Quantitative unique continuation for a parabolic equation

Abstract

We address the quantitative uniqueness properties of the solutions of the parabolic equation ${\partial }_{t}u-\Delta u=w_j(x,t){\partial }_{j}u+v(x,t)u$ where $v$ and $w$ are bounded. We prove that for solutions $u$, the order of vanishing is bounded by $C(\Vert v{\Vert }_{L^{\infty }}^{2/3}+\Vert w{\Vert }_{L^{\infty }}^2)$ matching the upper bound previously established in the elliptic case. in the elliptic case.

Indiana Univ. Math. J. (to appear)