Let $u$ be a weak solution to the conservation law:
$$ \begin{cases} u_t + F(u)_x = 0 & (x,t) \in \mathbb{R}\times(0, \infty) \\ u = g & \mathbb{R} \times \{0\} \end{cases} $$ Let us assume that $u$ is compactly supported in $\mathbb{R}\times [0,T]$ for all $T$, and that $F(0) = 0$. Prove then that for any fixed $t >0$ we have that: $$ \int_\mathbb{R}u(x,t) \mathrm{d}x = \int_\mathbb{R} g(x) \mathrm{d}x $$ My attempt: I know that for $u$ to be a continuous integral solution/weak solution, it must satisfy : $$ \int_0^\infty\int_{-\infty}^\infty u \phi_t + F(u) \phi_x \mathrm{d}x\mathrm{d}t + \int_{-\infty}^\infty g\phi \mathrm{d}x = 0 $$ For any $C^1$, compactly supported test function. How can I prove this identity?
