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

It is shown that a function $u$ satisfying $\mathrm{\mid }{\mathrm{\partial }}_{t}u+{\sum }_{i,j}{\mathrm{\partial }}_i(a^{ij}{\mathrm{\partial }}_{j}u)\mathrm{\mid }\le N(\mathrm{\mid }u\mathrm{\mid }+\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid })$, $\mathrm{\mid }u(x,t)\mathrm{\mid }\le N{e}^{N\mathrm{\mid }x{\mathrm{\mid }}^2}$ 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 $\mathrm{\mid }{\mathrm{\nabla }}_x{a}^{ij}(x,t)\mathrm{\mid }\le \frac{E}{\mathrm{\mid }x\mathrm{\mid }}$, where $E$ is less than a given number, and the conditions are in some sense optimal.