Abstract
In this paper, we consider the operator of parabolic type $$ Lu=\frac{\partial u}{\partial t}-\frac{\partial^{2}u}{\partial x^{2}}+q(x)u, $$ in the space $L_{2}(R^{2})$ with a greatly growing coefficient at infinity. The operator is originally defined on $C_{0}^{\infty}(R^{2})$, where $C_{0}^{\infty}(R^{2})$ is the set of infinitely differentiable and compactly supported functions. \noindent Assume that the coefficient $q(x)$ is a continuous function in $R=(-\infty, \infty)$, and it can be a strongly increasing function at infinity. \noindent The operator $L$ admits closure in space $L_{2}(R^{2})$, and the closure is also denoted by $L$. \noindent In the paper, we proved the bounded invertibility of the operator $L$ in the space $L_{2}(R^{2})$ and the existence of the estimate $$ \left \ \frac{\partial u}{\partial t}\right\ _{L_{2}(R^{2})}+\left \ \frac{\partial^{2} u}{\partial x^{2}}\right\ _{L_{2}(R^{2})}+\left \ q(x)u\right\ _{L_{2}(R^{2})} \leq C(\left \ Lu\right\ _{L_{2}(R^{2})}+\left \ u\right\ _{L_{2}(R^{2})}), $$ under certain restrictions on $q(x)$ in addition to the conditions indicated above. Example. $q(x)=e^{100 x }$,