ODEs whose finite-time solutions are not L^2 on their interval of definition Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be analytic and consider the ODE
$$x'(t)=f(x(t)).$$
It is well-known that if $(t_{min},t_{max})$ is the maximal domain of a solution $x$ and $t_{max}<\infty$, then
$$\lim_{t\to t_{max}}|x(t)|=\infty.$$
Let $t_0\in(t_{min},t_{max})$. What conditions on $f$ (appart from linearity) ensure that
$$\int_{t_0}^{t_{max}}|x(t)|^2dt=\infty,\quad\text{when }t_{max}<\infty?$$
 A: This  is  not  an  answer, but  is  a  comment.  (I  can not  give  comment  since  I  am under  50  reputation).
Linear  vector  fields  are  always  complete  vector  field  so they  do  not  satisfy your  condition.
But for  higher  order  polynomial  vector  field, I  guess that the  solutions  which are  not  a  complete  orbits, are  not  in $L ^2$.  My  motivation is  that according  to  an interesting  Paper  of  Chicone and  Sotomayor, the  solutions  escape  at  infinity  very  fast(exponentially)  since  there  is  a  hyperbolic  singularity  at  equator. 
On the  other  hand   your  question is  very  interesting  for  me  since it  implicitly   suggests  to  consider  some  different  function  spaces to  be  acted by $D_f$,  the  derivational  operator  associated  to  the  vector  field  $f$.
The  motivations  for  study of  this  derivational operator  is  explained in the  following  posts:
Does this function belong to $L^2(\mathbb{D})$?
Codimension of the range of certain linear operators
A: Consider the case $n=1$ (and, say, $f>0$, wlog), then $t_{max}<\infty$ iff $\int\frac{dx}{f(x)}<\infty$. Then $\int x(t)^2\ dt=\int\frac{x^2}{f(x)}\ dx$.
Since you didn't mention that simple case, I hope it may help.
