Comptes Rendus
no. 3
Partial differential equations
Null controllability for the semilinear heat equation in a non-smooth domain
[Nulle contrôlabilité de l'équation de la chaleur semilinéaire dans un domaine non régulier]

Présenté par : Gilles Lebeau

Tarik Ali-Ziane1 ; Zahia Ferhoune1 ; Ouahiba Zair1
1 Laboratoire AMNEDP, Faculté de Mathématiques, USTHB, BP 32, El-Alia, 16111 Alger, Algeria
Comptes Rendus. Mathématique, Volume 353 (2015) no. 3, pp. 229-234.

Dans ce travail, on donne un résultat de nulle contrôlabilité pour l'équation de la chaleur semi-linéaire dans un domaine borné de R2, polygonal ou fissuré. On suppose que la non-linearité croît moins vite que |s|log3/2⁡(1+|s|) quand |s|→∞, et on démontre le résultat par le théorème du point fixe de Schauder.

In this work we give a null-controllability result for the semi-linear heat equation in a polygonal or cracked bounded domain of R2. We suppose that the nonlinearity grows slower than |s|log3/2⁡(1+|s|) as |s|→∞ and then we prove our result by using Schauder's fixed point theorem.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.01.005
Tarik Ali-Ziane 1 ; Zahia Ferhoune 1 ; Ouahiba Zair 1

1 Laboratoire AMNEDP, Faculté de Mathématiques, USTHB, BP 32, El-Alia, 16111 Alger, Algeria 
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Tarik Ali-Ziane; Zahia Ferhoune; Ouahiba Zair. Null controllability for the semilinear heat equation in a non-smooth domain. Comptes Rendus. Mathématique, Volume 353 (2015) no. 3, pp. 229-234. doi : 10.1016/j.crma.2015.01.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.01.005/

[1] D.G. Aronson; J. Serrin Local behavior of solutions of quasilinear parabolic equations, Arch. Ration. Mech. Anal., Volume 25 (1967), pp. 81-122

[2] A.H. Belghazi; F. Smadhi; N. Zaidi; O. Zair Carleman inequalities for the two-dimensional heat equation in singular domains, Math. Control Relat. Fields, Volume 2 (2012) no. 4, pp. 331-359

[3] A. Doubova; A. Osses; J.-P. Puel Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients, ESAIM Control Optim. Calc. Var., Volume 8 (2002), pp. 621-661

[4] C. Fabre; J.-P. Puel; E. Zuazua On the density of the range of the semigroup for semilinear heat equations, Minneapolis, MN, 1992 (IMA Vol. Math. Appl.), Volume vol. 70, Springer, New York (1995), pp. 73-91

[5] E. Fernández-Cara; E. Zuazua Null and approximate controllability for weakly blowing-up semilinear heat equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 17 (2000) no. 5, pp. 583-616

[6] P. Grisvard Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, USA, 1985

[7] P. Grisvard Singularities in Boundary Value Problems, Recherches en mathématiques appliquées, Research in Applied Mathematics, vol. 22, Masson, Paris, 1992

[8] M.A. Moussaoui; B.K. Sadallah Régularité des coefficients de propagation de singularités pour l'équation de la chaleur dans un ouvert plan polygonal, C. R. Acad. Sci. Paris, Ser. I, Volume 293 (1981) no. 5, pp. 297-300

[9] E. Zuazua Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities, Control Cybern., Volume 28 (1999) no. 3, pp. 665-683

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