Geodesic
Increasing solutions of linear second-order equations with nonnegative characteristic form
Matematičeskie zametki, Tome 3 (1968) no. 2, pp. 171-178

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In a layer $H\{0$ we consider a linear second-order parabolic equation that degenerates on an arbitrary subset $\overline H$. It is assumed that the coefficient of the time derivative has a zero of sufficiently high order on the hyperplane $t=0$; as a consequence, the Cauchy problem will be unsolvable. The exact bounds are obtained of the permissible growth of the sought-for function when $|x|\to\infty$, ensuring a single-valued solution of the problem without initial data. 
@article{MZM_1968_3_2_a6,
     author = {A. S. Kalashnikov},
     title = {Increasing solutions of linear second-order equations with nonnegative characteristic form},
     journal = {Matemati\v{c}eskie zametki},
     pages = {171--178},
     publisher = {mathdoc},
     volume = {3},
     number = {2},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1968_3_2_a6/}
}
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A. S. Kalashnikov. Increasing solutions of linear second-order equations with nonnegative characteristic form. Matematičeskie zametki, Tome 3 (1968) no. 2, pp. 171-178. http://geodesic.mathdoc.fr/item/MZM_1968_3_2_a6/