Finite-time blow-up of an ordinary differential equation

A solution of an ordinary differential equation has 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 $x(0)=x_0>0$,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.