ODE flow of the Chafee-Infante problem

I'm currently listening to a lecture on dynamical systems. Unfortunately I am lacking some of the requirements for that course and thus ran into some problems with the latest problem set.

Show that the ODE flow $$ \frac{d^2}{dx^2}v + f(v) = 0 $$ for the Chafee-Infante nonlinearity $f(v)=v(1-v^2)$ is not global. [...]

So far I thought the ODE flow is a function mapping a point $p$ in the phase-space to the point $\Phi^t(p)$ where the solution starting at $p$ will be after time $t$. Is this not correct? Because the problem as stated above even has a well defined potential and will easily give you full solutions $v(x)$, $x\in(-\infty,\infty)$ for any starting point.

Or does my misconception lie with the word "global"? If $\Phi$ is defined for all points and all times, what else could "global" ask for?

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2022-07-25 20:46:03
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