Abstract
Uniqueness and existence theorems for the solution of the inverse problem for a degenerating parabolic equation with unbounded coefficients on a plane in conditions of integral observations are proven. Estimates of the solution with constants explicitly expresses via the input data of the problem are obtained.
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References
V. L. Kamynin, “On the solvability of the inverse problem for determining the right-hand side of a degenerate parabolic equation with integral observation,” Math. Notes 98 (5), 765–777 (2015).
M. Ivanchov and N. Saldina, “An inverse problem for strongly degenerate heat equation,” J. Inverse Ill-Posed Probl. 14 (5), 465–480 (2006).
P. Cannarsa, J. Tort, and M. Yamamoto, “Determination of source terms in a degenerate parabolic equation,” Inverse Probl. 26 (10), 105003 (2010).
Z. C. Deng, K. Qian, X. B. Rao, and L. Yang, “An inverse problem of identifying the source coefficient in degenerate heat equation,” Inverse Probl. Sci. Eng. 23 (3), 498–517 (2014).
N. Huzyk, “Inverse problem of determining the coefficients in degenerate parabolic equation,” Electron. J. Differ. Equations, No. 172, 1–11 (2014).
A. Kawamoto, “Inverse problems for linear degenerate parabolic equations by 'time-like' Carleman estimate,” J. Inverse Ill-Posed Probl. 23 (1), 1–21 (2015).
I. Bouchouev and V. Isakov, “Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets,” Inverse Probl. 15 (3), 95–116 (1999).
L. Jiang, Q. Chen, L. Wang, and J. E. Zhang, “A new well-posed algorithm to recover implied local volatility,” Quantitative Finance 3 (6), 451–457 (2003).
L. Jiang and Y. Tao, “Identifying the volatility of underlying assets from option prices,” Inverse Probl. 17 (1), 137–155 (2001).
A. I. Prilepko and D. G. Orlovskii, “Determination of a parameter in an evolution equation and inverse problems of mathematical physics I,” Differ. Uravn. 21 (1), 119–129 (1985).
A. I. Prilepko and D. G. Orlovskii, “Determination of a parameter in an evolution equation and inverse problems of mathematical physics II,” Differ. Uravn. 21 (4), 694–700 (1985).
J. R. Cannon and Y. Lin, “Determination of a parameter in some quasilinear parabolic differential equations,” Inverse Probl. 4 (1), 35–45 (1988).
J. R. Cannon and Y. Lin, “Determination of a parameter in Hölder classes for some semilinear parabolic differential equations,” Inverse Probl. 4 (3), 596–606 (1988).
A. B. Kostin, “Inverse problem for the heat equation with integral overdetermination,” in Inverse Problems for Mathematical Modeling of Physical Processes: Collected Research Papers (Mosk. Inzh.-Fiz. Inst., Moscow, 1991), pp. 45–49 [in Russian].
V. L. Kamynin and E. Francini, “An inverse problem for a higher-order parabolic equation,” Math. Notes 64 (5), 590–599 (1998).
A. I. Prilepko, D. G. Orlovsky, and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics (Marcel Dekker, New York, 2000).
S. N. Kruzhkov, “Quasilinear parabolic equations and systems with two independent variables,” Tr. Semin. im. I.G. Petrovskogo, No. 5, 217–272 (1979).
L. A. Lusternik and V. I. Sobolev, Elements of Functional Analysis (Gordon and Breach, New York, 1968; Vysshaya Shkola, Moscow, 1982).
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Original Russian Text © V.L. Kamynin, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 5, pp. 832–841.
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Kamynin, V.L. Inverse problem of determining the right-hand side in a degenerating parabolic equation with unbounded coefficients. Comput. Math. and Math. Phys. 57, 833–842 (2017). https://doi.org/10.1134/S0965542517050049
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DOI: https://doi.org/10.1134/S0965542517050049