Let $\epsilon>0$, consider the I.V.P of the viscous conservaion law $$\begin{cases}u_t+\partial_xf(u)=\epsilon u_{xx}\\ u(x,0)=u_0(x)\end{cases}$$ Show that there exists a unique solution $u\in L^{\infty}(\mathbb{R}\times[0,\infty))\cap C^{\infty}(\mathbb{R}\times (0,\infty))$ by the following steps:
Step 1: Define the map $u = A[v], \forall v\in L^{\infty}(\mathbb{R}\times [0,T])$ by $$\begin{cases}u_t+\partial_xf(v)=\epsilon u_{xx}\\ u(x,0)=u_0(x)\end{cases}$$ Show that there is $T>0$ which depends only on $\|u_0\|_{L^{\infty}}$ such that $A$ is a contraction mapping from $L^{\infty}(\mathbb{R}\times[0,T])\to L^{\infty}(\mathbb{R}\times[0,T])$; this implies uniqueness of the solution to the I.V.P.
Step 2: Show that any solution $u \in L^{\infty}(\mathbb{R}\times[0,T])$ satisfies the maximum principle $$\|u(\cdot,t)\|_{L^{\infty}}\leq \|u_0\|_{L^{\infty}}$$.
Step 3: Show that the I.V.P has a unique solution in $L^{\infty}(\mathbb{R}\times[0,\infty))$,moreover this solution is also in $C^{\infty}(\mathbb{R}\times (0,\infty))$.
Remark: Does anyone know what is the explicit form of the operator $A$ in Step 1? Why this iteration argument will work?
For step 3, is there any way to apply the maximum principle for the heat equation? (The issue is that the heat equation is defined over a finite time interval, but in this case we have an infinite time domain) Or so, is there any possibility to use a similar argument that shows the interior elliptic regularity for elliptic PDEs?