doi: 10.3934/mcrf.2020054

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

Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France

* Corresponding author

Received  January 2020 Revised  October 2020 Published  December 2020

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.

Citation: Laure Cardoulis, Michel Cristofol, Morgan Morancey. A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020054
References:
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15] R. GlowinskiJ.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems, A Numerical Approach, Cambridge University Press, Cambridge, 2008.  doi: 10.1017/CBO9780511721595.  Google Scholar
[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.   Google Scholar

[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.  Google Scholar

[18]

A. Lorenzi, An inverse problem for a semilinear parabolic equation, Ann. Mat. Pura Appl., 131 (1982), 145-166.  doi: 10.1007/BF01765150.  Google Scholar

[19]

A. Lunardi, Analytic Semigroups and Optima Regularity in Parabolic Problems, Modern Birkhauser Classics. Brikhauser/Springer Basel AG, Baser, 1995.  Google Scholar

[20]

S.-I. Nakamura, A note on uniqueness in an inverse problem for a semilinear parabolic equation, Nihonkai Math. J., 12 (2001), 71-73.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75pp.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15] R. GlowinskiJ.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems, A Numerical Approach, Cambridge University Press, Cambridge, 2008.  doi: 10.1017/CBO9780511721595.  Google Scholar
[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.   Google Scholar

[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.  Google Scholar

[18]

A. Lorenzi, An inverse problem for a semilinear parabolic equation, Ann. Mat. Pura Appl., 131 (1982), 145-166.  doi: 10.1007/BF01765150.  Google Scholar

[19]

A. Lunardi, Analytic Semigroups and Optima Regularity in Parabolic Problems, Modern Birkhauser Classics. Brikhauser/Springer Basel AG, Baser, 1995.  Google Scholar

[20]

S.-I. Nakamura, A note on uniqueness in an inverse problem for a semilinear parabolic equation, Nihonkai Math. J., 12 (2001), 71-73.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75pp.  Google Scholar

[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.  Google Scholar

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]

Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012

[2]

Mohamed Ouzahra. Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021081

[3]

K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 461-485. doi: 10.3934/naco.2020038

[4]

Agnid Banerjee, Ramesh Manna. Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021070

[5]

Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020

[6]

Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021055

[7]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021

[8]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[9]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065

[10]

Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005

[11]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066

[12]

Sergei Avdonin, Julian Edward. An inverse problem for quantum trees with observations at interior vertices. Networks & Heterogeneous Media, 2021, 16 (2) : 317-339. doi: 10.3934/nhm.2021008

[13]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[14]

Pavol Bokes. Exact and WKB-approximate distributions in a gene expression model with feedback in burst frequency, burst size, and protein stability. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021126

[15]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, 2021, 15 (3) : 387-413. doi: 10.3934/ipi.2020073

[16]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004

[17]

Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016

[18]

Yosra Soussi. Stable recovery of a non-compactly supported coefficient of a Schrödinger equation on an infinite waveguide. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021022

[19]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021032

[20]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

2019 Impact Factor: 0.857

Article outline

Figures and Tables

[Back to Top]