Consider the following hyperbolic equation for $u=u(t,x)$, $a\in\mathbb{R}$
$$ \frac{\partial u }{\partial t} + a\frac{\partial u }{\partial x} -u^2 = 1. $$
I want to show that there is no global solution, that is, there exists a $t_*$ such that the solution $u$ goes to infinity as $t\rightarrow t_*$. In order to do that I am thinking in showing that any two characteristic cruves intersec each other. My problem is to calculate the characteristic curves of the considered equation. Any help or idea is welcome!
Thank you.