Advanced Search
Article Contents
Article Contents

A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide

  • * Corresponding author

    * Corresponding author 
Abstract Full Text(HTML) Figure(3) Related Papers Cited by
  • In this article we consider the inverse problem of determining the diffusion coefficient of the heat operator in an unbounded guide using a finite number of localized observations. For this problem, we prove a stability estimate in any finite portion of the guide using an adapted Carleman inequality. The measurements are located on the boundary of a larger finite portion of the guide. A special care is required to avoid measurements on the cross-section boundaries which are inside the actual guide. This stability estimate uses a technical positivity assumption. Using arguments from control theory, we manage to remove this assumption for the inverse problem with a given non homogeneous boundary condition.

    Mathematics Subject Classification: Primary: 35R30; Secondary: 35K05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Representation of $ \Omega_l $ and $ \Omega_L $ in a 3D setting

    Figure 2.  Lighted lateral boundary from $ a $

    Figure 3.  A neighborhood of the lateral boundary

  • [1] A. BenabdallahM. CristofolP. Gaitan and M. Yamamoto, Inverse problem for a parabolic system with two components by measurements of one component, Applicable Analysis, 88 (2009), 683-709.  doi: 10.1080/00036810802555490.
    [2] M. BellassouedO. Imanuvilov and M. Yamamoto, Inverse problem of determining the density and two Lamé coefficients by boundary data, SIAM J. Math. Anal., 40 (2008), 238-265.  doi: 10.1137/070679971.
    [3] A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the Large of a Class of Multidimensional Inverse Problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272. 
    [4] L. Cardoulis and M. Cristofol, An inverse problem for the heat equation in an unbounded guide, Appl. Math. Lett., 62 (2016), 63-68.  doi: 10.1016/j.aml.2016.06.015.
    [5] M. ChoulliE. 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.
    [6] M. CristofolJ. GarnierF. Hamel and L. Roques, Uniqueness from pointwise observations in a multi-parameter inverse problem, Commun. Pure Appl. Anal., 11 (2012), 173-188.  doi: 10.3934/cpaa.2012.11.173.
    [7] M. Cristofol, I. Kaddouri, G. Nadin and L. Roques, Coefficient determination via asymptotic spreading speeds, Inverse Problems, 30 (2014), 035005, 16pp. doi: 10.1088/0266-5611/30/3/035005.
    [8] M. CristofolS. Li and E. Soccorsi, Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary, Math. Control and Relat. Fields, 6 (2016), 407-427.  doi: 10.3934/mcrf.2016009.
    [9] M. Cristofol and L. Roques, Biological invasions: Deriving the regions at risk from partial measurements, Math. Biosci., 215 (2008), 158-166.  doi: 10.1016/j.mbs.2008.07.004.
    [10] M. Cristofol and L. Roques, Stable estimation of two coefficients in a nonlinear Fisher-KPP equation, Inverse Problems, 29 (2013), 095007, 18pp. doi: 10.1088/0266-5611/29/9/095007.
    [11] M. Cristofol and L. Roques, Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations, Inverse Problems, 33 (2017), 095006, 12pp. doi: 10.1088/1361-6420/aa7a1c.
    [12] M. V. de Hoop, L. Qiu and O. Scherzer, Local analysis of inverse problems: Hölder stability and iterative reconstruction, Inverse Problems, 28 (2012), 045001, 16pp. doi: 10.1088/0266-5611/28/4/045001.
    [13] P. DuChateau and W. Rundell, Unicity in an inverse problem for an unknown reaction term in a reaction-diffusion equation, J. Differential Equations, 59 (1985), 155-164.  doi: 10.1016/0022-0396(85)90152-4.
    [14] H. EggerH. W. Engl and M. V. Klibanov, Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation, Inverse Problems, 21 (2005), 271-290.  doi: 10.1088/0266-5611/21/1/017.
    [15] R. GlowinskiJ.-L. Lions and  J. HeExact and Approximate Controllability for Distributed Parameter Systems, A Numerical Approach, Cambridge University Press, Cambridge, 2008.  doi: 10.1017/CBO9780511721595.
    [16] M. Gonzalez-Burgos and L. de Teresa, Some results on controllability for linear and nonlinear heat equations in unbounded domains, Adv. Differential Equations, 12 (2007), 1201-1240. 
    [17] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, American Mathematical Society, Providence, R.I., 1968.
    [18] A. Lorenzi, An inverse problem for a semilinear parabolic equation, Ann. Mat. Pura Appl., 131 (1982), 145-166.  doi: 10.1007/BF01765150.
    [19] A. Lunardi, Analytic Semigroups and Optima Regularity in Parabolic Problems, Modern Birkhauser Classics. Brikhauser/Springer Basel AG, Baser, 1995.
    [20] S.-I. Nakamura, A note on uniqueness in an inverse problem for a semilinear parabolic equation, Nihonkai Math. J., 12 (2001), 71-73. 
    [21] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
    [22] M. S. Pilant and W. Rundell, An inverse problem for a nonlinear parabolic equation, Comm. Partial Differential Equations, 11 (1986), 445-457.  doi: 10.1080/03605308608820430.
    [23] L. Roques, M. D. Chekroun, M. Cristofol, S. Soubeyrand and M. Ghil, Parameter estimation for energy balance models with memory, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140349, 20pp. doi: 10.1098/rspa.2014.0349.
    [24] L. Roques and M. Cristofol, On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation, Nonlinearity, 23 (2010), 675-686. 
    [25] L. Roques and M. Cristofol, The inverse problem of determining several coefficients in a nonlinear Lotka-Volterra system, Inverse Problems, 28 (2012), 075007, 12pp.
    [26] J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139.  doi: 10.1016/0022-0396(87)90043-X.
    [27] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75pp.
    [28] 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.  doi: 10.1051/cocv:2008043.
  • 加载中
Open Access Under a Creative Commons license



Article Metrics

HTML views(842) PDF downloads(264) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint