PDEs – Finite-Duration Solutions in Lipschitz-Kind Non-Scalar ODEs and PDEs

boundary value problemdynamical systemsfinite-durationinitial-value-problemspartial differential equations

Does Lipschitz-kind "non-scalar ODEs" and PDEs could stand having finite-duration solutions?

Intro

Recently I have found on these papers by Vardia T. Haimo (1985) Finite Time Controllers and Finite Time Differential Equations that there exists solutions of finite duration to differential equations, defining them here as:

Definition 1 – Solutions of finite-duration: the solution $y(t)$ becomes exactly zero at a finite time $T<\infty$ by its own dynamics and stays there forever after $(t\geq T\Rightarrow y(t)=0)$. 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. (Here I just pick the shorter name its look more natural to me for several I found: "finite-time", "finite-time-convergence", "finite-duration", "time-limited", "compact-supported time", "finite ending time", "singular solutions", "finite extinction time", among others).

The mentioned papers refers only to scalar autonomous ODEs of 1st and 2nd order, assuming that $T=0$ and that the right-hand-side of the ODE is at least class $C^1(\mathbb{R}\setminus\{0\})$, and with this it is mentioned that:

"One notices immediately that finite time differential equations cannot be Lipschitz at the origin. As all solutions reach zero in finite time, there is non-uniqueness of solutions through zero in backwards time. This, of course, violates the uniqueness condition for solutions of Lipschitz differential equations."

So, NO Lipschitz 1st or 2nd order scalar autonomous ODEs could stand solutions of finite duration, so classic linear models are discarded. Also are discarded classical solutions through Power Series which are analytical in the whole domain, since there is a non-zero measure compact set of points identical to zero, the only analytical function that could support this is the zero function, due the Identity Theorem.

This discard every example of solutions I saw in engineering, but fortunately through these questions here, here, and here, I have been able with the help of other users to find some examples, where I would like to share both of them (here, $\theta(t)$ is the Heaviside step function):

  1. This that is similar to the examples of the mentioned papers, where unfortunately no exact solution is given: $$ \dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1\quad\rightarrow\quad x(t)=\frac{1}{4}(2-t)^2\theta(2-t) \equiv \frac{1}{4}\left(1-\frac{t}{2}+\left|1-\frac{t}{2}\right|\right)^2$$
  2. Remembering now that uniqueness of solutions is not granted, focusing in positive real-valued solutions for a positive real-valued ending time $0<T<\infty$, and assuming a parameter $n>1$, the following family of solutions could be obtained: $$\dot{x} = -\sqrt[n]{x},\,x(0)>0,\,T>0\quad\rightarrow\quad x(t) = \left[\frac{n-1}{n}\left(T-t\right)\right]^{\frac{n}{n-1}}\theta(T-t) \equiv x(0) \left[ \frac{1}{2} \left( 1-\frac{t}{T} + \left| 1-\frac{t}{T} \right| \right)\right]^{\frac{n}{n-1}}$$

As can be seen, are nothing exotic, just a polynomial piecewise stitch with the zero function but in just a way the whole composition solves the differential equation in the whole domain.

Main question

Since no 1st and 2nd order scalar autonomous Lipschitz ODE could have a solution with a finite ending time, but Non-Lipschitz alternatives could through solutions that aren't necessarily unachievable in close form, I would like to know if Lipzchitz non-scalar ODEs and PDEs could have these kind of finite duration solutions, or, if conversely, it is also required to them of having at least a Non-Lipschitz point in time (maybe the extra dimensions made the "non-Lipschitz characteristic" not required – is what I wont to figure out).

Here I give emphasis to the time variable, since I already know that PDEs could have compact-supported solutions on the space variables, as is explained on this answer to other question I did… before, with scalar ODEs, there is no ambiguity, but for PDEs now I explicitly focus the question to the time variable (so that explains the "finite-duration", and not just compact-supported as I did on the mentioned question).

It is also the issue about treating the time variable as in a spacetime scheme $\mathbb{R}^n$, or in a classic parametrization scheme $\mathbb{R}^{n+1}$ which is mentioned in the cited answer (here I become a bit lost, so I don't going to give any restriction to see what you could tell me about).

What I believe

After founding this video where is explained that some PDEs could show finite-time blow-ups behavior, I start to wonder if some of this singularities could be due are the reciprocal, or have in their denominator, something is behaving as having a finite duration solution: I made that question here founding that the family of solutions of point $(2)$ indeed have reciprocals that will show a differential equation that behave as having finite-time blow-ups, but also in the comments the user [@CalvinKhor] give an example of a system that shows a finite-time blow-up behavior which reciprocal doesn't lead to a differential equation that admits finite-duration solutions.

So far, I have made a question here about the Euler's Disk toy which I think is an example of a system that is having a finite duration solutions since it sounds ends in finite time (and some angles are showing a finite-time blow-up behavior), but their motion equations, at least on this paper (eqs. 23, 24, and 25) are framed as a system of nonlinear ODEs (where I believe are Lipschitz, but I am not 100% sure), so I believe these kind of system exists. But so far I did not find any example for PDEs.

Added later – some attempts

So far everything I have tried have ending to be something I don't really know if it well defined (I made a related question here), but they are surely reaching zero in finite time so their phase-space should not been fully covered. Hope you can comment if for a Lipschitz kind of differential equation it is a required to cover or not their whole phase-space, because I am not sure if having a finite duration solution will go against it.

