= Finite-time blow-up of an ordinary differential equation
A solution of an <ordinary differential equation> has \b[finite-time blow-up] when it becomes unbounded on approach to a finite endpoint of its maximal interval of existence. For $\dot x=x^2$ and $x(0)=x_0>0$,
$$
x(t)=\frac{x_0}{1-x_0t},\qquad 0\leq t<1/x_0.
$$
The solution is smooth before $t=1/x_0$ but cannot be continued through that <time> as a finite real-valued solution. Smoothness of the <vector field> therefore does not by itself guarantee a solution for all future times.
This matters when using a <flow map> or <forward orbit>: their <time> domains must follow the solution's actual existence interval. In a physical model, blow-up can indicate that a neglected process becomes important before the formal singularity. That interpretation requires examining the model's assumptions, rather than identifying an infinite mathematical prediction with an observed infinite physical quantity.
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