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

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.

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 $$s=1$$.
• @Hamilton The solution is likely to be a discontinuity for other values of $p$ too. According to Rankine-Hugoniot, the shock speed is likely to be dependent on $p$ (see this post where the theory is presented). You may need to use the weak form of the PDE Jun 22 '20 at 8:29