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Inverse problem with nonlocal observation of finding the coefficient multiplying u t in the parabolic equation

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Abstract

We study the inverse problem of the reconstruction of the coefficient ϱ(x, t) = ϱ0(x, t) + r(x) multiplying u t in a nonstationary parabolic equation. Here ϱ0(x, t) ≥ ϱ0 > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L ∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫ T0 u(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.

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Authors and Affiliations

  1. National Research Nuclear University “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, 2016, published in Differentsial’nye Uravneniya, 2016, Vol. 52, No. 2, pp. 220–238.

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Kostin, A.B. Inverse problem with nonlocal observation of finding the coefficient multiplying u t in the parabolic equation. Diff Equat 52, 220–239 (2016). https://doi.org/10.1134/S0012266116020087

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