2nd Added Later

On the mentioned question here, through the comments and answers I become confident that the differential equation require at least on Singular point in time where the equation is Non-Lipschitz so uniqueness of solutions could be broken, which is required for the solution to become zero after a finite ending time on a non-zero measure domain of points, which act like stitching a piecewise section of the trivial zero function to the previous values of the solution.

As the example, generalizing something I use on the mentioned question, I found that If I pick a function $P(x-t)$ as the solutions for the wave equation $P_{xx}=P_{tt}$, then it is I made a function:
$$U(x,t)=P(x-t)\cdot\left[\frac{(n-1)}{n}(T-t)\right]^{\frac{n}{(n-1)}}$$
then it will solve the PDE:

$$U_{xx}-U_{tt}-\frac{2n}{(n-1)}\frac{U_{t}}{(T-t)}-\frac{n(2n-1)}{(n-1)^2}\frac{U}{(T-t)^2}=0$$

Then, since (i) the trivial solution $U(x,t)=0$ solves the differential equation, and (ii) the equation is Non-Lipschitz a the point $t=T$ fulfilling that $U(x,T)=U_t(x,T)=0$ for $n>1$, also the finite duration solution:
$$U^*(x,t)=P(x-t)\cdot\left[\frac{(n-1)}{n}(T-t)\right]^{\frac{n}{(n-1)}}\theta(T-t)$$
will be solving the same differential equation for $U(x,t)$ on the whole time domain $t \in \mathbb{R}$, and also, note that uniqueness is not longer hold since both $U$ and $U^*$ are valid solutions, at least I tested correctly for $n=2$ and $P(x-t)=\exp\left(1-\frac{1}{(1-(x-t)^2)}\right)$, and also for $U^*(x,t)=\left(1-(x-t)^2+|1-(x-t)^2|\right)^4\cdot\left(1-t+|1-t|\right)^2$ (plot here).

But against the intuition developed on the mentioned question, on this another question about physics of finite duration phenomena, an user named @ConnorBehan tells that there exists PDEs named fast diffusion equations where finite extinction times are achieved, so looking for this new term I found, indeed, appeared bunch of PDEs where is claimed finite ending times are achieved, like $\frac{d^2}{dx^2}\left(\sqrt{u}\right)=u_t$, but what is important to notice there, is that in every paper is claimed that uniqueness of solutions is hold at the same time is achieving a finite extinction time, which contradicts what is done so far on the other question.

So now I am totally lost again, and the papers are too abstract for my current knowledge, but what I believe it could be happening are:

  1. The equation aren't Lipschitz and the claims of uniqueness are being done only within the domain where the solutions still don't achieve the extinction time?
  2. Somehow uniqueness and Lipschitz-ness is holds on PDEs at the same time they can become zero forever after a finite time? (somehow against Picard-Lindelöf theorem which require the equation to be Lipschitz to hold the uniqueness part of it)
  3. Or they are just taking a piecewise section of a solution that becomes zero on a point in time and assuming is zero outside a domain they don't care about? So don't fulfilling the definition of solution of finite duration I am using here, because their solution don't solves the PDE after the extinction time (maybe is useful on their specific contexts).
  4. Maybe is other the situation and I don't getting the full idea of the papers.

Here I will list some of the papers I found:

Since are papers that works with frameworks that are similar to relativity or quantum mechanics, if they are really finite duration solutions in the context of this question they will be solving also this other questions I made here and here, but the issue of being Lipschitz or not and holding uniqueness of solutions made me think maybe they are not fulfilling the definition at the beginning of this question.

Hope you can help me what is going on on this examples.

Best Answer

Let's recall precisely Picard Lindelöf theorem where I pick up here the definition used in Wikipedia:

Let $D\subseteq \mathbb {R} \times \mathbb {R} ^{n}$ be a closed rectangle with $(t_{0},y_{0})\in D$. Let $f:D\to \mathbb {R} ^{n}$ be a function that is continuous in $t$ and Lipschitz continuous in $y$. Then, there exists some $\varepsilon \gt 0$ such that the initial value problem - IVP

$$\begin{cases}y^\prime(t)=f(t,y(t))\\ y(t_0)=y_0\end{cases}$$

has a unique solution $y(t)$ on the interval $[t_{0}-\varepsilon ,t_{0}+\varepsilon ]$

So, Picard Lindelöf theorem deals with IVP, and not with "Lipschitz-kind "non-scalar ODEs"

Let's consider the following IVP example

$$\begin{cases}y^\prime(t)=2 \sqrt{\lvert y(t) \rvert}\\ y(0)=y_0\end{cases}$$

If you take $y_0 \neq 0$, the map $(t,y) \mapsto 2 \sqrt{\lvert y \rvert}$ is locally Lipschitz around $y_0$ in $y$ and the IVP has a unique solution that indeed has a finite extinction time.

This is an example similar to "Extinction time for some nonlinear heat equations" - Louis A. Assalé, Théodore K. Boni, Diabate Nabongo", paragraph 2 with $p=1/2$.

The apparent contradiction is solved by noting:

  • That at times $t_1$ where $y(t_1)=0$, the map $(t,y) \mapsto 2 \sqrt{\lvert y \rvert}$ is not locally Lipschitz around that point.
  • And indeed, at that point, the IVP has several solutions (an infinity in fact), that are the maps $$y_a(t) = \begin{cases} 0 & t < a \\ (t-a)^2 & t \ge a \end{cases} $$ where $a \gt t_1$ and the always vanishing map.
  • If the map $(t,y) \mapsto f(t,y)$ is locally Lipschitz in $y$ at all points of its domain, then finite extension time solutions won't appear.
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