December  2021, 10(4): 837-859. doi: 10.3934/eect.2020094

Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions

Cadi Ayyad University, Faculty of Sciences Semlalia, LMDP, UMMISCO (IRD-UPMC) B.P. 2390, Marrakesh, Morocco

* Corresponding author: Lahcen Maniar

Received  February 2020 Revised  June 2020 Published  December 2021 Early access  September 2020

In this paper, we study an inverse problem for linear parabolic system with variable diffusion coefficients subject to dynamic boundary conditions. We prove a global Lipschitz stability for the inverse problem involving a simultaneous recovery of two source terms from a single measurement and interior observations, based on a recent Carleman estimate for such problems.

Citation: El Mustapha Ait Ben Hassi, Salah-Eddine Chorfi, Lahcen Maniar, Omar Oukdach. Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions. Evolution Equations and Control Theory, 2021, 10 (4) : 837-859. doi: 10.3934/eect.2020094
References:
[1]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.

[2]

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), 35-67.  doi: 10.1080/00036811.2010.534731.

[3]

M. Bellassoued and M. Yamamoto, Inverse source problem for a transmission problem for a parabolic equation, J. Inverse Ⅲ-Posed Probl., 14 (2006), 47-56.  doi: 10.1515/156939406776237456.

[4]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, 2$^nd$ edition, Birkhäuser Boston, Inc., Boston, MA, 2007, 159–165. doi: 10.1007/978-0-8176-4581-6.

[5]

I. BoutaayamouG. Fragnelli and L. Maniar, Inverse problems for parabolic equations with interior degeneracy and Neumann boundary conditions, J. Inverse Ⅲ-Posed Probl., 24 (2016), 275-292.  doi: 10.1515/jiip-2014-0032.

[6]

I. Boutaayamou, G. Fragnelli and L. Maniar, Lipschitz stability for linear parabolic systems with interior degeneracy, Electron. J. Differential Equations, (2014), No. 167, 26 pp.

[7]

I. Boutaayamou, A. Hajjaj and L. Maniar, Lipschitz stability for degenerate parabolic systems, Electron. J. Differential Equations, (2014), No. 149, 15 pp.

[8]

A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of class of multidimensional inverse problems, Soviet Math. Dokl., 24 (1981), 244-247. 

[9]

P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation, Inverse Problems, 26 (2010), 105003, 20 pp. doi: 10.1088/0266-5611/26/10/105003.

[10]

D. ChaeO. Y. Imanuvilov and S. M. Kim, Exact controllability for semilinear parabolic equations with Neumann boundary conditions, J. Dynam. Control Systems, 2 (1996), 449-483.  doi: 10.1007/BF02254698.

[11]

M. Cristofol and L. Roques, On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation, Nonlinearity, 23 (2010), 675-686.  doi: 10.1088/0951-7715/23/3/014.

[12]

J. Z. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions, Math. Biosci. Eng., 8 (2011), 503-513.  doi: 10.3934/mbe.2011.8.503.

[13]

A. FaviniJ. A. GoldsteinG. R. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), 1-19.  doi: 10.1007/s00028-002-8077-y.

[14]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1395-1446.  doi: 10.1137/S0363012904439696.

[15]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Note Series, Vol. 34, Seoul National University, Research Institute of Mathematics, Seoul, 1996.

[16]

G. C. Gal and L. Tebou, Carleman inequalities for wave equations with oscillatory boundary conditions and application, SIAM J. Control Optim., 55 (2017), 324-364.  doi: 10.1137/15M1032211.

[17]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. 

[18]

O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009.

[19]

V. Isakov, Inverse Problems for Partial Differential Equations, 2$^nd$ edition, Appl. Math. Sci. 127, Springer, New York, 2006.

[20]

V. Isakov, Inverse Source Problems, Mathematical Surveys and Monographs, Vol. 34, Amer. Math. Soc., Providence, RI, 1990. doi: 10.1090/surv/034.

[21]

J. Jost, Partial Differential Equations, 2$^nd$ edition, Springer, New York, 2007. doi: 10.1007/978-0-387-49319-0.

[22]

J. Jost, Riemannian Geometry and Geometric Analysis, 5$^th$ edition, Springer-Verlag, Berlin, 2008.

[23]

A. Khoutaibi and L. Maniar, Null controllability for a heat equation with dynamic boundary conditions and drift terms, Evol. Equ. Control Theory, 9 (2019), 535-559.  doi: 10.3934/eect.2020023.

[24]

A. Khoutaibi, L. Maniar, D. Mugnolo and A. Rhandi, Parabolic equations with dynamic boundary conditions and drift terms, preprint, 2019, arXiv: 1909.02377.

[25]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.

[26]

R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with a fluid, Tohoku Math. J., 35 (1932), 260-275. 

[27]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, New York-Heidelberg, 1972.

[28]

L. ManiarM. Meyries and R. Schnaubelt, Null controllability for parabolic equations with dynamic boundary conditions, Evol. Equ. Control Theory, 6 (2017), 381-407.  doi: 10.3934/eect.2017020.

[29]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.  doi: 10.1002/mma.590.

[30] E. M. Ouhabaz, Analysis of Heat Equations on Domains, LMS Monographs Series, 31, Princeton University Press, Princeton, NJ, 2005. 
[31]

M. E. Taylor, Partial Differential Equations. I. Basic Theory, 2$^nd$ edition, Applied Mathematical Sciences, Vol. 115, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.

