Geodesic
Inverse problem for the diffusion equation with overdetermination in the form of an external volume potential
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 9, pp. 1695-1702

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An initial-boundary value problem for the diffusion equation with an unknown initial condition is considered. Additional information used for determining the unknown initial condition is an external volume potential whose density is the Laplacian calculated for the solution of the initial-boundary value problem. Uniqueness theorems for the inverse problem are proved in the case when the spatial domain of the initial-boundary value problem is a spherical layer or a parallelepiped. 
@article{ZVMMF_2011_51_9_a9,
     author = {A. M. Denisov},
     title = {Inverse problem for the diffusion equation with overdetermination in the form of an external volume potential},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1695--1702},
     publisher = {mathdoc},
     volume = {51},
     number = {9},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_9_a9/}
}
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A. M. Denisov. Inverse problem for the diffusion equation with overdetermination in the form of an external volume potential. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 9, pp. 1695-1702. http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_9_a9/