Differential Equations

, Volume 51, Issue 5, pp 605–619 | Cite as

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

  • A. B. KostinEmail author
Partial Differential Equations


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.


Inverse Problem Parabolic Equation Monotone Operator Direct Problem Sobolev Embedding Theorem 
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© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

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

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