The entropy solution of the scalar conservation law
$$u_t + f(u)_x =0 \\ u(0,x)=u_0(x) $$
is defined as a function $u(t,x) \in L^{\infty}(\mathbb{R}^+\times\mathbb{R})$ that satisfies for any $\phi\in C^{\infty}_c(\mathbb{R}^+\times\mathbb{R} ; \mathbb{R}^+)$ and $ k \in \mathbb{R}$
$$ \int_{\mathbb{R}^+\times\mathbb{R}} |u-k|\phi_t + \hbox{sgn}(u-k)(f(u)-f(k))\phi_x dxdt + \int_{\mathbb{R}} u_0(x)\phi(0,x) dx \geq 0$$
In Denis Serre, "Systems of Conservation Laws 1" page 34, it is claimed that in order to ensure uniqueness of the entropy solutions it is necessary to enforce the integral inequality above to test functions which might be non zero in ${0} \times \mathbb{R}$, meaning that one would check reduce the validity of the above integral inequality to test function with support in $\mathbb{R}_{*}^+\times\mathbb{R}$ there could be an example of initial data for which there exists two entropy solutions. Anyone could point me to such an example !