Suppose blob $A$ spreads in all directions from any point it has already reached at speed $s$, starting from the point $a\in\mathbb{R}^n$. Suppose that $B$ spreads at $t$ from $b$. Once $A$ and $B$ collide with each other, they do not bleed into each other, they merely stop at the boundary. I am interested in the shape that $A$ and $B$ will finally take. The cases that $s=t$ or $s=0$ are easy to understand, but we are forced to formalise the intuitive idea in the general case. We will only given the properties defining $A$ in terms of $B$, but the converse is essentially identical. The evolution of the blob $A$ over time is given by $A_r$ for $r\in\mathbb{R}_+$. Each point in $A_r$ must be connected to $a$ by a path in $A_r$ of length $l$ such that $\frac{l}{s}\leq r$. Furthermore, if $x\in\mathbb{R}^n$ can be connected to $a$ by a path contained in of length $l$ with $\frac{l}{s}\leq r$, such that any path $P$ from $b$ to $x$ of length $m$ with $\frac{m}{s}<\frac{l}{s}$ intersects $A_r$ at a point $p$ with the length of $P$ up to $p$ greater than $r$, then $x\in A_r$. So we can talk about this as one object, $A$ will just be the disjoint union of $A_r$ for all $r$. Can we approximate or characterise $A$ and $B$ based on $a$, $b$, $s$ and $t$? Are they unique given these parameters? Is $\mathbb{R}^n=A\cup B$? Are there generalisations?
I've tagged this as differential geometry because it feels like the sort of problem that might use tools from differential geometry to solve, so I'm assuming that if there are well-studied generalisations, if they don't use more blobs or blobs with different shapes at the start, they're on different Riemannian manifolds. After all, at its core, it's about geodesics. Feel free to remove the tag.