  We consider a generalization of the model by Doering and Gadoua to
non-Markovian potential-switching generated by arbitrary renewal processes. For
the Markovian switching process, we extend the original results by Doering and
Gadoua by giving a complete description of the absorption process. For all
non-Markovian processes having the first moment of the waiting time
distributions, we get qualitatively the same results as in the Markovian case.
However, for distributions without the first moment, the mean first passage
time curves do not exhibit the resonant activation minimum. We thus come to the
conjecture that the generic mechanism of the resonant activation fails for
fluctuating processes widely deviating from Markovian.
