Consider the heat equation, $$\frac{\partial u}{\partial t} = \frac{\partial }{\partial x} \left(K_0(x) \frac{\partial u}{\partial x}\right)+Q(t)$$ $$u(t,x=1)=0$$ $$u(t,x=2)=1$$
where $K_{0}=x^2$
Now the question is under what conditions on the heat source $Q(t)$ does a steady state solution exist?
So far what I have done is expanded out to get this equation, $$-Q(t)=2x \frac{\partial u}{\partial x} + x^2 \frac{\partial ^2 u}{\partial x ^2}$$ Which according to wolfram alpha has solution, $$u(x,t) = -Q(t)ln(x)+\frac{c_1}{x}+c_2$$ from this it looks like there is always a steady state solution right?