Geodesic
On the Solvability of the Inverse Problem for Determining the Right-Hand Side of a Degenerate Parabolic Equation with Integral Observation
Matematičeskie zametki, Tome 98 (2015) no. 5, pp. 710-724.

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Existence and uniqueness theorems and theorems on stability under perturbations of the input data for solutions of the inverse problem for a degenerate parabolic equation in the plane with integral observation are obtained. The cases of bounded and unbounded coefficients are studied. Estimates of the solution with constants explicitly written out in terms of the input data of the problem are obtained.
Keywords: degenerate parabolic equation, solvability of the inverse problem, stability of solutions, perturbation of the input data, integral observation, maximum principle. 
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V. L. Kamynin. On the Solvability of the Inverse Problem for Determining the Right-Hand Side of a Degenerate Parabolic Equation with Integral Observation. Matematičeskie zametki, Tome 98 (2015) no. 5, pp. 710-724. http://geodesic.mathdoc.fr/item/MZM_2015_98_5_a5/

[1] S. N. Kruzhkov, “Kvazilineinye parabolicheskie uravneniya i sistemy s dvumya nezavisimymi peremennymi”, Tr. sem. im. I. G. Petrovskogo, 5, Izd-vo Mosk. un-ta, M., 1979, 217–272 | MR | Zbl

[2] A. I. Prilepko, D. G. Orlovskii, “Ob opredelenii parametra evolyutsionnogo uravneniya v obratnykh zadachakh matematicheskoi fiziki. I”, Differents. uravneniya, 21:1 (1985), 119–129 | MR | Zbl

[3] A. I. Prilepko, D. G. Orlovskii, “Ob opredelenii parametra evolyutsionnogo uravneniya v obratnykh zadachakh matematicheskoi fiziki .II”, Differents. uravneniya, 21:4 (1985), 694–700 | MR

[4] J. R. Cannon, Y. Lin, “Determination of a parameter $p(t)$ in some quasilinear parabolic differential equations”, Inverse Problems, 4:1 (1988), 35–45 | DOI | MR | Zbl

[5] J. R. Cannon, Y. Lin, “Determination of a parameter $p(t)$ in a Hölder classes for some semilinear parabolic equations”, Inverse Problems, 4:3 (1988), 595–606 | DOI | MR | Zbl

[6] V. L. Kamynin, “On convergence of the solutions of inverse problems for parabolic equations with weakly converging coefficients”, Elliptic and Parabolic Problems, Pitman Res. Notes Math. Ser., 325, Longman Sci. Tech., Harlow, 1995, 130–151 | MR | Zbl

[7] A. I. Prilepko, D. G. Orlovsky, I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Monogr. Textbooks Pure Appl. Math., 231, Marcel Dekker, New York, 2000 | MR | Zbl

[8] V. L. Kamynin, E. Franchini, “Ob odnoi obratnoi zadache dlya parabolicheskogo uravneniya vysokogo poryadka”, Matem. zametki, 64:5 (1998), 680–691 | DOI | MR | Zbl

[9] M. Ivanchov, N. Saldina, “An inverse problem for strongly degenerate heat equation”, J. Inverse Ill-Posed Probl., 14:5 (2006), 465–480 | DOI | MR | Zbl

[10] P. Cannarsa, J. Tort, M. Yamamoto, “Determination of source terms in a degenerate parabolic equation”, Inverse Problems, 26:10 (2010), 105003 | DOI | MR | Zbl

[11] Z.-C. Deng, L. Yang, “An inverse problem of identifying the coefficient of first-order in a degenerate parabolic equation”, J. Comput. Appl. Math., 235:15 (2011), 4404–4417 | DOI | MR | Zbl

[12] Z.-C. Deng, K. Qian, X.-B. Rao, L. Yang, G.-W. Luo, “An inverse problem of identifying the source coefficient in a degenerate heat equation”, Inverse Probl. Sci. Eng., 23:3 (2014), 498–517 | DOI | MR

[13] Z. Deng, L. Yang, “An inverse problem of identifying the radiative coefficient in degenerate parabolic equation”, Chin. Ann. Math. Ser. B, 35:3 (2014), 355–382 | DOI | MR | Zbl

[14] N. Huzyk, “Inverse problem of determining the coefficients in a degenerate parabolic equation”, Electron. J. Differential Equations, 2014, Paper No. 172 | MR | Zbl

[15] A. Kawamoto, “Inverse problems for linear degenerate parabolic equations by “time-like” Carleman estimate”, J. Inverse Ill-Posed Probl., 23:1 (2015), 1–21 | DOI | MR | Zbl

[16] I. Bouchouev, V. Isakov, “Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets”, Inverse Problems, 15:3 (1999), R95–R116 | DOI | MR | Zbl

[17] L. Jiang, Y. Tao, “Identifying the volatibility of underlying assets from option prices”, Inverse Problems, 17:1 (2001), 137–155 | DOI | MR | Zbl

[18] L. Jiang, Q. Chen, L. Wang, J. E. Zhang, “A new well-posed algorithm to recover implied local volatility”, Quant. Finance, 3:6 (2003), 451–457 | DOI | MR

[19] S. N. Kruzhkov, Nelineinye uravneniya s chastnymi proizvodnymi, Ch. 1, Izd-vo Mosk. un-ta, M., 1969

[20] L. A. Lyusternik, V. I. Sobolev, Kratkii kurs funktsionalnogo analiza, Vysshaya shkola, M., 1982 | MR | Zbl