Geodesic
Behavior of the solution of a parabolic equation on a characteristic
Matematičeskie zametki, Tome 12 (1972) no. 3, pp. 257-262.

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We consider the solution of a linear second-order parabolic equation with one spatial variable and a zero right side. We prove that since the solution decreases quite rapidly in the spatial variable as it approaches a particular point, it vanishes on the part of the characteristic joining the point to the boundary of the region in which the solution is defined. 
@article{MZM_1972_12_3_a4,
     author = {E. M. Landis},
     title = {Behavior of the solution of a parabolic equation on a characteristic},
     journal = {Matemati\v{c}eskie zametki},
     pages = {257--262},
     publisher = {mathdoc},
     volume = {12},
     number = {3},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_12_3_a4/}
}
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E. M. Landis. Behavior of the solution of a parabolic equation on a characteristic. Matematičeskie zametki, Tome 12 (1972) no. 3, pp. 257-262. http://geodesic.mathdoc.fr/item/MZM_1972_12_3_a4/