# Parabolic PDE

The quintessential example of a parabolic partial differential equation is the heat equation. Parabolic equations are often used to describe diffusive phenomenon, perhaps with drifts, and as such see wide applications in chemistry, biology, and engineering. The diffusion behavior also makes these equations connected to the statistics of random walks, and whereby these equations also see applications in mathematical finance. Yet another way to think of a parabolic partial differential equation is as the gradient flow associated to some energy functional, with the flow driving toward the minimization of said energy. From this point of view parabolic PDEs are often used in optimization and for studying geometric problems. Some famous examples include: the heat equation, reaction-diffusion equations such as the Allen-Cahn equation, the Black-Scholes equation in pricing model, the mean curvature flow and Ricci flow in geometry.

### Oscillatory Solutions to the Heat Equation

As we all learned in a first course in PDEs, for "reasonable" initial data $u_0$, the solution to the linear heat equation …