How can I prove the uniqueness in $[0, +\infty)\times (0,1)$ of the solution of a PDE as the following:
$\frac{\partial}{\partial t}v(t,x)=\frac{\partial}{\partial x}\Big((x-\frac{1}{2})v(t,x)\Big)$
$v(0,x)=g(x)$,
where $g\in C^\infty (\mathbb R)$ with support in the interval $(0,1)$.
@user268193 Uniqueness follows from the fact that the characteristics do not intersect and are in a one-to-one correspondence with the points of the parabolic boundary. The characteristics of your equation satisfy the ODE $$ dt=dx/(1/2-x) $$ which are exponential curves approaching $x=1/2$ as $t\to\infty$ and also $x=1/2$ itself. These curves do not intersect and, given any point $(t,x)\in[(0,\infty)\times(0,1)$, you can connect it along a characteristic to a unique point on either part of your boundary $(t,x)\in\{0\}\times[0,1]$ or $(0,\infty)\times\{0\}$ or $(0,\infty)\times\{1\}$. By uniqueness for ODEs along characteristics, the values of $u(t,x)$ are thus uniquely determined by the initial and boundary data.