Interpolation inequality at one time point for parabolic equations with time-independent coefficients and applications

In this paper, we study the H\"older-type interpolation inequality and observability inequality from measurable sets in time for parabolic equations either with L^p unbounded potentials or with electric potentials. The parabolic equations under consideration evolve in bounded C^{1,1} domains of R^N (N\geq3) with homogeneous Neumann boundary conditions. The approach for the interpolation inequality is based on a modified reduction method and some stability estimates for the corresponding elliptic operator.