  Motivated by previous works, we study semi-classical cosmological solutions
and the wave function of the Wheeler-DeWitt equation in the Bose-Parker-Peleg
model. We obtain the wave function of the universe satisfying the suitable
boundary condition of the redefined fields, which has not been considered in
previous works. For some limiting cases, the Wheeler-DeWitt equation is reduced
to the Liouville equation with a boundary, and its solution can be described by
well-known functions. The consistent requirement of the boundary condition is
related to the avoidance of the curvature singularity.
