Consider with the IC .
Show that a shock develops at a finite time and describe the weak solution.
The chracteristic line is .
To see this, note that the characteristics are so and .
Examine the middle of the region.
Now we know
So is the shock and the characteristics intersect at . Next we'd like to define (in this case) so that the pde is in the form .
The jump condition taken from the far left side to the far right side is:
Plug in the shock to get .
The weak solution is .