Some similarity solutions of the Navier-Stokes equations and related topics

We consider a semilinear equation arising from the Navier-Stokes equations - the governing equations of viscous fluid motion and related model equations. The solutions of the semilinear equation represent a certain class of exact solutions of the Navier-Stokes equations. Both the equation and our models have nonlocal terms. We will show that the nonlocality will play an intriguing role for the blow-up and/or global existence of the solutions and that the convection term, which is often neglected in the study of the blow-up problems, plays a very decisive role. In addition to our new contributions, open problems and known facts are surveyed.