Fourier's law describes heat flux when the temperature gradient is known. To determine how the temperature field itself changes with time, combine conduction with local energy conservation.

For a homogeneous solid with constant thermal properties and no internal heat generation,

$$\rho c\frac{\partial T}{\partial t}=k\nabla^2T.$$

Equivalently,

$$\frac{\partial T}{\partial t}=\alpha\nabla^2T,$$

where

$$\alpha=\frac{k}{\rho c}$$

is the thermal diffusivity.

What diffusivity means

Large thermal diffusivity means temperature differences spread through the material quickly. It increases with conductivity and decreases when the material stores more energy per unit volume.

Initial and boundary conditions

The differential equation does not determine one unique temperature history by itself. A transient problem also needs an initial temperature field and boundary conditions describing temperatures, heat fluxes or heat transfer at the surfaces.

Steady state

When temperature no longer changes with time,

$$\frac{\partial T}{\partial t}=0,$$

so the heat equation reduces to a spatial equilibrium equation.

The heat equation is a diffusion equation: local temperature differences are progressively smoothed by conductive energy transport.