heat equation

$$\frac{\partial u}{\partial t}={c}^{2}\cdot \frac{{\partial}^{2}u}{\partial {x}^{2}}$$ 


$${u}_{t}={c}^{2}\cdot {u}_{xx}$$ 

Where $u:{\mathbb{R}}^{2}\to \mathbb{R}$ is the function giving the temperature at time $t$ and position $x$ and $c$ is a real valued constant. This can be easily extended to 2 or 3 dimensions as

$${u}_{t}={c}^{2}\cdot ({u}_{xx}+{u}_{yy})$$ 

and

$${u}_{t}={c}^{2}\cdot ({u}_{xx}+{u}_{yy}+{u}_{zz})$$ 

Note that in the steady state, that is when ${u}_{t}=0$, we are left with the Laplacian of $u$: