March  2007, 19(1): 1-35. doi: 10.3934/dcds.2007.19.1

Finite-time blow-down in the evolution of point masses by planar logarithmic diffusion

1. 

Departamento de Matemáticas and ICMAT. Universidad Autónoma de Madrid, Cantoblanco. 28049 Madrid, Spain

Received  August 2006 Revised  February 2007 Published  June 2007

We are interested in a remarkable property of certain nonlinear diffusion equations, which we call blow-down or delayed regularization. The following happens: a solution of one of these equations is shown to exist in some generalized sense, and it is also shown to be non-smooth for some time $ 0 < t < t_1$, after which it becomes smooth and still nontrivial. We use the logarithmic diffusion equation to examine an example of occurrence of this phenomenon starting from data that contain Dirac deltas, which persist for a finite time. The interpretation of the results in terms of diffusion is also unusual: if the process starts with one or 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 concept of measure solution which seems natural for these diffusion processes.
Citation: Juan Luis Vázquez. Finite-time blow-down in the evolution of point masses by planar logarithmic diffusion. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 1-35. doi: 10.3934/dcds.2007.19.1
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