Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


IMPACT FACTOR 2017: 0.941
5-year IMPACT FACTOR: 0.953

CiteScore 2017: 0.91

SCImago Journal Rank (SJR) 2017: 0.461
Source Normalized Impact per Paper (SNIP) 2017: 1.022

Mathematical Citation Quotient (MCQ) 2017: 0.49

Online
ISSN
1569-3945
See all formats and pricing
More options …
Volume 25, Issue 3

Issues

Stability result for two coefficients in a coupled hyperbolic-parabolic system

Patricia Gaitan / Hadjer Ouzzane
Published Online: 2016-05-27 | DOI: https://doi.org/10.1515/jiip-2015-0017

Abstract

This work is concerned with the study of the inverse problem of determining two coefficients in a hyperbolic-parabolic system using the following observation data: an interior measurement of only one component and data of two components at a fixed time over the whole spatial domain. A Lipschitz stability result is proved using Carleman estimates.

Keywords: Inverse problems; Carleman estimates; coupled systems

MSC 2010: 35R30; 35M30; 35Q92

References

  • [1]

    P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system, Electron. J. Differential Equations 2000 (2000), Paper No. 22. Google Scholar

  • [2]

    C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d’approximation; application à l’équation de transport, Ann. Sci. Éc. Norm. Supér. (4) 3 (1970), 185–233. CrossrefGoogle Scholar

  • [3]

    L. Baudouin, M. de Buhan and S. Ervedoza, Global Carleman estimates for waves and application, Comm. Partial Differential Equation 38 (2013), no. 5, 823–859. Google Scholar

  • [4]

    L. Baudouin and J. P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems 18 (2002), 1537–1554. Google Scholar

  • [5]

    M. Bellassoued and M. Yamamoto, Carleman estimate with second large parameter for second order hyperbolic operators in a Riemannian manifold and applications in thermoelasticity cases, Appl. Anal. 91 (2012), no. 1, 35–67. CrossrefWeb of ScienceGoogle Scholar

  • [6]

    A. Benabdallah, M. Cristofol, P. Gaitan and M. Yamamoto, Inverse problem for a parabolic system with two components by measurments of one component, Appl. Anal. 88 (2009), no. 5, 1–28. CrossrefGoogle Scholar

  • [7]

    A. Benabdallah, P. Gaitan and J. Le Rousseau, Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation, SIAM J. Control Optim. 46 (2007), 1849–1881. CrossrefGoogle Scholar

  • [8]

    A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Soviet Math. Dokl. 17 (1981), 1–241. Google Scholar

  • [9]

    M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, IMA J. Math. Appl. Med. Biol. 10 (1993), 149–168. CrossrefGoogle Scholar

  • [10]

    M. Choulli, Une Introduction aux Problèmes Inverses Elliptiques et Paraboliques, Math. Appl. (Berlin) 65, Springer, Berlin, 2009. Google Scholar

  • [11]

    M. Cristofol, P. Gaitan and H. Ramoul, Inverse problems for a 2X2 reaction-diffusion system using a Carleman estimate with one observation, Inverse Problems 22 (2006), 1561–1573. Google Scholar

  • [12]

    M. Cristofol, P. Gaitan, H. Ramoul and M. Yamamoto, Identification of two coefficients with data of one component for a nonlinear parabolic system, Appl. Anal. 91 (2012), no. 11, 2073–2081. Web of ScienceCrossrefGoogle Scholar

  • [13]

    R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 6: Evolution Problems II, Springer, Berlin, 1993. Google Scholar

  • [14]

    C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, American Mathematical Society, Providence, 1998. Google Scholar

  • [15]

    A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lect. Notes Ser. 34, Seoul National University, Seoul, 1996. Google Scholar

  • [16]

    P. Gaitan and H. Ouzzane, Inverse problem for a free transport equation using Carleman estimates, Appl. Anal. 93 (2014), no. 5, 1073–1086. CrossrefWeb of ScienceGoogle Scholar

  • [17]

    A. Gerisch, M. Kotschote and R. Zacher, Well-posedness of a quasilinear hyperbolic-parabolic system arising in mathematical biology, NoDEA Nonlinear Differential Equations Appl. 14 (2007), no. 5–6, 593–624. Google Scholar

  • [18]

    O. Y. Imanuvilov, V. Isakov and M. Yamamoto, An inverse problem for the dynamical Lamé system with two sets of boundary data, Comm. Pure Appl. Math. 56 (2003), 1366–1382. Google Scholar

  • [19]

    O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems 14 (1998), 1229–1245. Google Scholar

  • [20]

    V. Isakov and N. Kim, Carleman estimates with second large parameter for second order operators, Sobolev Spaces in Mathematics. III: Applications in Mathematical Physics, Int. Math. Ser. 10, Springer, New York (2009), 135–159. CrossrefGoogle Scholar

  • [21]

    M. V. Klibanov, Inverse problems in the large and Carleman bounds, Differ. Equ. 20 (1984), 755–760. Google Scholar

  • [22]

    M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems 8 (1992), 575–596. Web of ScienceGoogle Scholar

  • [23]

    M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl. 21 (2013), 477–560. Google Scholar

  • [24]

    M. V. Klibanov and S. E. Pamyatnykh, Lipschitz stability of a non-standard problem for the non-stationary transport equation via Carleman estimate, Inverse Problems 22 (2006), 881–890. Google Scholar

  • [25]

    M. V. Klibanov and S. E. Pamyatnykh, Global uniqueness for a coefficient inverse problem for the non-stationary transport equation via Carleman estimate, J. Math. Anal. Appl. 343 (2008), no. 1, 352–365. Web of ScienceCrossrefGoogle Scholar

  • [26]

    M. V. Klibanov and A. A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 2004. Google Scholar

  • [27]

    M. Machida and M. Yamamoto, Global Lipschitz stability in determining coefficients of the radiative transport equation, Inverse Problems30 (2014), no. 3, Article ID 035010. Google Scholar

  • [28]

    H. Ouzzane, Inégalités de Carleman ; applications aux problèmes inverses et au contrôle de quelques problèmes d’évolution, Ph.D. thesis, Aix Marseille Université and Université d’Alger (USTHB), 2014. Google Scholar

  • [29]

    B. Wu and J. Liu, Conditional stability and uniqueness for determining two coefficients in a hyperbolic parabolic system, Inverse Problems 27 (2011), no. 7, Article ID 075013. Google Scholar

  • [30]

    M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems 25 (2009), no. 12, Article ID 123013. Google Scholar

  • [31]

    G. Yuan and M. Yamamoto, Lipschitz stability in the determination of the principal part of a parabolic equation, ESAIM Control Optim. Calc. Var. 15 (2009), 525–554. Google Scholar

About the article

Received: 2015-02-03

Revised: 2016-01-27

Accepted: 2016-03-30

Published Online: 2016-05-27

Published in Print: 2017-06-01


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 25, Issue 3, Pages 265–286, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2015-0017.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Nicolás Carreño, Roberto Morales, and Axel Osses
Inverse Problems, 2018, Volume 34, Number 8, Page 085005
[2]
Bin Wu and Jun Yu
IMA Journal of Applied Mathematics, 2017, Page hxw058

Comments (0)

Please log in or register to comment.
Log in