Coupling the rate of change (temporal dependence) of \( u \) with the flux density we have $$ \begin{equation*} \frac{\partial u}{\partial t} = -\mathrm{div}\mathbf{\rho}, \end{equation*} $$ which results in $$ \begin{equation*} \frac{\partial u}{\partial t}= D \mathrm{div} \mathbf{\nabla} u = D \Delta u, \end{equation*} $$ the diffusion equation, or heat equation.