degenerate parabolic equations with first order terms, in preparation

[10] Imanuvilov, O.Yu.; Yamamoto, M. Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., Volume 39 (2003) no. 2, pp. 227-274

[11] Lions, J.-L.; Magenes, E. Non-Homogeneous Boundary Value Problems and Applications, vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York/Heidelberg, 1972

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

[13] Opic, B.; Kufner, A. Hardy-Type Inequalities, Longman Scientific and Technical, Harlow, UK, 1990

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