function $\chi_{i}^{\varepsilon}-\left(\chi^{\varepsilon}_{i}\right)^{2}$ which satisfies satisfies pointwise for $(x,t)\in\Gamma_{t_1+C\varepsilon}^{t_2-C\varepsilon}$ and almost every $a\in\mathbb{R}$ $$\partial_{t}\left[\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^{2}\right]+f'(a)\partial_{x}\left[\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^{2}\right]=\partial_{a}q_{i}^{\varepsilon}+2\chi_{i}^{\varepsilon}\partial_{a}q_{i}^{\varepsilon}\,.$$ Integrating this in $\Gamma_{\bar{t}_{1}}^{\bar{t}}\times\mathbb{R}$ leads to $$\begin{gathered}
\label{asymptotic0}
\int_{\mathbb{R}}\int_{\Gamma_{\bar{t}_{1}}^{\bar{t}}}\partial_{t}\left[\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^{2}\right]+f'(a)\partial_{x}\left[\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^{2}\right]\,dx\,dt\,da\\=\int_{\mathbb{R}}\int_{\Gamma_{\bar{t}_{1}}^{\bar{t}}}2\chi_{i}^{\varepsilon}\partial_{a}q_{i}^{\varepsilon}\,dx\,dt\,da\,,\end{gathered}$$ where we made use of the fact, that $q_i^{\varepsilon}$ is compactly supported in $a$. For the left-hand side of (\[asymptotic0\]) one can compute following step by step (\[calculation0\]) and (\[calculation1\]) $$\begin{gathered}
\label{asymptotic1}
\int_{\mathbb{R}}\int_{\Gamma_{\bar{t}_{1}}^{\bar{t}}}\partial_{t}\left[\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^{2}\right]+f'(a)\partial_{x}\left[\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^{2}\right]\,dx\,dt\,da\\
=\int_{\mathbb{R}}\int_{\theta^{-}_{\varepsilon}(\bar{t})}^{\theta_{\varepsilon}^{+}(\bar{t})}\left[\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^2\right](x,\bar{t})\,dx\,da-\int_{\Lambda_{\bar{t}_1}^{\bar{t}}}\int_{\mathbb{R}}\begin{pmatrix}\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^2\\f'(a)\left[\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^2\right]\end{pmatrix}\cdot \tau\,da\,d\sigma\,.\end{gathered}$$ For the right-hand side of (\[asymptotic1\]) we observe $$\begin{gathered}
\label{asymptotic2}
 \lim_{\varepsilon\rightarrow0^+}\int_{\mathbb{R}}\int_{\theta^{-}_{\varepsilon}(\bar{t})}^{\theta_{\varepsilon}^{+}(\bar{t})}\left[\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^2\right](x,\bar{t})\,dx\,da\\=\int_{\mathbb{R}}\int_{\theta^{-}(\bar{t})}^{\theta^{+}(\bar{t})}\left[\chi_{i}-\left(\chi_{i}\right)^2\right](x,\bar{t})\,dx\,da\end{gathered}$$ and $$\begin{gathered}
\label{asymptotic3}
 \lim_{\varepsilon\rightarrow0^+}\int_{\Lambda_{\bar{t}_1}^{\bar{t}}}\int_{\mathbb{R}}\begin{pmatrix}\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^2\\f'(a)\left[\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^2\right]\end{pmatrix}\cdot \tau\,da\,d\sigma\\=\int_{\Lambda_{t_1}^{\bar{t}}}\int_{\mathbb{R}}\begin{pmatrix}\chi_{i}-\left(\chi_{i}\right)^2\\f'(a)\left[\chi_{i}-\left(\chi_{i}\right)^2\right]\end{pmatrix}\cdot \tau\,da\,d\sigma\,.\end{gathered}$$ Since $$\chi_i=\left(\chi_{i}\right)^2$$ the right-hand side of (\[asymptotic2\]) and (\[asymptotic3\]) are zero. Thus $$\lim_{\varepsilon\rightarrow0^+}\int_{\mathbb{R}}\int_{\theta^{-}_{\varepsilon}(\bar{t})}^{\theta_{\varepsilon}^{+}(\bar{t})}\left[\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^2\right](x,\bar{t})\,dx\,da-\int_{\Lambda_{\bar{t}_1}^{\bar{t}}}\int_{\mathbb{R}}\begin{pmatrix}\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^2\\f'(a)\left[\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^2\right]\end{pmatrix}\cdot \tau\,da\,d\sigma=0\,.$$ With (\[asymptotic1\]) one concludes $$\lim_{\varepsilon\rightarrow0^+}\int_{\mathbb{R}}\int_{\Gamma_{\bar{t}_{1}}^{\bar{t}}}\partial_{t}\left[\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^{2}\right]+f'(a)\partial_{x}\left[\chi_{i}^{\varepsilon}-\left(\chi_{i}^{\varepsilon}\right)^{2}\right]\,dx\,dt\,da=0\,.$$ Finally taking limits on both sides of (\[asymptotic0\]) we get $$\lim_{\varepsilon\to0^+}\int_{\Gamma_{t_{1}}^{\bar{t}}}\int_{\mathbb{R}}\chi^{\varepsilon}_{i}\partial_{a}q_{i}^{\varepsilon}\,da\,dx\,dt\quad\mbox{for}\quad i\in\{1,\,2\}\,,$$ as announced.

