Backward uniqueness for parabolic operators with variable coefficients in a half space

TL;DR

This paper proves that under certain optimal conditions on variable coefficients, solutions to a class of parabolic operators in a half space that vanish at initial time must be identically zero, establishing a backward uniqueness result.

Contribution

It establishes a backward uniqueness theorem for parabolic operators with variable coefficients under near-optimal Lipschitz and decay conditions.

Findings

Solutions vanish identically if initial data is zero and conditions are met.

Conditions on coefficients are nearly optimal and involve Lipschitz continuity with decay.

The result extends backward uniqueness to a broader class of variable coefficient operators.

Abstract

It is shown that a function $u$ satisfying $|\partial_tu+\sum_{i,j}\partial_i(a^{ij}\partial_ju)|\leq N(|u|+|\nabla u|)$, $|u(x,t)|\leq Ne^{N|x|^2}$ 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 $|\nabla_xa^{ij}(x,t)|\leq \frac{E}{|x|}$, where $E$ is less than a given number, and the conditions are in some sense optimal.