I know the steady state for $\frac{\partial f}{\partial t}+\nabla J=0$ (with ICs: $f(x,0)=0$, BCs: $J(0,t)=J(L,t)=0$) is when $\nabla J =0$, i.e. atomic flux divergence should be zero. This equivalent to say the total flux should be constant, i.e.
$$J=constant$$
For my problem, space is 1D and bounded from two ends and $J$ consists of two fluxes: $J=\alpha \nabla f+\beta \nabla g$. Equivalently, it can be seen as two opposite forces fighting against each other and balance in steady state. This is, the fluxes generated by the gradients of $f$ and $g$ have the same magnitude but opposite sign, i.e.: $\alpha \nabla f=-\beta \nabla g$. This leads to:
$$J=0$$
This has been giving me headache for a week that what is the necessary condition for the steady state? If $J=constant$ is the essential condition why in my problem $J\neq 0$ never happens (while it could). Basically, I always get $J=0$ (for any $\alpha, \beta, f$ and $g$).