We present results from the EDGE survey, a spatially resolved CO(1-0) follow-up to CALIFA, an optical Integral Field Unit (IFU) survey of local galaxies. By combining the data products of EDGE and CALIFA, we study the variation in molecular gas depletion time ($tau_{rm dep}$) on kiloparsec scales in 52 galaxies. We divide each galaxy into two parts: the center, defined as the region within $0.1 R_{25}$, and the disk, defined as the region between $0.1$ and $0.7 R_{25}$. We find that 14 galaxies show a shorter $tau_{rm dep}$ ($sim 1$ Gyr) in the center relative to that in the disk ($tau_{rm dep} sim 2.4$ Gyrs), which means the central region in those galaxies is more efficient at forming stars per unit molecular gas mass. This finding implies that the centers with shorter $tau_{rm dep}$ resemble the intermediate regime between galactic disks and starburst galaxies. Furthermore, the central drop in $tau_{rm dep}$ is correlated with a central increase in the stellar surface density, suggesting that a shorter $tau_{rm dep}$ is associated with molecular gas compression by the stellar gravitational potential. We argue that varying the CO-to-H$_2$ conversion factor only exaggerates the central drop of $tau_{rm dep}$.
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