Quantitative unique continuation for parabolic equations with neumann boundary conditions

In this paper, we establish a globally quantitative estimate of unique continuation at one time point for solutions of parabolic equations with Neumann boundary conditions in bounded domains. Our proof is mainly based on Carleman commutator estimates and a global frequency function argument, which is motivated by recent work [5]. As an application, we obtain an observability inequality from measurable sets in time for all solutions of the above equations.