Geodesic
On a~problem with nonlocal boundary condition for a~parabolic equation
Sbornik. Mathematics, Tome 74 (1993) no. 1, pp. 219-249

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Well-posed solvability is proved in an appropriate energy space of a boundary value problem with a nonlocal boundary condition for a one-dimensional parabolic equation; two-sided uniform estimates of the solution are obtained, which replace the maximum principle. The existence of an optimal control of the diffusion coefficient in the problem of minimizing the quality functional is established in the class of functions of bounded variation. 
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     title = {On a~problem with nonlocal boundary condition for a~parabolic equation},
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L. A. Muravei; A. V. Filinovskii. On a~problem with nonlocal boundary condition for a~parabolic equation. Sbornik. Mathematics, Tome 74 (1993) no. 1, pp. 219-249. http://geodesic.mathdoc.fr/item/SM_1993_74_1_a15/