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 $$\int_{t_0}^{t_{max}}|x(t)|^2dt=\infty,\quad\text{when }t_{max}<\infty?$$

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:

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.