I have tried to integrate this without getting to the right destiny!

$\displaystyle \int_{0}^{\infty} \int_{-\infty}^{\infty}[(u_t + f(u)_x)v]dxdt.$

where $\displaystyle v:\mathbb{R}\times [0,\infty) \rightarrow \mathbb{R}$ is smooth, with compact support.