  A general condition for sharp transition of decay rate from quantum to
thermal regimes is derived in dissipative tunneling models when position
dependent mass is involved. It is shown that the effect of dissipation in
general changes the order of the phase transition. Especially, for the models
with constant mass the Ohmic dissipation enlarges the range of parameters for
first-order phase transitions. In the case of second-order phase transition the
Ohmic dissipation suppresses the decay rate near the transition
temperature(T_c). For the super-Ohmic case the dissipation yields an opposite
effects to the Ohmic dissipation within exponential approximation.
