Examples of Finite-Duration solutions to Autonomous Ordinary Differential Equations ODEs? Examples of Finite-Duration solutions to Autonomous Ordinary Differential Equations ODEs?
Examples of the scalar versions:

*

*1st order: $\dot{x} = F(x)$

*2nd order: $\ddot{x} = F(x,\dot{x})$
I have found recently a paper Finite Time Differential Equations (V. T. Haimo - 1985), where its proved that there exists conditions under which first and second order scalar autonomous non-linear ODEs admits Finite-Duration Solutions, meaning here, that they becomes exactly zero due their own dynamics and remains forever on zero after this ending time (they reach zero "from the right", remember that the variable is "time"), so, they are different from thinking on a piecewise section of a solution of a common initial-value problem, neither solutions that are vanishing at infinity, and the same condition of being zero forever after an specific time, make these solutions to fail to fulfill the conditions of Uniqueness, discarding every Linear ODEs of having them (keep in mind this idea: NO LINEAR ODE stands Finite-Duration solutions), and also, since they achieve a point where is becomes the constant zero on a measurable set of points, these solutions cannot be analytical on the whole real line, so power series will have "issues" trying to match them.
But when asking for higher dimensions, as the amazing answer gave here, looks like compact-supported solution in the spatial variables exists (even for Linear Partial Differential Equations - PDEs), but for the time variable the story is totally different, is like there is a "fuzzy zone" trying to define a compact support in time since commonly time is a parametrization in $\mathbb{R}^{n+1}$ dimensions, or is considered as other variable in a space-time configuration, and so far I didn´t found yet any example of a finite-duration-solution to differential equations, not even in the scalar versions.
The closest functions I have found to a Finite-Duration solutions are bump functions $\in C_c^\infty$, which have  starts and endings at zero in a closed finite interval in the real line, but unfortunately I have only found one example that fulfill a Delay Differential Equation DDEs $\varphi'(t)=2\varphi(2t+1)-2\varphi(2t-1)$ so I don´t know how to apply the analysis if the cited paper (and also it don't have any known close form), and for other common bump-functions like $e^{-\frac{x^2}{(1-x^2)}},\,|x|\leq 1$ (or its extensions as non-piecewise definitions as are done here, $e^{-\frac{x^2}{(1-x^2)}}(1-x^2+|1-x^2|)$ as example, but the used "trick" introduce issues with the smoothness so maybe they lose their class $C_c^\infty$, and also introduce points where they are not well defined), I have only found non-autonomous nonlinear ODEs, which also don´t fit in the analysis of the paper (thinking here, in using an starting time different from where the bump function starts rising from zero, like $t=0$ at a point in the middle of the "bump").
Hope you can share examples of Finite-Duration Solutions with the autonomous ODE it fulfill (not piecewise sections of ordinary functions like the multiplication by a rectangular function), since they would help me to understand in a more tangible way how they behave.

Motivation/Discussion
Why are so few info on internet about Finite-Duration Solutions? Maybe they are just "too hard" to been researched about? Since I am not a mathematician, I would like to know if I am loosing my time if they are "known" because of being too hard to be solved under any approach.
This is an unknown topic?
Actually Wikipedia page for Autonomous Systems don’t talks about the existence of finite-duration solutions in any part (indeed, I recently added myself).
This is specially suspicious, since the description of the second order equations used on the cited paper is quite general, meaning that Uniqueness shouldn't be standing on a wide-spread kind of nonlinear ODEs. Here I would like to note that even if the analysis of the paper have some issues (there are many parts I don't even understand), all the previous and following characterization of these finite duration solutions still stands.
Note that being Autonomous imply that the system is time-invariant, so its law doesn't change with time.
Or Mathematicians/Physicists found it uninteresting?
At least I believe they are of huge importance: in classic mechanics systems, like a pendulum, they stop moving, so their solutions to the equations that describes their dynamics should be of Finite-Duration (so non Unique and also non-analytical), but commonly, for simplicity, friction is not considered (leading to solutions that vanishes at infinity), or they are studied numerically through their phase-space diagrams, where one can check that efectivelly the path dies at some time where $(x',\,x)=(0,\,0)$ (and also more smart and advanced approximations like Perturbation Theory, among others).
If these Finite Duration solutions are as restricted as “continuous and compact-supported functions”, which are always bounded, maybe other similar restrictions will rise for them, and I hope they could give more intuitive explanations to things as the "Least action principle" or the "2nd Law of Thermodynamics", like as a long-shot example, the "Ultraviolet Catastrophe" maybe explained by the Riemann-Lebesgue Lemma, just to give a few ideas...
From now on I will become more speculative, but as example, I have this question where "informally", I think could be argued that for second order systems like which are explained on the cited paper, actually their maximum rate of change will be always bounded, even when these solutions have an unlimited bandwith because they have finite extension in the time variable.
Other line of thought of why they are interesting, is thinking about the simpler form of Schrödinger Equation which is LINEAR (remember first paragraph), so I believe than studying finite-duration solutions could lead to deeply questions at least on physics: there is a huge mathematical difference between phenomena that last forever (like photons) which equations are linear (at least the basic ones, as also is the Maxwell's wave equation), and classic mechanics elements like a pendulum, where at least the exact solution to the nonlinear damped scenario is still unknown, and maybe, because is a solution of these not-so-known finite-duration kind... in this line, as example, it is interesting to see how a bump function as $e^{-\frac{x^2}{(1-x^2)}},\,|x|\leq 1$ behaves near $x=1$ Wolfram-Alpha, where the aggressive exponential behavior at the end (shared by all the bump functions I know so far), makes a lot of sense thinking on the "small-angle approximation" solution of the pendulum, wide known at least in engineering.
I hope you find them as interesting as I do.

