Abstract and Applied Analysis

An Inverse Source Problem for Singular Parabolic Equations with Interior Degeneracy

Khalid Atifi, Idriss Boutaayamou, Hamed Ould Sidi, and Jawad Salhi

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Abstract

The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.

Article information

Source
Abstr. Appl. Anal., Volume 2018 (2018), Article ID 2067304, 16 pages.

Dates
Received: 23 August 2018
Accepted: 28 November 2018
First available in Project Euclid: 10 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1547089412

Digital Object Identifier
doi:10.1155/2018/2067304

Mathematical Reviews number (MathSciNet)
MR3894337

Zentralblatt MATH identifier
07029283

Citation

Atifi, Khalid; Boutaayamou, Idriss; Ould Sidi, Hamed; Salhi, Jawad. An Inverse Source Problem for Singular Parabolic Equations with Interior Degeneracy. Abstr. Appl. Anal. 2018 (2018), Article ID 2067304, 16 pages. doi:10.1155/2018/2067304. https://projecteuclid.org/euclid.aaa/1547089412


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References

  • J. Vancostenoble, “Lipschitz stability in inverse source problems for singular parabolic equations,” Communications in Partial Differential Equations, vol. 36, no. 8, pp. 1287–1317, 2011.
  • O. Y. Imanuvilov and M. Yamamoto, “Lipschitz stability in inverse parabolic problems by the Carleman estimate,” Inverse Problems, vol. 14, no. 5, pp. 1229–1245, 1998.
  • P. Cannarsa, J. Tort, and M. Yamamoto, “Determination of source terms in a degenerate parabolic equation,” Inverse Problems, vol. 26, no. 10, 105003, 20 pages, 2010.
  • J.-P. Puel and M. Yamamoto, “On a global estimate in a linear inverse hyperbolic problem,” Inverse Problems, vol. 12, no. 6, pp. 995–1002, 1996.
  • A. L. Bukhgeim and M. V. Klibanov, “Global uniqueness of a class of multidimensional inverse problems,” Soviet Mathe-matics–-Doklady, vol. 24, pp. 244–247, 1981.
  • A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, vol. 34 of Lecture Notes Series, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, South Korea, 1996.
  • I. Boutaayamou, G. Fragnelli, and L. Maniar, “Carleman esti-mates for parabolic equations with interior degeneracy and Neumann boundary conditions,” Journal d'Analyse Mathé-matique, vol. 135, no. 1, pp. 1–35, 2018.
  • P. Cannarsa, P. Martinez, and J. Vancostenoble, “Carleman estimates for a class of degenerate parabolic operators,” SIAM Journal on Control and Optimization, vol. 47, no. 1, pp. 1–19, 2008.
  • G. Fragnelli and D. Mugnai, “Carleman estimates and observability inequalities for parabolic equations with interior degeneracy,” Advances in Nonlinear Analysis, vol. 2, no. 4, pp. 339–378, 2013.
  • I. Boutaayamou, G. Fragnelli, and L. Maniar, “Inverse problems for parabolic equations with interior degeneracy and Neumann boundary conditions,” Journal of Inverse and ILL-Posed Problems, vol. 24, no. 3, pp. 275–292, 2016.
  • I. Boutaayamou, G. Fragnelli, and L. Maniar, “Lipschitz stability for linear parabolic systems with interior degeneracy,” Electronic Journal of Differential Equations, No. 167, 26 pages, 2014.
  • J. Tort, “Determination of source terms in a degenerate para-bolic equation from a locally distributed observation,” Comptes Rendus MathÉmatique. AcadÉMIe des Sciences. Paris, vol. 348, no. 23-24, pp. 1287–1291, 2010.
  • Z.-C. Deng, K. Qian, X.-B. Rao, L. Yang, and G.-W. Luo, “An inverse problem of identifying the source coefficient in a degenerate heat equation,” Inverse Problems in Science and Engineering, vol. 23, no. 3, pp. 498–517, 2015.
  • X.-B. Rao, Y.-X. Wang, K. Qian, Z.-C. Deng, and L. Yang, “Numerical simulation for an inverse source problem in a degenerate parabolic equation,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 39, no. 23-24, pp. 7537–7553, 2015.
  • K. Atifi, Y. Balouki, El-H. Essoufi, and B. Khouiti, “Identifying Initial Condition in Degenerate Parabolic Equation with Singular Potential,” International Journal of Differential Equations, vol. 2017, Article ID 1467049, 17 pages, 2017.
  • G. Fragnelli and D. Mugnai, “Carleman estimates for singular parabolic equations with interior degeneracy and non smooth coefficients,” Advances in Nonlinear Analysis, 2016.
  • P. Cannarsa, P. Martinez, and J. Vancostenoble, “Global Carleman estimates for degenerate parabolic operators with applications,” Memoirs of the American Mathematical Society, vol. 239, no. 1133, ix+209 pages, 2016.
  • A. Hasanov, “Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: weak solution approach,” Journal of Mathematical Analysis and Applications, vol. 330, no. 2, pp. 766–779, 2007.
  • A. Bensoussan, G. da Prato, M. Delfour, and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäuser, 2nd edition, 2007.
  • G. Fragnelli and D. Mugnai, “Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations,” Memoirs of the American Mathematical Society, vol. 242, p. 1164, 2016.
  • A. Hasanov, P. DuChateau, and B. Pektas, “An adjoint problem approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equation,” Journal of Inverse and ILL-Posed Problems, vol. 14, no. 5, pp. 435–463, 2006.
  • Y. H. Ou, A. Hasanov, and Z. H. Liu, “Inverse coefficient prob-lems for nonlinear parabolic differential equations,” Acta Mathematica Sinica, vol. 24, no. 10, pp. 1617–1624, 2008.
  • A. Hasanov and V. G. Romanov, Introduction to inverse problems for differential equations, Springer, Heidelberg, Germany, 2017. \endinput