On Backward Uniqueness for the Heat Operator in Cones

Abstract

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