The heat equation is well understood. Given reasonable initial data, the solution exists for all time and is very well-behaved. If a nonlinear term is added then much more interesting things can happen. For example, for some values of p, the solution may blow up (ie, become infinite) at a finite time. The way in which this happens can be analysed, and the asymptotics of the solution near the blow up time specified quite precisely.
This project would involve learning about the linear and nonlinear heat equations, understanding the literature about blowup, and analysing some interesting examples in detail possibly involving new forms of blow up that have yet been fully described.