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 Mat. Zametki, 2008, Volume 84, Issue 1, Pages 48–58 (Mi mz4134)

On the Inverse Problem of Determining the Leading Coefficient in Parabolic Equations

V. L. Kamynin

Moscow Engineering Physics Institute (State University)

Abstract: We study the unique solvability of the inverse problem of determining the leading coefficient in the parabolic equation on the plane with coefficients depending on both time and spatial variables under the condition of integral overdetermination with respect to time. We obtain sufficient conditions for the unique solvability of the inverse problem. We present nontrivial examples of problems for which such conditions hold. It is shown that the imposed conditions necessarily hold if either the time interval is sufficiently large or the space interval on which the problem is considered is sufficiently small.

Keywords: parabolic equation, inverse problem for the parabolic equation, Poincaré–Steklov inequality, Schauder fixed-point theorem, maximum principle, compact operator

DOI: https://doi.org/10.4213/mzm4134

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English version:
Mathematical Notes, 2008, 84:1, 45–54

Bibliographic databases:

UDC: 517.956

Citation: V. L. Kamynin, “On the Inverse Problem of Determining the Leading Coefficient in Parabolic Equations”, Mat. Zametki, 84:1 (2008), 48–58; Math. Notes, 84:1 (2008), 45–54

Citation in format AMSBIB
\Bibitem{Kam08} \by V.~L.~Kamynin \paper On the Inverse Problem of Determining the Leading Coefficient in Parabolic Equations \jour Mat. Zametki \yr 2008 \vol 84 \issue 1 \pages 48--58 \mathnet{http://mi.mathnet.ru/mz4134} \crossref{https://doi.org/10.4213/mzm4134} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2451884} \transl \jour Math. Notes \yr 2008 \vol 84 \issue 1 \pages 45--54 \crossref{https://doi.org/10.1134/S0001434608070043} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000258855600004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-50849117454} 

• http://mi.mathnet.ru/eng/mz4134
• https://doi.org/10.4213/mzm4134
• http://mi.mathnet.ru/eng/mz/v84/i1/p48

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Kamynin V.L., “Unique solvability of the inverse problem of determination of the leading coefficient in a parabolic equation”, Differ. Equ., 47:1 (2011), 91–101
2. Xiao C., “Optimization method for the inverse coefficient problem of a parabolic equation”, Ceis 2011, Procedia Engineering, 15, eds. Ran C., Yang G., Elsevier Science BV, 2011, 4880–4884
3. Pan J., “On an overdetermined problem of determining parameter in a degenerate parabolic equation”, Lith. Math. J., 51:4 (2011), 533–542
4. Kamynin V.L., “Inverse problem of finding the coefficient of a lower derivative in a parabolic equation on the plane”, Differ. Equ., 48:2 (2012), 214–223
5. Fraguela A., Infante J.A., Ramos A.M., Rey J.M., “A Uniqueness Result for the Identification of a Time-Dependent Diffusion Coefficient”, Inverse Probl., 29:12 (2013), 125009
6. Pan J., “On an Over-Determined Problem of Free Boundary of a Degenerate Parabolic Equation”, Appl. Mat., 58:6 (2013), 657–671
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