Diffusion equation, heat equation
If we specialize to the heat equation,
we assume that the diffusion of heat through some
material is proportional with the temperature gradient
T(\mathbf{x},t)
and using
conservation of energy we arrive at the diffusion equation
\begin{equation*}
\frac{\kappa}{C\rho}\nabla^2 T(\mathbf{x},t) =\frac{\partial T(\mathbf{x},t)}{\partial t}
\end{equation*}
where
C is the specific heat and
\rho
the density of the material.
Here we let the density be represented by a
constant, but there is no problem introducing an explicit spatial dependence, viz.,
\begin{equation*}
\frac{\kappa}{C\rho(\mathbf{x},t)}\nabla^2 T(\mathbf{x},t) =
\frac{\partial T(\mathbf{x},t)}{\partial t}.
\end{equation*}