We consider the problem
\begin{align} \begin{cases} \frac{\partial u}{\partial t}+c(x,t)\frac{\partial u}{\partial x}=0\quad\quad\quad (x,t)\in \mathbb{R}\times\mathbb{R}_+\\u(x,0)=u_0(x)\quad\quad\quad\quad\quad x\in\mathbb{R}\end{cases}\end{align} where $c\in \mathcal{C}^1(\mathbb{R}\times\mathbb{R}_+,\mathbb{R})$ is Lipschitz in $\mathbb{R}\times\mathbb{R}_+$, $u_0\in \mathcal{C}^1(\mathbb{R},\mathbb{R})$ given. For all $(x_0,t_0)\in \mathbb{R}\times\mathbb{R}_+$, we consider the problem \begin{align} \begin{cases} \frac{d X}{d t}=c(X(t),t)\\X(t_0)=x_0\quad\quad\quad\quad\quad x\in\mathbb{R}\end{cases}\end{align} Could you please help me to show that $X(t_2,x_0,t_0)=X(t_2,X(t_1,x_0,t_0),t_1)$.
Thanks in advance.
