Consider the following nonlinear system: $\dot{x}_1=x_1^3x_2^2-\frac{x_1^7}{1+x_1^8}$, $\dot{x}_2=-kx_2$, where $k>0$.
In domain $D=\{(x_1,x_2)|\|x_1\|\leq 1, \|x_2\|\leq X\}$, I guess that the interval of the existence for this nonlinear system is not $[0,\infty)$, but I am not sure. Please help me to prove if this claim is true. Also, please help me to imply if this system demonstrates fnite-time escape of trajectories to infinity.