Decay at infinity of caloric functions within characteristic hyperplanes

Abstract

It is shown that a function $u$ satisfying, $|\Delta u+{\partial }_{t}u|\le M(|u|+|\nabla u|)$, $|u(x,t)|\le M{e}^{M|x{|}^2}$ in ${\mathbb{R}}^n\times [0,T]$ and $|u(x,0)|\le C_k{e}^{-k|x{|}^2}$ in ${\mathbb{R}}^n$ and for all $k\ge 1$, must vanish identically in ${\mathbb{R}}^n\times [0,T]$.