Letting $\varepsilon\to0^+$ in (\[result3\_for\_contraction\]) and using (\[asymptotic\_of\_q\]) leads to $$\label{last1}
 \int_{\mathbb{R}}\int_{\theta^{-}(\bar{t})}^{\theta^{+}(\bar{t})}\left(\chi_{1}-\chi_{2}\right)^{2}(x,\bar{t})\,dx\,da\\
\leq\int_{\Lambda_{t_1}^{\bar{t}}}\int_{\mathbb{R}}\begin{pmatrix}\left(\chi_{1}-\chi_{2}\right)^{2}\\f'(a)\left(\chi_{1}-\chi_{2}\right)^{2}\end{pmatrix}\cdot \tau\,da\,d\sigma\,.$$ We compute $$\label{last2}
 \int_{\mathbb{R}}\left(\chi(v_{1}(x,t);a)-\chi(v_{2}(x,t);a)\right)^{2}\,da=|v_{1}(x,t)-v_{2}(x,t)|\,,$$ and $$\label{last3}
 \int_{\mathbb{R}}\begin{pmatrix}\left(\chi_{1}-\chi_{2}\right)^{2}\\f'(a)\left(\chi_{1}-\chi_{2}\right)^{2}\end{pmatrix}\cdot \tau\,da=\begin{pmatrix}|w_{1}-w_{2}|\\\operatorname{sign}(w_1-w_2)(f(w_1)-f(w_2))\end{pmatrix}\cdot \tau\,.$$ Applying (\[last2\]) and (\[last3\]) in (\[last1\]) gives $$\label{last4}
 \int_{\theta^{-}(\bar{t})}^{\theta^{+}(\bar{t})}|v_{1}(x,\bar{t})-v_{2}(x,\bar{t})|\,dx
\leq\int_{\Lambda_{t_1}^{\bar{t}}}\begin{pmatrix}|w_{1}-w_{2}|\\\operatorname{sign}(w_1-w_2)(f(w_1)-f(w_2))\end{pmatrix}\cdot \tau\,d\sigma\,.$$ For the right hand side, we compute $$\begin{split}
 \int_{\Lambda_{t_1}^{\bar{t}}}&\begin{pmatrix}|w_{1}-w_{2}|\\\operatorname{sign}(w_1-w_2)(f(w_1)-f(w_2))\end{pmatrix}\cdot \tau\,d\sigma\\
&=\int_{s_1}^{s_2}|w_1(s,\gamma(s,t_1))-w_2(s,\gamma(s))|\\
&\quad+\partial_s\gamma(s,t_1) \left[\operatorname{sign}(w_1-w_2)(f(w_1)-f(w_2))\right](s,\gamma(s,t_1)\,ds\\
&=\int_{s_1}^{s_2}|w_1(s,\gamma(s,t_1))-w_2(s,\gamma(s))|\cdot(1\pm\partial_s\gamma(s,t_1)f'(\alpha))\\
&\leq C \int_{s_1}^{s_2}|w_1(s,\gamma(s,t_1))-w_2(s,\gamma(s))|\,ds\,,
\end{split}$$ for a function $\alpha$. From (\[last4\]) we obtain $$\int_{\theta^{-}(\bar{t})}^{\theta^{+}(\bar{t})}|v_{1}(x,\bar{t})-v_{2}(x,\bar{t})|\,dx
\leq C\int_{\Lambda_{t_1}^{\bar{t}}}|w_1(s,\gamma(s,t_1))-w_2(s,\gamma(s))|\,d\sigma$$ as claimed.

Blow up at the points of negative density. {#section_blow_up}
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In this section we aim to prove the following lemma

\[lemma\_positiv\_density\] Let $u\in L^{\infty}(\mathbb{R}\times[0,T))$ be a weak solution of 