Are there physics models that accurately handle the assumption of having solutions that achieve finite ending times?
Intro
Recently I learned on the answers and comments of this QUESTION that the only possible way for a differential equation to having solutions that achieve a finite ending time, is by their differential equation having a singular point in time (non-Lipschitz) where uniqueness of solutions could be broken. To be precise about what I mean with a finite ending time I am using this:
Definition - Solutions of finite duration: the solution $x(t)$ becomes exactly zero at a finite time $T<\infty$ by its own dynamics and stays there forever after $(t\geq T\Rightarrow x(t)=0)$. So, they are different of just a piecewise section made by any arbitrarily function multiplied by rectangular function: it must solve the differential equation in the whole domain. As example: $\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$ could be solved by $x(t) = \frac{1}{4}\left(1-\frac{t}{2}+\left|1-\frac{t}{2}\right|\right)^2$. Second order systems' examples on the mentioned question.
This get me shocked since every model I learned on engineering were "made" to always holding uniqueness of solutions, but conversely, every day experience of classical systems is that they indeed stops moving due their dynamics at some finite time (leaving outside here movement because of thermal noise which is random and "unrelated" to classical systems' differential equation): intuition tells me that exist the time when I stop my car engine, when my feet stops touching the floor, when I turn on and off a light switch, among others.
But Why this kind of solutions should matter, or Why maybe holding uniqueness could worth been abandoned sometimes (since at best leads to solutions that "vanishes at infinity" - which could even lead to causality issues), is explained in detail on the mentioned question, so I left explanations out of here since will get the question too long. I know that this topic could be controversial, so please take a look there first there so we don't be repeating the same discussions here: there the discussion is focused on the mathematics point of view, here I would like to understand if this finite duration solutions could be a tool that "make sense" to be used on physics: for example, What are the implications of achieving a finite ending time on time-symmetry?
Question
I beforehand understand that models are made to being practical of use explaining real data, and non-uniqueness could made a big mess complicating things (which is a huge point against them), but also since nowadays models are squeezed to the very possible drop trying find new answers that helps to explain reality "unexpected results" (like made classical intuition about quantum entanglement, as one of many examples), I would like to know if considering solutions of finite duration could help to understand better some of these phenomena.
So, I would like to know if there are any current examples of models in physics that have singular points where finite duration solutions could be introduced, or otherwise, why the assumption of having solutions that are achieving finite ending times could be a "mistaken intuition" for modeling physics phenomena (Here I am not stating that every phenomena should achieve an "end" in finite time, but instead if make sense to model "some phenomena" in this way).
As example, on the mentioned questioned is found that a sufficient condition (but not necessarily required), for an autonomous system $\ddot{x}=F(\dot{x},x)$ to having solutions of finite duration is fulfilling these two properties:
- The differential equation support the trivial zero solution, and
- The differential equation also has at least one singular point in time $t=T<\infty$ where $x(T)=\dot{x}(T)=0$
