[@Da]). Precisely we are going to show

\[properties\_of\_entropy\_solutions\] Let $v_1\in L^{\infty}(\mathbb{R}\times(0,T))$ be a weak solution of (\[scl\]). Then there exists a constant $\lambda_0>0$, depending on $f$ and $\|v_1\|_{\infty}$, such that, for any domain $\Gamma_{t_1}^{t_2}$ satisfying $0<\hat{\lambda}\leq \lambda_0$, the entropy solution $v\in L^{\infty}(\Gamma_{t_{1}}^{t_{2}})$ of (\[modified\_scl\]) with boundary condition $v_{1}\in L^{\infty}(\Lambda_{t_{1}}^{t_{2}})$ satisfies $$\label{boundary_asymptotic}
\lim_{\varepsilon\to0^+}\int_{s_1}^{s_2-\varepsilon}\left|v(s,\gamma(s,t_1+\varepsilon))-v_1(s,\gamma(s,t_1))\right|\,ds=0\,,$$ where $$s_1=-\frac{t_2-t_1}{\hat{\lambda}}-\delta\quad\mbox{and}\quad s_2=\frac{t_2-t_1}{\hat{\lambda}}+\delta\,.$$ Moreover $$\label{Linfty_bound}
             \|v\|_{\infty}\leq \|v_{1}\|_{\infty}\,$$ and there exists a constant $C>0$, depending only on $\|v\|_1$ and $\hat{\lambda}$, such that $$\label{control_of_the_defect_measure}
              \int_{\Gamma_{t_{1}}^{t_{2}}}q(x,t,a)\, da\, dx\, dt\leq C(t_{2}-t_{1})\,.$$ Let now $w_1,\,w_2\in L^{\infty}(\mathbb{R}\times(0,T))$ be weak solutions of (\[scl\]). Then there there exists a constant $\lambda_1>0$ depending on $f$ and $\max\{\|w_1\|_{\infty},\|w_2\|_{\infty}\}$ such that, for any domain $\Gamma_{t_1}^{t_2}$ satisfying $0<\hat{\lambda}\leq \lambda_1$ and any choice of two entropy solutions respectively $v_1\in L^{\infty}(\Gamma_{t_{1}}^{t_{2}})$ with boundary condition $w_{1}\in L^{\infty}(\Lambda_{t_{1}}^{t_{2}})$ and $v_2\in L^{\infty}(\Gamma_{t_{1}}^{t_{2}})$ with boundary condition $w_2\in L^{\infty}(\Lambda_{t_{1}}^{t_{2}})$ the following holds : for any $t\in(t_1,t_2)$ and a constant $C>0$ depending on $\Gamma_{t_1}^{t_2}$ and $\max\{\|w_1\|_{\infty},\|w_2\|_{\infty}\}$: $$\label{contraction_inequality}
 \int_{\theta^{-}(t)}^{\theta^{+}(t)}|v_1(x,t)-v_2(x,t)|\, dx\leq C\int_{\Lambda_{t_{1}}^{t_{2}}}|w_1-w_2|\, d\sigma\,,$$ where $$\theta^{\pm}(t)=\pm\frac{t-t_1}{\hat{\lambda}}\pm\delta\,.$$

Inequality (\[contraction\_inequality\]) implies in particular the uniqueness of the entropy solution for a given initial data $w$ on $\Lambda_{t_1}^{t_2}$ issued from a weak solution to (\[scl\]).

We start to prove (\[boundary\_asymptotic\]). Let $R>0$ such that $$R+f(\pm R)\geq0\,.$$ We choose $\lambda_0$ such that $$\label{def_lambda0}
\lambda_0^{-1}=\max\left\{\left|f'(R+1+\|v_1\|_{\infty})\right|,\left|f'(-R-1-\|v_1\|_{\infty})\right|\right\}\,.$$ We 