Suppose we have a laterally insulated metal rod of length 1, parametrized by length, starting with 0 at the left hand endpoint. Suppose the insulation is not perfect and there is lateral heat loss. Suppose left hand endpoint is kept at temperature -2 and right hand endpoint is perfectly insulated.

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 are the correct ones:
$u(0,t) = -2, \quad u_x(1,t)=0$
$u(0,t) = 0, \quad u(1,t)=-2$
$u_x(0,t) = -2, \quad u_x(1,t)=-2$
$u(0,t) = -2, \quad u(1,t)=0$
$u_x(0,t) = -2, \quad u(1,t)=0$
$u_x(0,t) = -2, \quad u_x(1,t)=0$