Hi all.

I am having trouble with the following problem:

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

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

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


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