The lectures will describe how to study dynamics leading up to singular limits in partial differential equations (finite-time blow-up, rupture, quenching, etc). For many problems, the intermediate asymptotics before singularity formation are given by self-similar solutions. Topics to be covered will include: determining similarity solutions for nonlinear PDEs, 0th, 1st and 2nd kind similarity solutions, stability analysis of self-similar dynamics, geometric effects and conserved quantities, and numerical methods for problems with finite-time singularities.
The material should be accessible to anyone with a basic background in partial differential equations.
Applications include problems in fluid dynamics, nonlinear diffusion, geometric evolution equations and models from mathematical biology.