Increasing solutions of linear second-order equations with nonnegative characteristic form

In a layer $H\{0<t\le T,\ x\in R^n\}$ 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.