Suppose we have a laterally insulated metal rod of length 1, parametrized by length, starting with 0 at the left hand endpoint. Suppose that there is heat removed from the rod (heat sink), independent of time, but depending on position. Both endpoints are insulated.

Which of the following equations could govern 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 are the correct ones:
$u(0,t) = 0, \quad u_x(1,t)=0$
$u(0,t) = 0, \quad u(1,t)=u_x(1,t)$
$u_x(0,t) = 0, \quad u_x(1,t)=0$
$u(0,t) = 0, \quad u(1,t)=0$
$u_x(0,t) = u(0,t), \quad u(1,t)=0$
$u_x(0,t) = u(0,t), \quad u_x(1,t)=0$