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

Previous Article
Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws
MCRF Home
This Issue
Next Article
Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control

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

Abstract
Related Papers

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 $\partial_tu-\Delta_x u=F(x,t,\sigma(t),u(x,t))$.

Keywords: Parabolic equation, determination of time-depend coefficient, stability estimate., semi-linear parabolic equation, inverse problem.

Mathematics Subject Classification: 35R3.

Citation: Mourad Choulli, Yavar Kian. Stability of the determination of a time-dependent coefficient in parabolic equations. Mathematical Control & 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. Google Scholar
[2]	J. R. Cannon and S. Pérez Esteva, A note on an inverse problem related to the 3-D heat equation,, in, 77 (1986), 133. Google Scholar
[3]	J. R. Cannon and Y. Lin, Determination of a parameter p(t) in some quasilinear parabolic differential equations,, Inverse Problems, 4 (1988), 35. Google Scholar
[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. doi: 10.1016/0022-247X(90)90414-B. Google Scholar
[5]	M. Choulli, An abstract inverse problem,, J. Appl. Math. Stoc. Ana., 4 (1991), 117. doi: 10.1155/S1048953391000084. Google Scholar
[6]	M. Choulli, Abstract inverse problem and application,, J. Math. Anal. Appl., 160 (1991), 190. doi: 10.1016/0022-247X(91)90299-F. Google Scholar
[7]	M. Choulli, "Une Introduction aux Problèmes Inverses Elliptiques et Paraboliques,", Mathématiques & Applications (Berlin), 65 (2009). doi: 10.1007/978-3-642-02460-3. Google Scholar
[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. doi: 10.3934/cpaa.2006.5.447. Google Scholar
[9]	M. Choulli and M. Yamamoto, Some stability estimates in determining sources and coefficients,, J. Inv. Ill-Posed Problems, 14 (2006), 355. doi: 10.1163/156939406777570996. Google Scholar
[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. doi: 10.1007/s00013-011-0329-z. Google Scholar
[11]	G. Eskin, Inverse hyperbolic problems with time-dependent coefficients,, Commun. PDE, 32 (2007), 1737. doi: 10.1080/03605300701382340. Google Scholar
[12]	G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect,, J. Math. Phys., 49 (2008). doi: 10.1063/1.2841329. Google Scholar
[13]	S. Itô, "Diffusion Equations,", Transaction of Mathematical Monographs, 114 (1992). Google Scholar
[14]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1968). Google Scholar
[15]	A. Lorenzi and E. Sinestrari, "Stability Results for a Partial Integrodifferential Equation,", Proc. of the Meeting, (1987). Google Scholar
[16]	A. Lorenzi and E. Sinestrari, An inverse problem in the theory of materials with memory,, J. Nonlinear Anal., 12 (1988), 1317. doi: 10.1016/0362-546X(88)90080-6. Google Scholar
[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. Google Scholar
[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. Google Scholar
[19]	R. Salazar, "Determination of Time-Dependent Coefficients for a Hyperbolic Inverse Problem,", Ph.D. Thesis, (2010). Google Scholar

show all references

References:

