  We study the influence of density-dependent symmetry energy at high densities
in simulations of core-collapse supernovae, black hole formation and
proto-neutron star cooling by extending the relativistic mean field (RMF)
theory used for the Shen EOS table. We adopt the extended RMF theory to examine
the density dependence of the symmetry energy with a small value of the slope
parameter $L$, while the original properties of the symmetric nuclear matter
are unchanged. In order to assess matter effects at high densities, we perform
numerical simulations of gravitational collapse of massive stars adopting the
EOS table at high densities beyond $10^{14}$ g/cm$^3$ with the small $L$ value,
which is in accord with the experimental and observational constraints, and
compare them with the results obtained by using the Shen EOS. Numerical results
for 11.2M$_{\odot}$ and 15M$_{\odot}$ stars exhibit minor effects around the
core bounce and in the following evolution for 200 ms. Numerical results for
40M$_{\odot}$ and 50M$_{\odot}$ stars reveal a shorter duration toward the
black hole formation with a smaller maximum mass for the small $L$ case.
Numerical simulations of proto-neutron star cooling over 10 s through neutrino
emissions demonstrate increasing effects of the symmetry energy at high
densities. Neutrino cooling drastically proceeds in a relatively long timescale
with high luminosities and average energies with the small symmetry energy.
Evolution toward the cold neutron star is affected because of the different
behavior of neutron-rich matter while supernova dynamics around core bounce
remains similar in less neutron-rich environments.
