Inverse problem of determining the source function in a degenerate parabolic equation with a divergent principal part on a plane
https://doi.org/10.26583/vestnik.2025.5.4
EDN: LMVJHR
Abstract
We study the linear inverse problem of determining the unknown, time-dependent right-hand side (source function) in a one-dimensional parabolic equation with a weakly degenerate principal part defined in divergence form. The additional observation condition is specified in integral form. Sufficient conditions are established under which a solution to the inverse problem under consideration exists and is unique. No restrictions are imposed on the value of T or the size of the domain, i.e. the proven theorems are of a global nature. The solution is understood in the generalized sense according to Sobolev; in particular, the unknown source function is sought in the space L2(0, T). The equation’s coefficients may depend on both the time and spatial variables. Degeneracy of the equation is also permitted in both the time and spatial variables. Proofs of the existence and uniqueness theorems for the solution of the inverse problem are based on the study of the unique solvability of the corresponding direct problem, which is also new and of independent interest. When studying the unique solvability of the inverse problem, it is reduced to studying the solvability of a certain operator equation, using general theorems of functional analysis
Keywords
inverse problems of determining the source function,
degenerate parabolic equations,
integral observation
Supporting agencies: The work was carried out with the support of the Priority-2030 Program of NRNU MEPhI
About the Authors
V. I. Kamynin
National Research Nuclear University MEPhI
Russian Federation
Russian Federation
O. V. Nagornov
National Research Nuclear University MEPhI
Russian Federation
Russian Federation
References
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Review
For citations:
Kamynin V.I., Nagornov O.V. Inverse problem of determining the source function in a degenerate parabolic equation with a divergent principal part on a plane. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2025;14(5):414-423. (In Russ.) https://doi.org/10.26583/vestnik.2025.5.4. EDN: LMVJHR
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