On Backward Uniqueness for the Heat Operator in Cones

Consider the system $|\partial_tu+\Delta u|\leq M(|u|+|\nabla u|)$, $|u(x,t)|\leq Me^{M|x|^2}$ in $\mathcal{C}_{\theta}\times[0,T]$ and $u(x,0)=0$ in $\mathcal{C}_{\theta}$, where $\mathcal{C}_{\theta}$ is a cone with opening angle $\theta$. L. Escauriaza constructed an example to show that such system has a nonzero bounded solution when $\theta<90^\circ$, and it's conjectured that the system has only zero solution for $\theta>90^\circ$. Recently Lu Li and V. \v{S}ver\'{a}k \cite{LlS} proved that the claim is true for $\theta>109.5^\circ$. Here we improve their result and prove that only zero solution exists for this system when $\theta>99^\circ$ by exploring a new type of Carleman inequality, which is of independent interest.