[1]	J. R. Cannon and S. Pérez Esteva, An inverse problem for the heat equation,, Inverse Problems, 2 (1986), 395. Google Scholar
[2]	J. R. Cannon and S. Pérez Esteva, A note on an inverse problem related to the 3-D heat equation,, in, 77 (1986), 133. Google Scholar
[3]	J. R. Cannon and Y. Lin, Determination of a parameter p(t) in some quasilinear parabolic differential equations,, Inverse Problems, 4 (1988), 35. Google Scholar
[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. doi: 10.1016/0022-247X(90)90414-B. Google Scholar
[5]	M. Choulli, An abstract inverse problem,, J. Appl. Math. Stoc. Ana., 4 (1991), 117. doi: 10.1155/S1048953391000084. Google Scholar
[6]	M. Choulli, Abstract inverse problem and application,, J. Math. Anal. Appl., 160 (1991), 190. doi: 10.1016/0022-247X(91)90299-F. Google Scholar
[7]	M. Choulli, "Une Introduction aux Problèmes Inverses Elliptiques et Paraboliques,", Mathématiques & Applications (Berlin), 65 (2009). doi: 10.1007/978-3-642-02460-3. Google Scholar
[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. doi: 10.3934/cpaa.2006.5.447. Google Scholar
[9]	M. Choulli and M. Yamamoto, Some stability estimates in determining sources and coefficients,, J. Inv. Ill-Posed Problems, 14 (2006), 355. doi: 10.1163/156939406777570996. Google Scholar
[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. doi: 10.1007/s00013-011-0329-z. Google Scholar
[11]	G. Eskin, Inverse hyperbolic problems with time-dependent coefficients,, Commun. PDE, 32 (2007), 1737. doi: 10.1080/03605300701382340. Google Scholar
[12]	G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect,, J. Math. Phys., 49 (2008). doi: 10.1063/1.2841329. Google Scholar
[13]	S. Itô, "Diffusion Equations,", Transaction of Mathematical Monographs, 114 (1992). Google Scholar
[14]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1968). Google Scholar
[15]	A. Lorenzi and E. Sinestrari, "Stability Results for a Partial Integrodifferential Equation,", Proc. of the Meeting, (1987). Google Scholar
[16]	A. Lorenzi and E. Sinestrari, An inverse problem in the theory of materials with memory,, J. Nonlinear Anal., 12 (1988), 1317. doi: 10.1016/0362-546X(88)90080-6. Google Scholar
[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. Google Scholar
[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. Google Scholar
[19]	R. Salazar, "Determination of Time-Dependent Coefficients for a Hyperbolic Inverse Problem,", Ph.D. Thesis, (2010). Google Scholar

[1]	Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014
[2]	Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661
[3]	Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367
[4]	Mourad Choulli, El Maati Ouhabaz, Masahiro Yamamoto. Stable determination of a semilinear term in a parabolic equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 447-462. doi: 10.3934/cpaa.2006.5.447
[5]	Út V. Lê. Contraction-Galerkin method for a semi-linear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 141-160. doi: 10.3934/cpaa.2010.9.141
[6]	Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761
[7]	Anne Mund, Christina Kuttler, Judith Pérez-Velázquez. Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5695-5707. doi: 10.3934/dcdsb.2019102
[8]	Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307
[9]	J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176
[10]	Yongqin Liu. The point-wise estimates of solutions for semi-linear dissipative wave equation. Communications on Pure & Applied Analysis, 2013, 12 (1) : 237-252. doi: 10.3934/cpaa.2013.12.237
[11]	Jianhai Bao, Xing Huang, Chenggui Yuan. New regularity of kolmogorov equation and application on approximation of semi-linear spdes with Hölder continuous drifts. Communications on Pure & Applied Analysis, 2019, 18 (1) : 341-360. doi: 10.3934/cpaa.2019018
[12]	Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014
[13]	Nguyen Thieu Huy, Vu Thi Ngoc Ha, Pham Truong Xuan. Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2103-2116. doi: 10.3934/cpaa.2016029
[14]	Qiang Du, Jiang Yang, Zhi Zhou. Analysis of a nonlocal-in-time parabolic equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 339-368. doi: 10.3934/dcdsb.2017016
[15]	Patrick Martinez, Judith Vancostenoble. The cost of boundary controllability for a parabolic equation with inverse square potential. Evolution Equations & Control Theory, 2019, 8 (2) : 397-422. doi: 10.3934/eect.2019020
[16]	Yavar Kian. Stability of the determination of a coefficient for wave equations in an infinite waveguide. Inverse Problems & Imaging, 2014, 8 (3) : 713-732. doi: 10.3934/ipi.2014.8.713
[17]	Wenqiang Zhao. Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2499-2526. doi: 10.3934/dcdsb.2018065
[18]	Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883
[19]	Michael Winkler. Nontrivial ordered ω-limit sets in a linear degenerate parabolic equation. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 739-750. doi: 10.3934/dcds.2007.17.739
[20]	José M. Amigó, Isabelle Catto, Ángel Giménez, José Valero. Attractors for a non-linear parabolic equation modelling suspension flows. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 205-231. doi: 10.3934/dcdsb.2009.11.205

2018 Impact Factor: 1.292

American Institute of Mathematical Sciences