Suppose we have substance moving in a stream with convection, the stream is parametrized by distance from the source. Suppose at $x=0$, the substance has concentration -2. There is no boundary point at the right hand end point.

Which equation most likely governs this setup:
$u_t=\alpha^2 u_{xx}$
$u_{tt}=\alpha^2 u_{xx}$
$u_t=\alpha^2 u_{xx}-3 u_x$
$u_t=\alpha^2 u_{xx}-3 u$
$u_{tt}+u_{xx}=0$
$u_t=\alpha^2 u_{xx}-3 x$
$u_t=\alpha^2 u_{xx}-3 t$

Which boundary conditions would work:
$u(0,t) = 0$
$u(0,t) = -2$
$u_x(0,t) = -2$
$u_x(0,t) = 0$