= Solution
Put $h=|u-v|$, $q=Q(u,v)$ and $h_0=|u_0-v_0|$. The <mean value theorem> gives $|q|\le Mh$ on the bounded state range. Fix a final time $t$ and let $L(\tau)=a-M(t-\tau)$ and $R(\tau)=b+M(t-\tau)$. Choose a smooth increasing function $\rho$ equal to zero on $(-\infty,0]$ and one on $[1,\infty)$. For small $\delta>0$, define
$$
W_\delta(\tau,x)=\rho\left(\frac{x-L(\tau)}\delta\right)\rho\left(\frac{R(\tau)-x}\delta\right).
$$
Its two nonnegative <derivative> contributions imply $(W_\delta)_\tau+M|(W_\delta)_x|\le0$. Therefore $h(W_\delta)_\tau+q(W_\delta)_x\le0$.
Use the nonnegative test $\varphi(\tau,x)=\eta_\epsilon(\tau)W_\delta(\tau,x)$, where $\eta_\epsilon(0)=1$, it is one up to just before $s$, decreases smoothly to zero near $s$, and remains zero afterward. The <Kato inequality for scalar conservation laws> gives
$$
-\int\eta_\epsilon'(\tau)\!\int hW_\delta\,dx\,d\tau\le\int h_0(x)W_\delta(0,x)\,dx,
$$
because the other interior contribution is nonpositive. At Lebesgue times $s$, the temporal <approximate identity> tends to $\int h(s)W_\delta(s)$; take a countable sequence $\delta\downarrow0$ and use dominated convergence on the bounded cone. This yields
$$
\boxed{\int_{a-M(t-s)}^{b+M(t-s)}|u(s,x)-v(s,x)|dx\le\int_{a-Mt}^{b+Mt}|u_0(x)-v_0(x)|dx}
$$
for almost every $s\in[0,t]$. This is <local L1 contraction for scalar conservation laws>. It uses only local integrability, so the bounded data need not have finite global $L^1$ <norm>. If $M=0$, the flux difference is zero and the same proof uses a stationary interval.
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