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Inverse problem of finding the coefficient of u in a parabolic equation on the basis of a nonlocal observation condition

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Abstract

We consider the problem of reconstructing the coefficient c(x) multiplying u(x, t) in a parabolic equation. To find it, in addition to initial and boundary conditions, we pose a nonlocal observation condition of the form \(\int_0^T {u(x,t)} d\mu (t) = \chi (x)\) where the function χ(x) and the measure dµ(t) are known. We obtain sufficient conditions for the uniqueness and solvability of this problem, which have the form of easy-to-verify inequalities. We present examples of inverse problems for which the assumptions of our theorems are necessarily satisfied and an example of a problem that has a nonunique solution.

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Authors and Affiliations

  1. National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow, Russia

    A. B. Kostin

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  1. A. B. Kostin
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Correspondence to A. B. Kostin.

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Original Russian Text © A.B. Kostin, 2015, published in Differentsial’nye Uravneniya, 2015, Vol. 51, No. 5, pp. 596–610.

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Kostin, A.B. Inverse problem of finding the coefficient of u in a parabolic equation on the basis of a nonlocal observation condition. Diff Equat 51, 605–619 (2015). https://doi.org/10.1134/S0012266115050043

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