Imagine you have two volatilities, the second which is "activated" when the stock crosses a barrier called $p_b$. The present price is $p_1$. ($p_b>p_1$).
This can be used to price options after a crash when it's assumed that options are more expensive in the short term and are reverting over the long term.
There is also a time parameter for the ratio of time for the duration first volatility ($t_1$) with the time duration until expiration ($t_2$).
I've tried pricing such an option and it's quite messy and involves some improvisation for tricky situations such as when the barrier is the same as the present price or when the time duration $t_1$ is very small relative to $t_2$.
This is the best I have done so far. It still has inconsistencies where the call exceeds the present price for very specific instances.
When the volatilities are equal, the equation reduces to Black-Scholes.
If the first volatility is very flat and the second is greater, the stock --- as $t$ progresses --- will eventually assume the longer-run volatility, but the option is slightly cheaper than simply pricing it with the long-run because the lesser short-run volatility is like friction. Likewise, if the finite frame volatility is greater than the longer run, then we should get a small "boost" for the call price from the short-run even as the option assumes the long-run lesser volatility.