My attempts so far
On the mentioned paper, is given as first order ODE example an equation of the form:
$$ \dot{x} = -\text{sgn}(x)\sqrt{|x|},\,\,x(0)=1 \tag{A. 1}$$
where Wolfram-Alpha fails to find a solution.
But for example, for the different differential equation:
$$ \dot{y} = -\sqrt{y},\,\,y(0)=1 \tag{A. 2}$$
where Wolfram-Alpha is able to find the solution:
$$ y(t) = \frac{1}{4}\left(t-2\right)^2 \tag{A. 3}$$
Using this last solution as a reference, I believe that a finite-duration solution to eq. A. 1 is:
$$ x(t) = \frac{1}{4}\left(\left|1+\frac{t}{2}\right|+\left|1-\frac{t}{2}\right|-|t|\right)^2 \tag{A. 4}$$
where it can be seen in Wolfram-Alpha that at least for positive time $t>0$ the solutions is indeed solving the equation.I don´t have enough background to formally prove that it is indeed the finite-duration solution, so if you can, I hope you can help me to prove it.
From its plot it can be seen that after time $t=2$ the solution is indeed becoming zero forever, so the triangular function trick is working $x(t) \cong \Lambda^2\left(\frac{t}{2}\right)$, but it also rises the issue on the definition of the differential equation because of its solution being zero forever after the ending time: think of eq. A. 1 in the form $ \frac{x'}{\text{sgn}(x)\sqrt{|x|}} = -1$, at first sight there is no issues, but since it admits solutions of finite-duration, somehow after its final time this problem could be ignored (I hope, please refer to the paper for formality, since I don´t have enough background to explain why it still can be sustained the equation even if their exists a division by zero).
But I still been interesting that the "trick" could be used to find the finite-duration solution (if I am right), since it gives a solution that is not a section of a bump function, as I proposed before and explain here.

Later I have figure out the following: given that the previous result only works for $t>0$, I have tried another alternative solutions:
$$ x(t) = \frac{1}{4}\left(\left|1+\frac{t}{2}\right|+\left|1-\frac{t}{2}\right|-|t|\right)^2\,\theta(t) \tag{A. 5}$$
with $\theta(t)$ the standard unitary step function, which is different from zero only on $t\in (0,\,2)$ (plot here), but this time trying to verify if it is solving eq. A.1 becomes a big mess on Wolfram-Alpha, mainly because now there is a discontinuity at $t=0$ where the eq. A.1 is not being matched:

*

*$\frac{d}{dt}\left(\frac{1}{4}\left(\left|1+\frac{t}{2}\right|+\left|1-\frac{t}{2}\right|-|t|\right)^2\,\theta(t)\right)=\begin{cases} \frac{t-2}{2},\, 0<t<2 \\ 0,\,t<0\vee t\geq 2 \\ \text{indeterminate},\,\text{otherwise} \end{cases}$ (see Wolfram-Alpha).

*$\text{sgn}(x)\sqrt{|x|} = \begin{cases} 1-\frac{t}{2},\, 0\leq t \leq 2 \\ 0,\, \text{otherwise}\end{cases}$ (see Wolfram-Alpha).

