Finite-time blow-down in the evolution of point masses by planar logarithmic diffusion
Juan Luís Vázquez · Discrete and Continuous Dynamical Systems · 2007
We are interested in a remarkable property of certain nonlineardiffusion equations, which we call blow-down or delayedregularization. The following happens: a solution of one of theseequations is shown to exist in some generalized sense, and it isalso shown to be non-smooth for some time $ 0 < t < t_1$, after which itbecomes smooth and still nontrivial. We use the logarithmicdiffusion equation to examine an example of occurrence of thisphenomenon starting from data that contain Dirac deltas, whichpersist for a finite time. The interpretation of the results interms of diffusion is also unusual: if the process starts with oneor several point masses surrounded by a continuous distribution,then the masses decay into the medium over a finite period of time.The study of the phenomenon implies consideration of a new conceptof measure solution which seems natural for these diffusionprocesses.