Let $n\in\mathbb{N}$, let $T>0$, and let $U\subset\mathbb{R}^n$ be an [open set](/page/Open%20Set). Let $\mathcal{L}^n$ denote $n$-dimensional [Lebesgue measure](/page/Lebesgue%20Measure), and let $L^2(U)$, $H^1(U)$, and $H_0^1(U)$ denote real-valued function spaces over $(U,\mathcal{B}(U),\mathcal{L}^n)$. Suppose