Growth assumption and example of finite (arbitrarily small) time blow up for ODE

Question

Consider the following ODE initial value problem \begin{align*} &\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\ &\Phi(0,x) = x, & x \in \mathbb{R}^N. \end{align*}

We say that $\Phi: [0,T] \times \mathbb{R}^N \to \mathbb{R}^N$ is the flow of the ODE (as in this paper) if it solves it in some sense.

We assume that the vector field $\boldsymbol{F}:[0,T]\times \mathbb{R}^N \to \mathbb{R}^N$ is Sobolev and such that that $$(*) \qquad \frac{|\boldsymbol{F}|}{1+|x|} \in L^1\left([0,T]; L^1(\mathbb{R}^N) \right) + L^1\left([0,T]; L^\infty(\mathbb{R}^N) \right),$$ that is, there exist \begin{align*} &\boldsymbol{F}_1 \in L^1\left([0,T]; L^1(\mathbb{R}^N) \right)\\ &\boldsymbol{F}_2 \in L^1\left([0,T]; L^\infty(\mathbb{R}^N) \right) \end{align*} such that $$\frac{\boldsymbol{F}}{1+|x|} = \boldsymbol{F}_1 + \boldsymbol{F}_2.$$

In an answer to Quantitative finite speed of propagation property for ODE (cone of dependence), it has been remarked that the flow $\Phi$ can blow up in finite (and arbitrarily small) time if the $F_1\neq 0$.

1. Can you provide an example of such flow that blows up in finite (and arbitrarily small) time?

2. Why is this not in contrast with the fact that assumption (*) is used in the existence and uniqueness result of Theorem 30 (page 23) of this paper?

3. In the theorem cited in the previous point, is assumption (*) key for existence or uniqueness?

Answer 1

The definition of $\Phi$ you are considering is different from the one in the paper, which is Definition 13:

Given a certain class $\mathcal L_b$ of measure-valued solutions of the continuity equation (see the paper for the details), $\Phi(t,x)$ is a $\mathcal L_b$-lagrangian flow of $b$ starting from a measure $\bar \mu$ at time 0 if

(a) for $\bar\mu$-a.e. $x$ the function $\Phi(\cdot, x)$ is absolutely continuous on $[0,T]$ and solves $\Phi(t,x) = x + \int_0^t F(s,\Phi(s,x)) \,ds$ for all $t\in[0,T]$.

(b) the image $\mu_t := \Phi(t,\cdot)_\sharp \bar \mu$ of $\bar\mu$ under the map $\Phi(t,\cdot)$ belongs to the class $\mathcal L_b$.

Probably this is the reason why the suggested example leads to confusion. More precisely, let me answer the questions:

1. Consider for $N=1$ a small modification of the classical blow-up-in-finite-time example $\dot x = x^2$: take $F(t,x) = x^2$ for $x$ in some small neighbourhood of $(1-t)^{-1}$ and set $F(t,x)=0$ for the other $x$ (so that $\int \frac{|F(t,x)|}{1+|x|} \, dx < 1$). Then $\Phi(t,1) = (1-t)^{-1}$ escapes to infinity in time 1.

2. It is not in contrast with Theorem 30 of the paper because their definition of $\Phi$ is different (see above). The point is that $\Phi(t,\cdot)$ satisfies the ODE for a.e. $x$, not for all $x$.

3. In Theorem 30 the assumption (*) is needed both for existence and uniqueness. This is evident since the proof of Theorem 30 relies on the proofs of Theorems 16 and 19, both of which are based on the Comparison Principle (Theorem 26), where the assumption (*) is used.