Assume $M\subset \mathbb R^n$ is a hypersurface, I move $M$ in normal direct, i.e. $$ \partial_t x = f(x)\nu(x) \tag{1} $$ where $x$ is the position vector, $\nu$ is normal vector, $f(x)$ is speed. Now I want to change the speed, for example, for a given function $\psi (t)\ge 0$, consider $$ \partial_t x = (f(x)+\psi(t))\nu \tag{2} $$ For same initial hypersurface $M$, I use $M_1(t)$ and $M_2 (t)$ denote the result of (1) and (2) at time $t$. Then I guess for any $t_2\ge 0$ there is $t_1\ge 0$ such that $$ M_1(t_1) =C M_2(t_2) $$ I mean the result of (1) at time $t_1$ is similar to the result of (2) at $t_2$. They have the same shape, just size is different. Is this guess right ?
I get this question from the normalizing of mean curvature flow, generally, we normalize the area, but if we normalize the volume enclosed by hypersurface, the equivalence equals to the above question.