Are there physics models that accurately handle the assumption of having solutions that achieve finite ending times? 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:

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*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 $x(T)=\dot{x}(T)=0$
 A: Thanks to the comment by @ConnorBehan about the fast difussion equation, I have been able to find another topics on physics where solutions of finite duration are studied, like sublinear damping, nonlinear Schrödinger equation, nonlinear heat equation, among others... I left here some of the papers I have found:

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*"A note on the dynamics of an oscillator in the presence of strong friction" - H.Amann & J.I.Diaz

*"A conservation law with spatially localized sublinear damping" - Christophe Besse & Sylvain Ervedoza

*"Behavior of Solutions of Second-Order Differential Equations with Sublinear Damping" - J. Karsai & J. R. Graef

*"Extinction time for some nonlinear heat equations" - Louis A. Assalé, Théodore K. Boni, Diabate Nabongo

*"Stability of the separable solution for fast diffusion" - James G. Berryman & Charles J. Holland 

*"Degenerate parabolic equations with general nonlinearities" - Robert Kersner

*"Nonlinear Heat Conduction with Absorption: Space Localization and Extinction in Finite Time" - Robert Kersner

*"Finite extinction time for a class of non-linear parabolic equations" - Gregorio Diaz & Ildefonso Diaz

*"Finite Time Extinction by Nonlinear Damping for the Schrödinger Equation" - Rémi Carles, Clément Gallo

*"Classification of extinction profiles for a one-dimensional diffusive Hamilton–Jacobi equation with critical absorption" - Razvan Gabriel Iagar, Philippe Laurençot

*"Fast diffusion flow on manifolds of nonpositive curvature" - M. Bonforte, G. Grillo, J. Vázquez
On these papers, are found examples for classic mechanics and quantum mechanics, but I haven't found yet an example in relativistic mechanics (Standard or General), but the last paper use the framework of Differential Topology/Geometry it is also used on SR/GR... don't know if maybe the topic is too new in this area... I found also this another paper:

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*"Finite extinction time for the solutions to the Ricci flow on certain three-manifolds" - Grisha Perelman
were later I found the author is indeed Grigori Perelman (see the last paper $P03b$), the mathematician that refuse the Field Medal ($2006$) and the Millennium Problems' prize ($2010$) for solving the Poincaré conjecture... so maybe the topic have not yet enough work to easy find related papers on google (actually, I am asking this here).
