Physical meaning of one dimensional heat equation. Consider heat conduction in a $1$-D material.  The temperature distribution  $u$ after sufficiently long time can be modeled by $$-(a(x)u_x)_x = f(x), \qquad x \in [0,1]$$ where $a$ is heat conductivity, $f$ denotes internal heat sources.
Now i want to understand how this equation is modeled? Can anyone explain it plz? Also what is the difference between this equation and general heat equation?

You can start from the derivation of the general heat equation in one dimension. For example, you can look at the https://en.wikipedia.org/wiki/Heat_equation. You just need to account for the fact that the heat conductivity is position dependent. You should get:
$$\partial_tu-\partial_x(a(x)\partial_xu)=f(x)$$
Now all you need to do is assume that the system reaches an equilibrium when $t\rightarrow\infty$. If the system is in equilibrium, it does not change with time, so $\partial_tu=0$
