Uniqueness classes for solutions of the Cauchy problem for linear equations with rapidly increasing coefficients

Uniqueness classes, and also nonuniqueness classes, are found for solutions of the Cauchy problem for equations or the form $\displaystyle\frac{\partial u}{\partial t}=\sum^n_{k=0}q_k(x)\frac{\partial^ku}{\partial x^k}$ in which the growth $q_0(x)$ as $|x|\to\infty$ is sufficiently rapid, the growth of the other coefficients is “subordinate” to that of $q_0(x)$, and the classes depend on $q_0(x)$.