My research projects have been on the well-posedness and the blow-up problems for the evolutionary partial differential equations.
Roughly speaking, an initial value problem for an evolutionary differential equation is said to be well-posed in some space H if for any initial data in H, there exists a solution to the initial value problem which depends continuously on the initial data. If the solutions exist forever, then this problem is said to be globally well-posed in H. If the solutions fail to exist after some finite time, then the problem is only said to be locally well-posed.
Usually, the reason for a local solution failing to exist globally is because a certain norm of the solution tends to infinity at some finite time. This phenomenon is called blow-up and it is interesting to study the related issues, such as the lifespan estimate, the blow-up behavior and so on.