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June  2013, 3(2): 143-160. doi: 10.3934/mcrf.2013.3.143

## Stability of the determination of a time-dependent coefficient in parabolic equations

 1 LMAM, UMR 7122, Université de Lorraine, Ile du Saulcy, 57045 Metz, cedex 1, France 2 UMR-7332, Aix Marseille Université, Centre de Physique Théorique, Campus de Luminy, Case 907, 13288 Marseille, cedex 9, France

Received  February 2012 Revised  May 2012 Published  March 2013

We establish a Lipschitz stability estimate for the inverse problem consisting in the determination of the coefficient $\sigma(t)$, appearing in a Dirichlet initial-boundary value problem for the parabolic equation $\partial_tu-\Delta_x u+\sigma(t)f(x)u=0$, from Neumann boundary data. We extend this result to the same inverse problem when the previous linear parabolic equation is changed to the semi-linear parabolic equation $\partial_tu-\Delta_x u=F(x,t,\sigma(t),u(x,t))$.
Citation: Mourad Choulli, Yavar Kian. Stability of the determination of a time-dependent coefficient in parabolic equations. Mathematical Control and Related Fields, 2013, 3 (2) : 143-160. doi: 10.3934/mcrf.2013.3.143
##### References:
 [1] J. R. Cannon and S. Pérez Esteva, An inverse problem for the heat equation, Inverse Problems, 2 (1986), 395-403. [2] J. R. Cannon and S. Pérez Esteva, A note on an inverse problem related to the 3-D heat equation, in "Inverse Problems" (Oberwolfach, 1986), Internat. Schriftenreihe Numer. Math., 77, Birkhäuser, Basel, (1986), 133-137. [3] J. R. Cannon and Y. Lin, Determination of a parameter p(t) in some quasilinear parabolic differential equations, Inverse Problems, 4 (1988), 35-45. [4] J. R. Cannon and Y. Lin, An inverse problem of finding a parameter in a semi-linear heat equation, J. Math. Anal. Appl., 145 (1990), 470-484. doi: 10.1016/0022-247X(90)90414-B. [5] M. Choulli, An abstract inverse problem, J. Appl. Math. Stoc. Ana., 4 (1991), 117-128. doi: 10.1155/S1048953391000084. [6] M. Choulli, Abstract inverse problem and application, J. Math. Anal. Appl., 160 (1991), 190-202. doi: 10.1016/0022-247X(91)90299-F. [7] M. Choulli, "Une Introduction aux Problèmes Inverses Elliptiques et Paraboliques," Mathématiques & Applications (Berlin), 65, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02460-3. [8] M. Choulli, E. M. Ouhabaz and M. Yamamoto, Stable determination of a semilinear term in a parabolic equation, Commun. Pure Appl. Anal., 5 (2006), 447-462. doi: 10.3934/cpaa.2006.5.447. [9] M. Choulli and M. Yamamoto, Some stability estimates in determining sources and coefficients, J. Inv. Ill-Posed Problems, 14 (2006), 355-373. doi: 10.1163/156939406777570996. [10] M. Choulli and M. Yamamoto, Global existence and stability for an inverse coefficient problem for a semilinear parabolic equation, Arch. Math. (Basel), 97 (2011), 587-597. doi: 10.1007/s00013-011-0329-z. [11] G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Commun. PDE, 32 (2007), 1737-1758. doi: 10.1080/03605300701382340. [12] G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect, J. Math. Phys., 49 (2008), 18 pp. doi: 10.1063/1.2841329. [13] S. Itô, "Diffusion Equations," Transaction of Mathematical Monographs, 114, AMS, Providence, RI, 1992. [14] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, American Math. Soc., Providence, RI, 1968. [15] A. Lorenzi and E. Sinestrari, "Stability Results for a Partial Integrodifferential Equation," Proc. of the Meeting, Volerra Integrodifferential Equations in Banach Spaces, Trento, Pitman, London, 1987. [16] A. Lorenzi and E. Sinestrari, An inverse problem in the theory of materials with memory, J. Nonlinear Anal., 12 (1988), 1317-1335. doi: 10.1016/0362-546X(88)90080-6. [17] A. I. Prilepko and D. G. Orlovskiĭ, Determination of evolution parameter of an equation and inverse problems in mathematical physics. I, Translations from Diff. Uravn., 21 (1985), 119-129, 182. [18] A. I. Prilepko and D. G. Orlovskiĭ, Determination of evolution parameter of an equation and inverse problems in mathematical physics. II, Translations from Diff. Uravn., 21 (1985), 694-701, 735. [19] R. Salazar, "Determination of Time-Dependent Coefficients for a Hyperbolic Inverse Problem," Ph.D. Thesis, University of California, Los Angeles, 2010.

