Geodesic
On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination condition
Matematičeskie zametki, Tome 77 (2005) no. 4, pp. 522-534

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We study the unique solvability of the inverse problem of determining the righthand side of a parabolic equation whose leading coefficient depends on both the time and the spatial variable under an integral overdetermination condition with respect to time. We obtain two types of condition sufficient for the local solvability of the inverse problem as well as study the so-called Fredholm solvability of the inverse problem under consideration. 
@article{MZM_2005_77_4_a4,
     author = {V. L. Kamynin},
     title = {On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination condition},
     journal = {Matemati\v{c}eskie zametki},
     pages = {522--534},
     publisher = {mathdoc},
     volume = {77},
     number = {4},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_77_4_a4/}
}
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V. L. Kamynin. On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination condition. Matematičeskie zametki, Tome 77 (2005) no. 4, pp. 522-534. http://geodesic.mathdoc.fr/item/MZM_2005_77_4_a4/