We draw analogies between self-similar, focusing dynamics in nonlinear partial differential equations (PDEs) and macroscopic dynamic features of the glass transition. In particular, we explore the divergence of the appropriate relaxation times in the case of hard spheres as the limit of random close packing is approached. We illustrate the analogy in the critical case, and suggest a description that can capture the onset of dynamic self-similarity in both phenomena. (C) 2003 Elsevier B.V. All rights reserved.