Uniqueness in the Cauchy Problem for the Hermite Heat Equation

We prove the uniqueness of solutions to the Cauchy problem for the Hermite heat equation as follows: Let U(x, t) be a continuous function on R × [0, T] satisfying the following (∂∂t−∂2∂x2+x2)U(x,t)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\frac {\partial }{\partial t} - {\frac {\partial ^{2}}{\partial {x}^{2}}} + x^{2})U(x,t) = 0$\end{document} in R × (0, T),For every 𝜖 > 0, there exists a constant C := C(𝜖) > 0 and 0 < α > 1 such that supx∈R|U(x,t)|≤Cexp[(𝜖/t)α]for0<t<T,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sup\limits_{x\in\textbf{R}}|U(x,t)|\leq C\exp [(\epsilon/t)^{\alpha}] \ \text{for} \ 0<t<T, $$\end{document}U(x, 0) = 0 for x ∈ R.