IVP of quasi-linear transport equation
The proof is fine, since the statement "the slope of the chords of $a\circ h$ is non-negative" implies "$a\circ h$ is non-decreasing".
Now, what happens if $a\circ h$ is decreasing somewhere? Using the mean value theorem over $[s_1,s_2]$, we rewrite the intersection $(1)$ in OP as $$ y = -\frac{s_2-s_1}{a\circ h(s_2)-a\circ h(s_1)} = -\frac{1}{(a\circ h)'(s)} $$ for some $s \in [s_1,s_2]$. Now, we obtain that the smallest $y>0$ at which characteristics intersect is given by $$ y_b = \inf_{s\in\Bbb R} \left( -\frac{1}{(a\circ h)'(s)}\right) = \frac{-1}{\inf_{s\in \Bbb R} (a\circ h)'(s)}\, . $$ This is the $y$ where a shock wave occurs (breaking time).
There are several other proofs than the previous geometric argument (cf. e.g. (1)), which is based on the intersection of characteristics and the mean value theorem. Let us summarize a few methods:
- Dependence to initial data. We differentiate the expression of characteristics $x = x_0 + a\circ h(x_0)\, y$ with respect to the initial abscissa $x_0$. This derivative vanishes at $y = -1/(a\circ h)'(x_0)$. The smallest such $y$ is the breaking time.
- Characteristic evolution of $u_x$. We differentiate the PDE with respect to $x$, and we set $q=u_x$, $q' = q_y + a(u) q_x$. Hence, $q' + a'(u)\, q^2 = 0$, which solution is $q(y) = q_0/(1+q_0 k y)$, where $q_0 = h'(x_0)$ and $k = a'(u) = a'\circ h(x_0)$ is constant along the characteristics. Finally, $q$ blows up at $y = -1/(q_0 k) = -1/(a\circ h)'(x_0)$.
- Implicit function theorem. The implicit equation $u = h(x-a(u) y)$ can be solved for $u$ as a differentiable function of $x$ and $y$ for $y$ small enough. Differentiating w.r.t. $x$, one has $u_x = q_0/(1+q_0 k y)$ with the same notations as above.
(1) P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves. SIAM, 1973. doi:10.1137/1.9781611970562.ch1