Rewriting the heat diffusion equation with temperature dependent diffusion coefficient to include joule heating.

I am modelling heat flow in a solid round copper conductor with a set area. I plan to discretize and solve numerically in Python. However, I only have a curve fit for thermal conductivity and specific heat. I'm hoping this won't be a problem as I am not seeking an analytical solution.

Here is where I have started:

$$\frac{\partial u}{\partial t} = \frac{k(u)}{\rho C(u)}\frac{\partial^2 u}{\partial x^2}$$

Where $k(u)$ and $C(u)$ are the thermal conductivity and specific heat, respectively. $\rho$ is density.

Now, in the initial estimation of conductor heat load, in watts, I used the following:

$$\frac{dQ}{dt} = I^2r\frac{L}{A} + k\frac{A}{L}\Delta u$$

Where $I$ is electric current, $r$ is resistivity, $k$ is thermal conductivity, $\Delta u$ is temperature change, and $L$ and $A$ are length and area of the conductor, respectively.

I interpreted this answer to mean that the cold side of the conductor is seeing $\frac{dQ}{dt}$ joules per second (or watts), some of which is coming from the temperature difference in the thermally anchored ends, $k\frac{A}{L}\Delta u$, and some which is coming from the current passing through, $I^2r\frac{L}{A}$.

Heat is related to temperature change by the following equation:

$$Q=mc\Delta T$$

Is this relationship what will allow me to incorporate the resistive heating $I^2r\frac{L}{A}$ into the original pde? Do I take the time derivative of both sides?

My related background: BSEE and one ode class. I have read quite a bit about discretizing pde's and also about solving 2nd order linear pde's.

From all the searching I've done it seems like nonlinear pdes are very problem specific in what works. While an analytic solution would be nice (if possible), I'd be quite happy with a non linear pde that incorporates resistive heating. From there I feel I could find a numerical solution.

$$\rho \cdot C(u)\frac{\partial u}{\partial t} = k(u)\cdot\frac{\partial^2u}{\partial x^2}$$