Consider the nonlinear transport equation

u_t + c(u)u_x = 0; xR; t > 0

with initial condition u(0; x) = f(x).

1. Show that if c and f are both increasing or both decreasing, then there is no shock.

2. Using c(u) = u and

f(x)=1,x<01+x,0<x<12,x>1

find the solution. Draw the characteristics.