On Backward Uniqueness for the Heat Operator in Cones

Abstract

Consider the system $|{\partial }_{t}u+\Delta u|\le M(|u|+|\nabla u|)$, $|u(x,t)|\le M{e}^{M|x{|}^2}$ 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. Šverák \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.

15 pages

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