Here are some examples of a PDE solutions on a domain based on a heart-shaped curve.

Excitability

Models like FitzHugh–Nagumo are the basis for a lot of work in cardiac electrophysiology and related areas. So let’s solve these PDEs in a heart!

Load this FitzHugh–Nagumo simulation and enjoy. You can click and select 3D to get a 3D view of the beating heart.
Try this growing simulation for a crude implementation of these equations on a growing domain.

Chemotaxis

The next example is based on the Keller–Segel model for chemotaxis.

Load this Keller–Segel simulation and enjoy!
Try this growing simulation for a crude implementation of these equations on a growing domain.

Warm hearted

By plotting vector fields on top of the solutions to PDEs, as exemplified in the heat equation, we can visualise how heat flows from a source as it traces out a heart in this interactive simulation.