/sci/ - Science & Math

/lit/ here, hope this is the right thread.
I would love some advice for a sci-fi short story I'm developing, as I haven't dealt with any maths/physics since I graduated high-school over a decade ago.
The story has a plot device that involves matter expanding concentrically/spherically at a rapid pace. I want to express that the rate of expansion declines as a function of area (already) covered, where the initial speed/rate is significant (~1% c) but the decline also 'speeds up' so that by the time the total area covered reaches 50km the speed drops down very fast to speed of sound levels and then remains constant (or rather any further decline becomes so miniscule that it seems practically constant by human standards).
In the story this is supposed to function like a law of nature, so I want to express it as a formula. Any help would be much appreciated. As I said I'm not familiar with math/physics, so please excuse the poor explanation...