Decay at infinity of caloric functions within characteristic hyperplanes
  It is shown that a function $u$ satisfying, $|\Delta u+\partial_tu|\le M(|u|+|\nabla u|)$, $|u(x,t)|\le Me^{M|x|^2}$ in $\R^n\times [0,T]$ and $|u(x,0)|\le C_ke^{-k|x|^2}$ in $\R^n$ and for all $k\ge 1$, must vanish identically in $\R^n\times [0,T]$. 
L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega