  A time dependent variational approach is considered to derive the equations
of movement for the $\lambda \phi^4$ model. The temporal evolution of the model
is performed numerically in the frame of the Gaussian approximation in a
lattice of 1+1 dimensions given non homogeneous initial conditions (like
bubbles) for the classical and quantum parts of the field which expands. A
schematic model for the initial conditions is presented considering the model
at finite fermionic density. The non zero fermionic density may lead either to
the restoration of the symmetry or to an even more asymmetric phase. Both kinds
of situations are considered as initial conditions and the eventual differences
in early time dynamics are discussed. In the early time evolution there is
strong energy exchange between the classical and quantum parts of the field as
the initial configuration expands. The contribution of the quantum fluctuations
is discussed especially in the strong coupling constant limit. The continuum
limit is analyzed.
