  Using the exact Bethe ansatz solution of the Hubbard model and Luttinger
liquid theory, we investigate the density profiles and collective modes of
one-dimensional ultra-cold fermions confined in an optical lattice with a
harmonic trapping potential. We determine a generic phase diagram in terms of a
characteristic filling factor and a dimensionless coupling constant. The
collective oscillations of the atomic mass density, a technique that is
commonly used in experiments, provide a signature of the quantum phase
transition from the metallic phase to the Mott-insulator phase. A detailed
experimental implementation is proposed.
