On backward uniqueness for the heat operator in cones

Consider the system |∂tu+Δu|≤M(|u|+|∇u|)|∂tu+Δu|≤M(|u|+|∇u|), |u(x,t)|≤MeM|x|2|u(x,t)|≤MeM|x|2 in Cθ×[0,T]Cθ×[0,T] and u(x,0)=0u(x,0)=0 in CθCθ, where CθCθ is a cone with opening angle θ  . L. Escauriaza constructed an example to show that such system has a nonzero solution when θ<90°θ<90°, and it's conjectured that the system has only zero solution when θ>90°θ>90°. Lu Li and V. Šverák proved that the conjecture is true when θ>109.5°θ>109.5°. Here we improve their result and prove that the conjecture is true when θ>99°θ>99° by exploring a new type of Carleman inequality, which is of independent interest.