Hi all.

I am having trouble with the following problem:

Given the PDE: U_t + f(u)u_x = 0,
where f is a differentiable function,
and the IVP: u(x,0) = h(x), h another function.

(a) Show that u satisfies the implicit equation: u = h(x - f(u)t).

(b) Show that the solution becomes singular at some positive time t, unless f(h(s)) is a non-decreasing function of s.


I can derive the solution in (a), but (b) has me totally perplexed. Thanks in advance!