I have to do a final project for my PDE subject and last year I did one about Game Theory (specifically, Prisonner's Dilemma and Snowdrift game) for my ODE subject, which the rest of the students enjoyed and which my teacher thought it was a fun approach to the contents of the subject. I want to do a similar thing, but every Game Theory problem I have seen only involves ODEs.
Until now we have seen first order PDEs (both linear and non-linear) and second order (hyperbolic, parabolic and elliptic) PDEs. Also wave equation and heat equation.
What would be some fun PDE related problems to discuss?
(I hope I put the right tags, because it is the first time I ask a soft question).
If you would like to continue your line of inquiry involving game theory, but with PDEs instead of ODE, you are referred to "Mean Field Games". This is where you will see PDE related to optimal control theory in games. This is a very very active field of research.
Maybe this would help to get you started:
http://www.science4all.org/article/mean-field-games/
https://terrytao.wordpress.com/2010/01/07/mean-field-equations/
As usual obtaining explicit solution in optimal control can be difficult. But certain types of problems are nice. Look into Linear Quadratic mean field game.
Just a side remark, you will find all types of math in game theory: quantum mechanics, information theory, differential geometry and physics. This is a rapidly evolving field.