If you haven’t already read Neal Stephenson’s Seveneves I highly recommend you do. The basic premise is that the moon breaks apart and causes the eventual destruction of the earth’s surface.
The moon blew up without warning and for no apparent reason.
Obviously, the moon blowing up is bad, but scientist in the book quickly figure out that’s it’s really, really bad. So bad, that the earth will become uninhabitable.
In the novel, one day the moon breaks up into 7 roughly equal-sized pieces. These pieces continue peacefully orbiting the Earth for a while, and eventually two pieces collide. This collision causes a piece to fragment, making future collisions more likely. The process repeats, at what Stephenson says is an exponential rate, until the Earth is under near-constant bombardment from meteorites, wiping out (nearly) all life on Earth.
One of the events depicted in the book is the ‘white sky’, where the number of impacts grows exponentially until they are near-constant. Plotted on a logarithmic scale, there does appear to be an initial exponential phase in the simulation, lasting a relatively short time. Soon after, the impact rate slows but not to zero, with impacts occurring every so often for the remainder of the simulation. This is another scenario predicted in the book, where the planet takes many thousands of years to cool down after the initial heavy bombardment.