  We derive general equations for the nonlinear relaxation time of Brownian
diffusion in randomly switching potential with a sink. For piece-wise linear
dichotomously fluctuating potential with metastable state, we obtain the exact
average lifetime as a function of the potential parameters and the noise
intensity. Our result is valid for arbitrary white noise intensity and for
arbitrary fluctuation rate of the potential. We find noise enhanced stability
phenomenon in the system investigated: the average lifetime of the metastable
state is greater than the time obtained in the absence of additive white noise.
We obtain the parameter region of the fluctuating potential where the effect
can be observed. The system investigated also exhibits a maximum of the
lifetime as a function of the fluctuation rate of the potential.
