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Hadamard well-posedness for a structure acoustic model with a supercritical source and damping terms
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 |
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.
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. Boutaayamou, G. 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. Chae, O. 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. Favini, J. A. Goldstein, G. 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. Maniar, M. 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] |
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. Boutaayamou, G. 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. Chae, O. 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. Favini, J. A. Goldstein, G. 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. Maniar, M. 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] |
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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