Abstract
We consider the inverse problem of determining the time-dependent thermal diffusivity that is equal to zero at the initial moment of time. We establish conditions for the existence and uniqueness of a classical solution of the problem under consideration.
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References
A. G. Podgaev, “On boundary-value problems for some quasilinear parabolic equations with nonclassical degeneration,” Sib. Mat. Zh., 28, No. 2, 129–139 (1987).
L. A. Caffarelli and A. Friedman, “Continuity of the density of a gas flow in a porous medium,” Trans. Amer. Math. Soc., 252, 99–113 (1979).
V. P. Glushko, “Degenerate linear differential equations,” Differents. Uravn., 4, No. 11, 1957–1966 (1968).
A. S. Kalashnikov, “On growing solutions of linear equations of the second order with nonnegative characteristic form,” Mat. Zametki, 3, No. 2, 171–178 (1968).
A. S. Kalashnikov, “A problem without initial conditions in the classes of growing solutions for some linear degenerate parabolic systems of the second order. I,” Vestn. Mosk. Univ., Ser. Mat., Mekh., No. 2, 42–48 (1971).
A. S. Kalashnikov, “A problem without initial conditions in the classes of growing solutions for some linear degenerate parabolic systems of the second order. II,” Vestn. Mosk. Univ., Ser. Mat., Mekh., No. 3, 3–9 (1971).
O. A. Oleinik and E. V. Radkevich, “Equations of the second order with nonnegative characteristic form,” in: VINITI Series in Mathematical Analysis [in Russian], VINITI, Moscow (1971), pp. 7–252.
T. D. Dzhuraev, “On boundary-value problems for linear parabolic equations degenerating at the boundary of a domain,” Mat. Zametki, 12, No. 5, 643–652 (1972).
V. P. Glushko, “On the solvability of mixed problems for parabolic equations of the second order with degeneration,” Dokl. Akad. Nauk SSSR, 207, No. 2, 266–269 (1972).
A. V. Glushak and S. D. Shmulevich, “On some well-posed problems for parabolic equations of higher order degenerating in time variable,” Differents. Uravn., 22, No. 6, 1065–1068 (1986).
O. I. Voznyak and S. D. Ivasyshen, “A Cauchy problem for parabolic systems with degeneration on the initial hyperplane,” Dopov. Nats. Akad. Nauk Ukr., No. 6, 7–11 (1994).
M. M. Gadzhiev, “Inverse problem for a degenerate elliptic equation,” in: Application of Methods of Functional Analysis to Equations of Mathematical Physics [in Russian], Novosibirsk (1987), pp. 66–71.
M. Ivanchov, Inverse Problems for Equations of Parabolic Type, VNTL Publ. (2003).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1563–1570, November, 2005.
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Ivanchov, M.I., Saldina, N.V. Inverse problem for the heat equation with degeneration. Ukr Math J 57, 1825–1835 (2005). https://doi.org/10.1007/s11253-006-0032-6
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DOI: https://doi.org/10.1007/s11253-006-0032-6