Observability inequality, the interpolation inequality and the spectral inequality for the degenerate parabolic equation in R

Abstract

This paper investigates the interrelationships between the observability inequality, the H\"older-type interpolation inequality, and the spectral inequality for the degenerate parabolic equation in $\mathbb{R}$. We elucidate the distinctive properties of observable sets pertaining to the degenerate parabolic equation. Specifically, we establish that a measurable set in $\mathbb{R}$ fulfills the observability inequality when it exhibits $\gamma$-thickness at a scale $L$, where $\gamma>0$ and $L>0$.