The role of self-similarity in singularities of PDE's (0812.1339v1)

Abstract: We survey rigorous, formal, and numerical results on the formation of point-like singularities (or blow-up) for a wide range of evolution equations. We use a similarity transformation of the original equation with respect to the blow-up point, such that self-similar behaviour is mapped to the fixed point of a \textit{dynamical system}. We point out that analysing the dynamics close to the fixed point is a useful way of characterising the singularity, in that the dynamics frequently reduces to very few dimensions. As far as we are aware, examples from the literature either correspond to stable fixed points, low-dimensional centre-manifold dynamics, limit cycles, or travelling waves. For each "class" of singularity, we give detailed examples.