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##### References:
 [1] J. R. Cannon and S. Pérez Esteva, An inverse problem for the heat equation, Inverse Problems, 2 (1986), 395-403. [2] J. R. Cannon and S. Pérez Esteva, A note on an inverse problem related to the 3-D heat equation, in "Inverse Problems" (Oberwolfach, 1986), Internat. Schriftenreihe Numer. Math., 77, Birkhäuser, Basel, (1986), 133-137. [3] J. R. Cannon and Y. Lin, Determination of a parameter p(t) in some quasilinear parabolic differential equations, Inverse Problems, 4 (1988), 35-45. [4] J. R. Cannon and Y. Lin, An inverse problem of finding a parameter in a semi-linear heat equation, J. Math. Anal. Appl., 145 (1990), 470-484. doi: 10.1016/0022-247X(90)90414-B. [5] M. Choulli, An abstract inverse problem, J. Appl. Math. Stoc. Ana., 4 (1991), 117-128. doi: 10.1155/S1048953391000084. [6] M. Choulli, Abstract inverse problem and application, J. Math. Anal. Appl., 160 (1991), 190-202. doi: 10.1016/0022-247X(91)90299-F. [7] M. Choulli, "Une Introduction aux Problèmes Inverses Elliptiques et Paraboliques," Mathématiques & Applications (Berlin), 65, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02460-3. [8] M. Choulli, E. M. Ouhabaz and M. Yamamoto, Stable determination of a semilinear term in a parabolic equation, Commun. Pure Appl. Anal., 5 (2006), 447-462. doi: 10.3934/cpaa.2006.5.447. [9] M. Choulli and M. Yamamoto, Some stability estimates in determining sources and coefficients, J. Inv. Ill-Posed Problems, 14 (2006), 355-373. doi: 10.1163/156939406777570996. [10] M. Choulli and M. Yamamoto, Global existence and stability for an inverse coefficient problem for a semilinear parabolic equation, Arch. Math. (Basel), 97 (2011), 587-597. doi: 10.1007/s00013-011-0329-z. [11] G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Commun. PDE, 32 (2007), 1737-1758. doi: 10.1080/03605300701382340. [12] G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect, J. Math. Phys., 49 (2008), 18 pp. doi: 10.1063/1.2841329. [13] S. Itô, "Diffusion Equations," Transaction of Mathematical Monographs, 114, AMS, Providence, RI, 1992. [14] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, American Math. Soc., Providence, RI, 1968. [15] A. Lorenzi and E. Sinestrari, "Stability Results for a Partial Integrodifferential Equation," Proc. of the Meeting, Volerra Integrodifferential Equations in Banach Spaces, Trento, Pitman, London, 1987. [16] A. Lorenzi and E. Sinestrari, An inverse problem in the theory of materials with memory, J. Nonlinear Anal., 12 (1988), 1317-1335. doi: 10.1016/0362-546X(88)90080-6. [17] A. I. Prilepko and D. G. Orlovskiĭ, Determination of evolution parameter of an equation and inverse problems in mathematical physics. I, Translations from Diff. Uravn., 21 (1985), 119-129, 182. [18] A. I. Prilepko and D. G. Orlovskiĭ, Determination of evolution parameter of an equation and inverse problems in mathematical physics. II, Translations from Diff. Uravn., 21 (1985), 694-701, 735. [19] R. Salazar, "Determination of Time-Dependent Coefficients for a Hyperbolic Inverse Problem," Ph.D. Thesis, University of California, Los Angeles, 2010.
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