Asymptotic solution of conservation law $u_t + (\frac{1}{p}|u|^p)_x = 0$

Question

For $1 < p < 2$, let $u_p$ be a unique solution to

$u_t + \left(\frac{1}{p}|u|^p\right)_x = 0$,

for initial condition

$u_0(x) = \begin{cases} 1 \quad x > 0, \\ 0 \quad x < 0. \end{cases}$

I now want to determine the solution $u(t, x) = \lim \limits_{p \to 1} u_p(t, x)$ of the limit problem

$u_t + (|u|)_x = 0$

with same same initial condition. My problem is that I don't really know how to get started with this or which ansatz to use. I tried using the Lax-Oleinik formula given by the Evans, but got no results.

Answer 1

Given that the initial data is non-negative, let's assume that $u$ is non-negative. Therefore, we are left with $$ u_t + u_x = 0 ,\qquad u(x,0) = u_0(x) $$ where $u_0$ is the step function. The solution $u = u_0(x-t)$ is indeed non-negative, therefore this solution is the one obtained for $p=1$. Note that this solution is a contact discontinuity $$ u(x,t) = \left\lbrace \begin{aligned} 1 \qquad x> st,\\ 0 \qquad x< st. \end{aligned} \right. $$ with speed $s=1$.