Entropy solution - Burgers' equation Why doesn't the following problem have a solution for $t\ge1$? 
$u_{t}+uu_{x}=0\\
u(0,x)=-x$.
The characteristics don't intersect and they cover the whole space above t=1. 

All characteristics intersect at $(t,x) = (1, 0)$.
Indeed, the characteristic starting from $(0, x_0)$ is $x(t) = x_0 - x_0 \, t = (1-t) x_0$.
