TITLE: Distinguishing between Laplace's equation and heat equation?
QUESTION [5 upvotes]: I have a pde $$\frac{\partial^2 u}{\partial x^2}=\frac{1}{c^2} \frac{\partial u}{\partial t}$$
I have been told this is a heat equation. Why? 
What are the distinguishing features between the heat equation and the Laplace equation?

REPLY [2 votes]: The Laplace equation applies when your length scale is small so it can be assumed $c\rightarrow\infty$ and in this case the diffusion equation becomes the Laplace equation. This is a key physical point.
Because there is no time derivative in the Laplace equation, it suggests information travels infinitely fast which is physically impossible. So every physical situation where the Laplace  equation applies is actually another equation like the wave equation or the diffusion equation.
It can also be viewed as the steady state version of the heat equation.