Solving $x'=-\text{sgn}(x)\sqrt{|x|}$: Uniqueness of solutions of finite duration

Question

Show that $$x(t) = \frac{\text{sgn}(x(0))}{4}\left(2\sqrt{|x(0)|}-t\right)^2\cdot\theta\!\left(2\sqrt{|x(0)|}-t\right)$$ is a solution to$$x'=-\text{sgn}(x)\sqrt{|x|}.$$ Is the solution unique?

This question was modified since I was able to find the answer to the first question.

Here, $\theta(t)$ is the Heaviside step function, and $\text{sgn}(x)$ is the Sign function.

This is a continuation of this question. By using the answer given by @KBS, I could find solutions with the same "structure" as the mentioned answer if $x(0)>0$, but for initial conditions $x(0)<0$, I struggle because the term $(2\sqrt{x(0)}-t)^2$ is, first, insensible to sign changes, and secondly, becomes complex, so I added an absolute value function to the initial condition, and later just pasted a term $\text{sgn}(x(0))$ in order to "make" the same solutions I find in the "slope-field" of wolfram-Alpha for negative initial values, leading to:

$$x(t) = \frac{\text{sgn}(x(0))}{4}\left(2\sqrt{|x(0)|}-t\right)^2\cdot\theta\!\left(2\sqrt{|x(0)|}-t\right)$$ which fulfill as I require:

1. $$\lim\limits_{t \to 0} x(t) \equiv x(0)$$ for any $x(0)\in \mathbb{R}$ (there is a discontinuity when arriving to $x(0^+)$ and $x(0^-)$ but since $x(0^+)\equiv x(0^-)\equiv 0$ I think there is no issues at all: checked here and then here.
2. It have a finite extinction time $T = 2\sqrt{|x(0)|}\geq 0$ where the system stop moving at zero $x(t) = 0,\ \forall t\geq T$.

But when I insert the function $x(t)$ into the equation: $$x'+\text{sgn}(x)\sqrt{|x|} = 0$$ I cannot make the terms to become zero: from the differentiation side I got some Dirac delta functions as shown here, and from the nonlinear term I cannot match the exponent in the sign functions outside and within the square root as shown here, so I don't know if the solution that we have found is indeed a solution to the differential equation "formally" speaking (at every point and real-valued initial condition).

• Is the $x(t)$ that we obtained indeed a formal solution to $x'+\text{sgn}(x)\sqrt{|x|} = 0$? (SOLVED)
• Is this solution unique? at least in the domain $t \in [0,\ T)$

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Added Later

Later I remembered this answer by @md2perpe where show that since $x\delta(x)=0$ in this situation where solution $x(t)=f(t)\theta(t)$ is such as:

1. $(f\theta)'=f'\theta+f\theta'=f'\theta+f\delta =f'\theta$ since $f(t)$ is a polynomial with the same argument as the Dirac's delta function $\delta(t)$, it happens to be $f\delta \equiv (c-t)^n\delta(c-t)=0$.
2. its equivalent $\text{sgn}(f\theta)\sqrt{|f\theta|} = \text{sgn}(f)\sqrt{|f|}\,\theta$

Then I do could match the left-hand-side $f'\theta$ with the right-hand-side $\text{sgn}(f)\sqrt{|f|}\,\theta$ at least while $t<T$, meaning that $x(t)$ is indeed a solution of finite duration (this, since the trivial solution $x(t)=0$ satisfies the differential equation after $t\geq T$). You could see it's numerical verification on Desmos.

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Update 2

@CalvinKhor have sugest me this comment by @ThomasRichard in MathOverflow suggesting a real life physical model that fits the differential equation: "It might be useful to mention that the ODE $y′=−\sqrt{y^{+}}$, for which non uniqueness holds, actually models a real physical system. $y(t)$ represents the water height in a pierced cylindrical bucket. And the non uniqueness of the Cauchy problem with initial condition $y(0)=0$ is natural in this model: if you know the bucket is empty at time $0$, it is hard to tell if there was water in it before and when it got empty."

And later, @CalvinKhor also shared the explanation in this other comment, which I like to share since I found it really interesting and suits here:

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Motivation (added due requirement by @RodrigodeAzevedo in the comments)

About two years ago I accidentally found these papers about V. T. Haimo: Finite Time Differential Equations and Finite Time Controllers, and realized from them that no analytic function, so no non-piecewised defined power series, neither no linear ODE, could accurately represent a dynamic system that stop moving from its own, and as engineer I felt cheated since everything I was taught could become a power series, and daily life phenomena do stops at finite times. So I started to make myself the easier examples I can and try to solve them in close form, since very few info there is about these solutions of finite duration, at least grouped as one specific topic, and the mentioned papers don't show closed form solutions neither (this question example is a modification of an example on the first paper).

Answer 1

IVPs for $x'(t)= -{\rm sgn}(x(t))\sqrt{|x(t)|}$ are uniquely solvable to the right, since $x \mapsto -{\rm sgn}(x)\sqrt{|x|}$ is monotone decreasing on $\mathbb{R}$. More general, let $f:\mathbb{R} \to\mathbb{R}$ be monotone decreasing, and let $x_1,x_2:[0,T) \to \mathbb{R}$ be solutions of $x'(t)=f(x(t))$, $x(0)=x_0$. Set $d(t):= (x_1(t)-x_2(t))^2$ $(t \in [0,T))$. Then $$ d'(t) = 2(x_1(t)-x_2(t))(f(x_1(t))- f(x_2(t))) \le 0 \quad (t \in [0,T)). $$ Hence $d$ is decreasing and $\ge 0$, and $d(0)=0$. Thus $d(t)=0$, that is $x_1(t)=x_2(t)$ $(t \in [0,T))$.