I am trying to find a weak entropy solution for the conservation law
$\partial_tu+\partial_x(u^5)=0 \text{ for } (x,t)\in\mathbb{R}\times(0,\infty)$
$u(x,0)=u_0(x) \text{ for } x\in\mathbb{R}$
where
$u_0(x)= \begin{cases} -1 & x< 0 \\ 0 & x>0 \end{cases}$
The characteristics $(X,T)$ cross instantaneously at $(0,0)$ which we can remedy by putting a shock $\sigma(t)=t$ there, which satisfies the Rankine-Hugoniot condition. This then gives us
$u(x,t)=\begin{cases} -1 & x<t \\ 0 & x>t \end{cases}$
which is a weak solution since it satisfies the conservation law piecewise, and at the discontinuity it satisfies the Rankine-Hugoniot condition. However this is not an entropy solution since there is no constant $C>0$ for which
$u(x+z,t)-u(x,t)\leq C\left(1+\frac{1}{t}\right)z$
is satisfied for all $x\in\mathbb{R}$ and for all $z,t>0$. For instance, if we fix $t$ and say that $x<t$, then we can write $t=x+\varepsilon$ for some $\varepsilon>0$. Then if we say $z=2\varepsilon$, it follows that $x+z>t$ and
$u(x+z,t)-u(x,z)=0-(-1)=1>C\left(1+\frac{1}{x+\varepsilon}\right)2\varepsilon=C\left(2\varepsilon+\frac{2\varepsilon}{x+\varepsilon}\right)$
for any $C$ if we make $\varepsilon$ small enough (since the right hand side tends to zero).
Of course I will have to try for another weak solution here, but it isn't clear to me what I should do. I have no option but to insert the shock $\sigma(t)$, and after that I feel like it's a guessing game where I put a shock/rarefaction wave/whatever else to make this an entropy solution. Any pointers?
