  We present a study of spatially-resolved star formation histories (SFHs) for
60 $z\sim2.3$ main-sequence, star-forming galaxies selected from the MOSDEF
spectroscopic survey in the GOODS-N field, with median stellar mass
$\log(M_\star/M_\odot)=9.75$ and spanning the range
$8.6<\log(M_\star/M_\odot)<11.5$. Photometry is decomposed into a central and
outer spatial component using observed $z_\mathrm{F850LP}-H_\mathrm{F160W}$
colors. The Prospector code is used to model spectral energy distributions for
the center, outskirt, and integrated galaxy using HST/ACS and WFC3,
Spitzer/IRAC, and ground-based photometry, with additional constraints on
gas-phase metallicity and spectroscopic redshift from MOSDEF spectroscopy. For
the low-resolution bands, spatially-resolved photometry is determined with an
iterative approach. The reconstructed SFHs indicate that the majority of
galaxies with $\log(M_\star/M_\odot)<10.5$ are observed while their central
regions undergo relatively recent ($<100$ Myr) bursts of star formation, while
the outskirts have a smooth, quasi-steady SFH that gently increases towards the
redshift of observation. The enhanced star formation activity of the central
parts is broadly consistent with the idea that it is produced by highly
dissipative gas compaction and accretion. The wide range of central densities
and sizes observed in the sample suggests that for the selected galaxies such a
process has started but is still far from being completed. The implication
would be that selecting star-forming galaxies at cosmic noon frequently
includes systems in an "evolved" evolutionary phase where the centers have
recently started a burst of star formation activity that will likely initiate
inside-out quenching in the next several hundred million years.