But, from another happy accident, I found that the following solution indeed solves eq. A.1 in the whole real line:
$$x(t) = \frac{1}{4}\left(1-\frac{t}{2}+\left|1-\frac{t}{2}\right|\right)^2\tag{A. 6}$$
which plot can be seen here and it is, at least numerically, fulfilling being a solution of eq. A.1 Wolfram-Alpha, but again, I cannot formally prove it, so any help or comment will be usefull, but at least following what it is said on Wikipedia page for ODEs about Local Existence and Uniqueness, I believe the results are right.
But as the mentioned paper states uniqueness of solution is not granted since equation A. 1 is not a Lipschitz ODE, so I am also wondering if equations A. 4 and A. 6 are actually two different solutions to the Initial Value Problem (IVP) of eq. A. 1: I am not fully aware if a solution of IVP must matched only from time $t_0$ onward or also from the real line that comes before $t_0$, but if is only required onward, non uniqueness could be delivering an extra degree of freedom or flexibility to the solutions, since they could behave as they want after matching the solution between $t_0$ and $t_F$, and I have another question here where I am asking about this, as another example of an IVP, $y(t) = \sqrt{e^{\frac{1-t}{1+t}}-1}$ lives only on the reals in a close interval, and I don´t know if finite duration solution could become complex after the initial time $t_0$.
 A: An illustrative example of a system with a finite-duration solution would be Norton’s dome: If you roll a ball with the right speed up this dome, it will come to rest at the top after a finite time (without friction). At this point, the behaviour becomes unclear: The ball staying at the top or rolling down in any direction are admissible solutions in classical mechanics.
However, Norton’s dome is only a thought experiment describing a “point-singular” scenario. Nothing remotely peculiar happens once you change any parameter ever so slightly (under- or overshooting the ball, not aiming straight for the dome’s top, slightly changing the dome’s shape). Thus there are no relevant real parallels. And that’s not even considering friction.
Mind that this different from many other pathologic examples or singularities. While those are usually not reached exactly in reality either or mechanisms beyond a model kick in to prevent them, excessive behaviour already happens in their vicinity. For example if a continuous model is clearly inconsistent at a specific point, we can usually conclude that it will behave unrealistically in the vicinity.
So, unless somebody surprisingly finds a real application of such ODEs, I am not surprised that there is little interest in them. The original paper you cite comes from robotics, in which one might design a system to exhibit such a behaviour. But then even such a system would only approximate the behaviour and one can obtain the desired properties in other ways and without dissecting a mathematical oddity.
A: So far, following the example given in this answer, it looks like, at least, if a scalar differential equation initial value problem has a Non-Lipschitz point $T>0$ where happens to be true that $x(T)=\dot{x}(T)=0$ and the differential equation stands the trivial zero solution, then is possible to make finite-duration solutions by "stitching" an nontrivial solution for $t<T$ with the zero function for $T\geq t$ (this was noticed in this answer to another question I did).
After asking these questions here and here, I believe that the family of Non-Lipschitz ordinary differential equations for integer $n>1$:
$$\dot{x} = -\sqrt[n]{x},\,x(0)>0,\,T>0$$
by integrating $\int \frac{dx}{\sqrt[n]{x}} = \frac{n}{n-1}x^{\frac{n-1}{n}}$ and considering the integration constant as an ending time $T$, this differential equations could stand the finite-duration solutions:
$$x(t) = \left[\frac{n-1}{n}\left(T-t\right)\right]^{\frac{n}{n-1}}\theta(T-t)$$
with $\theta(t)$ the Heaviside's standard unitary step function.
Even so, I am not completely sure but following the style of solutions presented in this paper, I believe the same solutions $x(t)$ will be solutions also for the differential equation:
$$\dot{x} = -\text{sgn}(x)\sqrt[n]{x},\,x(0)>0,\,T>0$$
Here I believe there is caution since when $\frac{n}{n-1}$ is even, the order $(T-t)$ or $(t-T)$ doesn't matter on the solutions, but it will be important when taking the derivative so it could indicate that the same solutions answer $\dot{x}=\sqrt[n]{x}$ when it is false.
Also, it could be useful to note that:
$$\begin{array}{r c l}
x(t) & = & \left[\frac{n-1}{n}\left(T-t\right)\right]^{\frac{n}{n-1}}\theta(T-t) \\
& \equiv & \left[\frac{n-1}{n}\left(T-t\right)\theta(T-t)\right]^{\frac{n}{n-1}} \\
& \equiv & \left[\frac{n-1}{n}T\left(1-\frac{t}{T}\right)\theta(1-\frac{t}{T})\right]^{\frac{n}{n-1}} \\
& = & x(0)\left[\left(1-\frac{t}{T}\right)\theta(1-\frac{t}{T})\right]^{\frac{n}{n-1}} \\
& \equiv & x(0) \left[ \frac{1}{2} \left( 1-\frac{t}{T} + \left| 1-\frac{t}{T} \right| \right)\right]^{\frac{n}{n-1}} \\
& = & \left[ \frac{(n-1)}{2n}T \left( 1-\frac{t}{T} + \left| 1-\frac{t}{T} \right| \right)\right]^{\frac{n}{n-1}} \\
\end{array}$$

added later
Looks like it still works for any real-valued $n>1$ not needing it to be integer.
