Is there any example of PDE whose projection of 2 distinct characteristics curve intersect?

Question

I know that for any PDE if we consider its any 2 distinct characteristics they never intersect. I also know that for semilinear pde projection of 2 distinct characteristics they never intersect.(I do not know how to prove this fact. Just read in lecture notes. Please if Possible give me reference of proof).

Is there is an example of PDE such that projection of 2 distinct characteristics both intersect?

Any Help will be appreciated

Answer 1

Consider the inviscid Burgers' equation: $$u_t+u u_x=0$$ with the initial condition $u_0 = 1, x<0$ and $u_0=0, x>0$. To the left of $0$, characteristics move with speed $1$ and to the right of $0$, characteristics move with speed $0$, so they will eventually inetersect