Backward uniqueness for parabolic operators with variable coefficients in a half space

Abstract

It is shown that a function $u$ satisfying $|{\partial }_{t}u+{\sum }_{i,j}{\partial }_i(a^{ij}{\partial }_{j}u)|\le N(|u|+|\nabla u|)$, $|u(x,t)|\le N{e}^{N|x{|}^2}$ in ${\mathbb{R}}_{+}^n\times [0,T]$ and $u(x,0)=0$ in ${\mathbb{R}}_{+}^n$ under certain conditions on $\{a^{ij}\}$ must vanish identically in ${\mathbb{R}}_{+}^n\times [0,T]$. The main point of the result is that the conditions imposed on $\{a^{ij}\}$ are of the type: $\{a^{ij}\}$ are Lipschitz and $|{\nabla }_x{a}^{ij}(x,t)|\le \frac{E}{|x|}$, where $E$ is less than a given number, and the conditions are in some sense optimal.

32 pages