[32]

J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations, Comm. Partial Differential Equations, 36 (2011), 1287-1317.  doi: 10.1080/03605302.2011.587491.

[33]

J. L. Vazquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type, J. Differential Equations, 250 (2011), 2143-2161.  doi: 10.1016/j.jde.2010.12.012.

[34]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75 pp. doi: 10.1088/0266-5611/25/12/123013.

show all references

References:
[1]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.

[2]

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), 35-67.  doi: 10.1080/00036811.2010.534731.

[3]

M. Bellassoued and M. Yamamoto, Inverse source problem for a transmission problem for a parabolic equation, J. Inverse Ⅲ-Posed Probl., 14 (2006), 47-56.  doi: 10.1515/156939406776237456.

[4]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, 2$^nd$ edition, Birkhäuser Boston, Inc., Boston, MA, 2007, 159–165. doi: 10.1007/978-0-8176-4581-6.

[5]

I. BoutaayamouG. Fragnelli and L. Maniar, Inverse problems for parabolic equations with interior degeneracy and Neumann boundary conditions, J. Inverse Ⅲ-Posed Probl., 24 (2016), 275-292.  doi: 10.1515/jiip-2014-0032.

[6]

I. Boutaayamou, G. Fragnelli and L. Maniar, Lipschitz stability for linear parabolic systems with interior degeneracy, Electron. J. Differential Equations, (2014), No. 167, 26 pp.

[7]

I. Boutaayamou, A. Hajjaj and L. Maniar, Lipschitz stability for degenerate parabolic systems, Electron. J. Differential Equations, (2014), No. 149, 15 pp.

[8]

A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of class of multidimensional inverse problems, Soviet Math. Dokl., 24 (1981), 244-247. 

[9]

P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation, Inverse Problems, 26 (2010), 105003, 20 pp. doi: 10.1088/0266-5611/26/10/105003.

[10]

D. ChaeO. Y. Imanuvilov and S. M. Kim, Exact controllability for semilinear parabolic equations with Neumann boundary conditions, J. Dynam. Control Systems, 2 (1996), 449-483.  doi: 10.1007/BF02254698.

[11]

M. Cristofol and L. Roques, On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation, Nonlinearity, 23 (2010), 675-686.  doi: 10.1088/0951-7715/23/3/014.

[12]

J. Z. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions, Math. Biosci. Eng., 8 (2011), 503-513.  doi: 10.3934/mbe.2011.8.503.

[13]

A. FaviniJ. A. GoldsteinG. R. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), 1-19.  doi: 10.1007/s00028-002-8077-y.

[14]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1395-1446.  doi: 10.1137/S0363012904439696.

[15]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Note Series, Vol. 34, Seoul National University, Research Institute of Mathematics, Seoul, 1996.

[16]

G. C. Gal and L. Tebou, Carleman inequalities for wave equations with oscillatory boundary conditions and application, SIAM J. Control Optim., 55 (2017), 324-364.  doi: 10.1137/15M1032211.

[17]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. 

[18]

O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009.

[19]

V. Isakov, Inverse Problems for Partial Differential Equations, 2$^nd$ edition, Appl. Math. Sci. 127, Springer, New York, 2006.

[20]

V. Isakov, Inverse Source Problems, Mathematical Surveys and Monographs, Vol. 34, Amer. Math. Soc., Providence, RI, 1990. doi: 10.1090/surv/034.

[21]

J. Jost, Partial Differential Equations, 2$^nd$ edition, Springer, New York, 2007. doi: 10.1007/978-0-387-49319-0.

[22]

J. Jost, Riemannian Geometry and Geometric Analysis, 5$^th$ edition, Springer-Verlag, Berlin, 2008.

[23]

A. Khoutaibi and L. Maniar, Null controllability for a heat equation with dynamic boundary conditions and drift terms, Evol. Equ. Control Theory, 9 (2019), 535-559.  doi: 10.3934/eect.2020023.

[24]

A. Khoutaibi, L. Maniar, D. Mugnolo and A. Rhandi, Parabolic equations with dynamic boundary conditions and drift terms, preprint, 2019, arXiv: 1909.02377.

[25]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.

[26]

R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with a fluid, Tohoku Math. J., 35 (1932), 260-275. 

[27]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, New York-Heidelberg, 1972.

[28]

L. ManiarM. Meyries and R. Schnaubelt, Null controllability for parabolic equations with dynamic boundary conditions, Evol. Equ. Control Theory, 6 (2017), 381-407.  doi: 10.3934/eect.2017020.

[29]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.  doi: 10.1002/mma.590.

[30] E. M. Ouhabaz, Analysis of Heat Equations on Domains, LMS Monographs Series, 31, Princeton University Press, Princeton, NJ, 2005. 
[31]

M. E. Taylor, Partial Differential Equations. I. Basic Theory, 2$^nd$ edition, Applied Mathematical Sciences, Vol. 115, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.

[32]

J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations, Comm. Partial Differential Equations, 36 (2011), 1287-1317.  doi: 10.1080/03605302.2011.587491.

[33]

J. L. Vazquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type, J. Differential Equations, 250 (2011), 2143-2161.  doi: 10.1016/j.jde.2010.12.012.

[34]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75 pp. doi: 10.1088/0266-5611/25/12/123013.

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