I want to prove that there is a unique classical solution (if it exists) to
$$ \begin{cases} \partial_tu+A(u)\partial_xu=0, \quad t\geq 0, \quad x\in\mathbb{R}\\ u(x,0)=u_0(x), \end{cases} $$ where $A\in C^1(\mathbb{R})$. To do so, I take two solutions $u,v$ build their difference $d=u-v$, and note that it solves $$ \partial_td+A(u)\partial_x d+(A(u)-A(v))\partial_xv=0. $$ and according to Lax' "Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves" (Eq. 1.4), one can see from this uniqueness since $|A(u)-A(v)|\leq C |d|$. What is the estimate that I need here, and can you sketch how to obtain it? I feel like we need a Gronwall like estimate. If I multiply with $d$ and integrate in space, I get an equation for the derivative of $\|d\|_{L^2}$, but I don't know what to do with the term $\int A(u)(\partial_x d )d$.
I guess I could also show uniqueness by the method of characteristics, but I want to understand this approach.
