Backward uniqueness for solutions of linear parabolic equations

Abstract:

We address the backward uniqueness property for the equation $u_t-\Delta u = w_j\partial _{j}u+v u$ in ${\mathbb R}^n\times (T_0,0]$. We show that under rather general conditions on $v$ and $w$, $u|_{t=0}=0$ implies that $u$ vanishes to infinite order for all points $(x,0)$. It follows that the backward uniqueness holds if $w=0$ and $v\in L^{\infty }([0,T_0],L^p({\mathbb R}^n))$ when $p>n/2$. The borderline case $p=n/2$ is also covered with an additional continuity and smallness assumption.