ОN INVERSE PROBLEM OF DETERMINING THE ABSORPTION COEFFICIENT IN THE PARABOLIC EQUATION UNDER THE CONDITION OF FINAL OBSERVATION
https://doi.org/10.26583/vestnik.2024.5.5
EDN: NGQCYO
Abstract
We consider the nonlinear inverse problem of determining the $x$-dependent lower coefficient in a uniformly parabolic equation with many spatial variables. The coefficients of the equation can depend on both the time and spatial variables and are assumed to be bounded, but generally speaking, discontinuous. However (in contrast to the papers of other authors), there are no restrictions on the signs of the lower coefficients of the equation and its right-hand side. As an additional condition, we put the condition of the final (at the final moment of time) observation. The solution of the inverse problem is understood in a generalized sense and is sought in Sobolev classes. We establish two types of sufficient conditions under which a generalized solution of the inverse problem exists and is unique. We also give the example of the inverse problem for which the results proved in the work are valid. It is noted that the solution of the specified problem exists and is unique either if the time interval on which the problem is considered is sufficiently large (and the domain of spatial variables is fixed) or if the domain of spatial variables is sufficiently small (and the time period is fixed).
Keywords
Supporting agencies: The work was carried out with the support of the Priority-2030 Program of the National Research Nuclear University MEPhI.
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For citations:
Kamynin V.L. ОN INVERSE PROBLEM OF DETERMINING THE ABSORPTION COEFFICIENT IN THE PARABOLIC EQUATION UNDER THE CONDITION OF FINAL OBSERVATION. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2024;13(5):329-339. (In Russ.) https://doi.org/10.26583/vestnik.2024.5.5. EDN: NGQCYO
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