Nonlinear integrated optic devices using the quasi-phase matching may require the periodic domain inversion in ferroelectric crystals. The periodic domain inversion in the Z-cut $LiNbO_3$ substrate is obtained by Ti film deposition and heat-treatment. As a result of experiment, the evolution of three types of domain inversion(i.e., $Li_2O$ outdiffused domain inversion, Ti lateral-diffused domain inversion and Ti indiffused domain inversion in turn) is elucidated. To compare properties of domain inversions, waveguides were fabricated by proton exchange and annealing. It is shown that Ti lateral-diffused domain inversion induces the surface-guiding effect less than Ti indiffused domain inversion. Therefore, Ti lateral-diffused domain inversion may be preferable in fabricating periodic domain gratings with a long period. In Appendix, the domain inversion using external electric fields is tried in the Y-cut $LiNbO_3$ substrate.