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.
2026-03-27 09:37:57.1774604277
Entropy solution - Burgers' equation
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All characteristics intersect at $(t,x) = (1, 0)$.
Indeed, the characteristic starting from $(0, x_0)$ is $x(t) = x_0 - x_0 \, t = (1-t) x_0$.