3 BACKWARD UNIQUENESS FOR PARABOLIC OPERATORS WITH VARIABLE COEFFICIENTS IN A HALF SPACE
Jie Wu, Liqun Zhang · 2016
It is shown that a function $u$ satisfying $|\partial_tu+\sum_{i,j}\partial_i(a^{ij}\partial_ju)|\leq N(|u|+| abla u|)$, $|u(x,t)|\leq Ne^{N|x|^2}$ 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 $| abla_xa^{ij}(x,t)|\leq \frac{E}{|x|}$, where $E$ is less than a given number, and the conditions are in some sense optimal.