Singularities
The basic questions about singularities (their formation, propagation, and removal, and regularity of solutions) are the same as for linear PDE, but as usual much harder to study. In the linear case one can just use spaces of distributions, but nonlinear PDEs are not usually defined on arbitrary distributions, so one replaces spaces of distributions by refinements such as Sobolev spaces.
An example of singularity formation is given by the Ricci flow: Hamilton showed that while short time solutions exist, singularities will usually form after a finite time. Perelman's solution of the Poincaré conjecture depended on a deep study of these singularities, where he showed how to continue the solution past the singularities.
Read more about this topic: List Of Nonlinear Partial Differential Equations, Methods For Studying Nonlinear Partial Differential Equations